Formalization Of Theories of Design Using : First Order Logic Augmented With A Causal Calculus ; And Entropy of a Design Defined In Terms Of The Function , Behavior and Structure Ontology

A symbolic logical framework (L) consisting of first order logic augmented with a causal calculus has been provided to formalize, axiomatize and integrate theories of design. L is used to represent designs in the FunctionBehavior-Structure (FBS) ontology in a single, widely applicable language that enables the following: seamless integration of representations of function, behavior and structure; and generality in the formalization of theories of design. FRs, constraints, structure and behavior are represented as sentences in L. FRs are represented (as abstractions of behavior) in the form of existentially quantified sentences, the instantiation of whose individual variables yields the representation of behavior. This enables the logical implication of FRs by behavior, without recourse to apriori criteria for satisfaction of FRs by behavior. Functional decomposition is represented to enable lower level FRs to logically imply the satisfaction of higher level FRs. The theory of whether and how structure and behavior satisfy FRs and constraints is represented as a formal proof in L. Important general attributes of designs such as solution-neutrality of FRs, probability of satisfaction of requirements and constraints (calculated in a Bayesian framework using Monte Carlo simulation), extent and nature of coupling, etc. have been defined in terms of the representation of a design in L. The entropy of a design is defined in terms of the above attributes of a design, based on which a general theory of what constitutes a good design has been formalized to include the desirability of solution-neutrality of (especially higher level) FRs, high probability of satisfaction of requirements and constraints, wide specifications, low variability and bias, use of fewer attributes to specify the design, less coupling (especially circular coupling at higher levels of FRs), parametrization, standardization, etc..

The relationship between the science of design and frameworks of the practice of design can be seen to be analogous to the relationship between the natural sciences and engineering. The natural sciences provide knowledge and theories about objects and phenomena of the natural world that form the basis for the design of systems to transform the natural world from one state to another. Similarly, the science of design should provide knowledge and theories about the general artifacts and processes of design, that should then form the basis for the development of the practice of design. Scientific method (e.g. Peirce (1878), Fann (1970), Popper (2005)) for the natural sciences is generally characterized by the construction of formal, axiomatic theories involving the following: logical induction and/or abduction to frame axioms or premises; deduction (from axioms or premises) to explain observations and make predictions; and induction (in the form of tests of hypotheses) to negate or corroborate predictions, thereby falsifying or corroborating a scientific theory. The success of the scientific method in the natural sciences is, at least partly, due to the construction of integrated, axiomatic theories that explain a number of phenomena in a variety of domains, based on a relatively small number of axioms.
Similarly, the development of the science of design needs the development of formal, axiomatic, integrated theories that describe, explain, predict and allow the negation or corroboration of propositions about designs (e.g. in terms of their functions, structure and behavior) and design processes (that are used to develop design artifacts). Theories of design could pertain to the following: a) What constitutes a design; b) How a given design brings about a transformation in its external environment (i.e., theories about how a design satisfies its requirements and constraints); c) What is a good design; c) What constitutes a design process; d) How does a design process yield a design; e) What is a good design process; etc.. Since the above types of theories of design are interrelated (e.g. how a design satisfies its requirements is likely to influence the assessment of the quality of the design, and a good design process would be expected to yield good design artifacts) a unified and standard formal language and logic is required to express, integrate and axiomatize the various types of theories of design mentioned above.
The question then arises as to what might be a suitable formal logic (or logics) for the design sciences. The answer to this depends on the general types of representation and reasoning needed in the above types of theories of design. Simon (1996) broadly defined the activity of designing as devising courses of action to change existing situations to preferred ones, and posited that the design artifact forms the interface between its internal and external environments. A theory of how a design brings about a desired change in its environment needs a definition of a design in terms of its constituent elements, and their relationships with one another and with the environment, i.e., an ontology of design artifacts and processes. A product design process involves various activities executed in an iterative and integrated manner (Pahl and Beitz (1988), Hubka and Eder (1988), Gero (1990), Suh (1990), Ullrich and Eppinger (1995), Otto and Wood (2001)), including the following: Defining the requirements and constraints that the product (e.g. automobiles, computers, etc.) should satisfy; Defining the structure of the product components; Specifying and/or anticipating the behavior of the product; and Evaluating the product for the satisfaction of its requirements and constraints. Pahl and Beitz (1988) describe a hierarchical approach of defining the functions of the product and choosing devices to execute the functions until a fully detailed product is realized. They describe the functions of the product as transformations of input materials, signals and energy into output materials, signals and energy. Individual functions are achieved using working principles (mechanisms), and the overall hierarchy of functions being achieved by the integration of working principles into a working structure. The behavior of the working principles and working structures is implicit in the representations or descriptions of the working principles and structures. Hubka and Eder (1988) proposed that the transformation of the environment from a given state to a desired state is achieved through a transformation system consisting of inputs, a transformation process using a transformation technology, and various operators (human, technical, information and management systems) that together enable the transformation process to achieve the desired state of the environment. The artifact that is designed itself is a technical system consisting of the following: inputs (materials, energy, information); functions (capabilities of the product that produce the desired effects on the environment); organs (system of parts that achieves a given function); constructional parts; chains of actions caused by organs under the given inputs; outputs; and effects on the environment. The hierarchy of the transformation system consists in each operator itself being a transformation system. Suh (1990) proposed that the requirements of a product be analyzed into functional requirements (FRs) and constraints, and that design parameters be identified to satisfy FRs in such a manner that the resulting design is functionally uncoupled or decoupled design and has the least information content, i.e., the highest probability of satisfaction of FRs and constraints. Gero (1990) proposed the Function-Behavior-Structure (FBS) ontology wherein: functions define the purpose of the product; the structure defines the organization of the components of the product; the behavior of the product is derived from its structure, and verified against expected behavior derived from the functions of the product. Gero (1990) represents functions, in a language of commands, as combinations of verbs and nouns (e.g. "provide daylighting"), and behavior using the language of qualitative physics (Bobrow (1984)). Umeda, et al. (1990Umeda, et al. ( , 1996 proposed a Function-Behavior-State diagram, wherein the state of the product subsumes its structure. They represented functions as intentions to perform an action in a certain manner (e.g. "to move table precisely"), and described the causal decomposition of functions (analysis of a function into causally related sub-functions) to guide the selection of devices whose behavior satisfies the causal relationships among functions. They represented behavior using the qualitative physics of processes (Forbus (1984)). They assessed the satisfaction of functions using an apriori mapping between functions and expected behavior, simulating the behavior of the structure of the design, and evaluating whether the simulated behavior satisfies expected behavior. Vescovi, et al. (1993) represented functions in their Casual Functional Representation Language (CFRL) as logical combinations of causal process descriptions (CPDs), i.e., networks of temporally and causally related nodes, wherein each node is defined in terms of conditions to be satisfied by expected behavior, in a certain physical context and by a certain device. The above representation of functions constituted an abstraction of expected behavior. They represented behavior as temporally or causally ordered trajectories of states, simulated using qualitative physics (Forbus (1984(Forbus ( , 1990). The evaluation of satisfaction of functions by behavior was implemented by verifying whether the states constituting behavior satisfy the conditions for behavior specified by corresponding nodes constituting the functions. They also used operators such as ALWAYS and SOMETIMES to characterize the extent of matching between CPDs and behavior. Goel, at al. (2009), in their Structure-Behavior-Function (SBF) framework, extended CFRL to include the concept of substances (attributes of objects such as angular momentum), bidirectional causality, etc.. Gero's (1990) ontology correspond to functions in the ontologies of Pahl and Beitz (1988), Otto andWood (2001), Vescovi, et al. (1993) and Umeda, et al. (1990) and to organs in Hubka and Eder (1988). Structure in Gero's (1990) ontology corresponds to the concept of structure in the ontologies of Pahl and Beitz (1988), Otto andWood (2001), Vescovi, et al. (1993), to the concept of state as a generalization of structure in Umeda, et al. (1990) and to the concept of component structure of Hubka and Eder (1988). Behavior in Gero's (1990) ontology corresponds to the concept of behavior in Vescovi, et al. (1993) and Umeda, et al. (1990), to the concept of action sequences in Hubka and Eder (1988), and is implicit in the concept of working principle of Pahl and Beitz (1988). Thus the FBS ontology is apparent in a variety of representations of design, as described above.

Functions in
A central question pertaining to any design is: "How will the design satisfy its requirements and constraints?".
An explanation of how a design will satisfy its requirements and constraints can be seen as a theory of satisfaction of requirements and constraints by the structure and behavior of the design. Formalization of theories of design also requires a formalization of the theory of how a design satisfies its requirements and constraints. Such a theory would enable the evaluation of whether a design makes logical sense and its ability to satisfy its requirements and constraints. A theory of satisfaction of requirements and constraints by a design needs a formal language and logic for its representation. In the FBS ontology, the use of different languages for representing functions and behavior, respectively, was partially motivated by the preference for retaining the flexibility of choosing a language to simulate behavior (e.g. qualitative physics) independently of the choice of language to represent functions (Vescovi, et al. (1993)). However, using different languages to represent functions and behavior, respectively, requires the apriori specification of criteria for the satisfaction of functions, either in the language of behavior as in Gero (1990) or in the language of functions as in Vescovi, et al. (1993).
On the other hand, behavior, by its very definition, could be shown to satisfy functions without the need for pre-specified criteria for satisfaction of functions (e.g. the behavior of an internal combustion engine generating 75 horse power of mechanical energy at 2000 cycles per minute, by definition, satisfies the function of generating mechanical energy). This kind of satisfaction of functions by behavior is a logical implication of functions by behavior. Behavior can be shown to logically imply functions, without recourse to apriori criteria for satisfaction of functions by behavior, only in a common language used to represent both functions and behavior, accompanied by a logic that enables formal proofs of satisfaction of functions by behavior. Further, the modeling of behavior in Gero (1990), Vescovi, et al. (1993 and Umeda, et al. (1990) is based on qualitative physics, which is suited to model the behavior of physical systems (Forbus (1984)), but not the behavior of any general system. Thus there is a need for a single, standard, formal language and logic to represent, generally and in an integrated manner, functions, behavior, structure and theories of how structure and behavior realize functions.
Product design theories often identify general attributes of a good design, e.g.: solution-neutrality of functions; independence of functional requirements; functional uncoupling or decoupling (modularity); standardization, symmetry; minimal information content; etc. ( (Pahl and Beitz (1998), Suh (1990), Ullrich and Eppinger (1995)). Suh (1990) organized attributes of a good design into an axiomatic system, wherein a subset of these attributes are identified as axioms (e.g. independent functional requirements, minimal information content) and others are derived from these axioms (e.g. functional uncoupling or decoupling). In order to integrate various attributes of a good design proposed by various theories of design into a cogent, unified system, the general attributes of a good design have to be interpreted in terms of a common or standard representation of a design. The FBS ontology, given its use various theories of design as described above, is a good candidate for such a common or standard representation of design. Further, the general attributes of a good design, such as those mentioned above, have to be organized into a system of propositions in a manner that enables the following: a) assessment of internal consistency of the system of propositions; b) identification of axiomatic and derived propositions; and c) identification of additional propositions entailed by the existing axiomatic and derived propositions. Apart from formally unifying diverse theories of what constitutes a good design, such a formal theory might also yield additional desirable attributes of a design not contained in the original set of attributes represented in the theory.
Theories of design processes could be descriptive (e.g. they might describe how the activity of designing is carried out) or prescriptive (e.g. they might describe how the activity of designing should be carried out) (Finger and Dixon (1989), Chakrabarti and Blessing (1995)). Prescriptive design theories could be expressed as a mixture of descriptions (e.g. definition of a function in (Pahl and Beitz (1988)) as well as prescriptions (instructions or imperative sentences) (e.g. ensure the solution-neutrality of functions (Pahl and Beitz (1988)).
For example, the Axiomatic Design Theory (ADT) of Suh (1990) often uses prescriptive language such as "Maintain the independence of Functional Requirements" to describe a good design process. Logics whose sentences are imperatives (commands) are modal logics (logics that qualify the truth value of a proposition, using qualifiers such as "necessarily true", "should be true" (language of commands), "intended to be true", "will be true in the future", etc.) (Cresswell, et al. (2012)). Also, the logic of the language of functions of Gero (1990) (language of commands), Umeda, et al. (1990) (language of intentions), Vescovi, et al. (1993) (use of operators such as ALWAYS and SOMETIMES) and Suh (1991) (language of commands), as described above, can all be considered as forms of modal logic. Simon (1996) pointed out paradoxes that arise in imperative logic (the logic of commands) when directly combined with the rules of declarative logic. For example, the imperative in ADT of "Maintain the independence of FRs" when represented in symbolic logical form Maintain(Independence(FRs)) could paradoxically imply Maintain (Independent(FRs) v ~Independent(FRs)), since in declarative logic any proposition A logically implies A or ~A. Such paradoxes can often be found in modal logic (e.g. Sometimes(A) implies Sometimes(A or Not A) if rules of declarative logic are directly applied, which is a contradiction, since "A or Not A" is always true in declarative logic). To avoid such paradoxes Simon (1996) proposed that predicate (first order) logic (Boolos and Jeffrey (1989) is adequate for the application of the scientific method in design. He proposed that imperative statements be reduced to declarative statements through the use of utility functions. For example, if a utility function for a design is proposed that carries a higher value for a design whose FRs are independent, then the imperative "Maintain the independence of FRs" can be reduced to stating that a design whose FRs are independent has a higher utility. Thus, since utility functions themselves can be adequately represented and reasoned about within the framework of predicate (i.e. first order) logic, predicate logic is sufficient to formalize the science of design (Simon (1996)).
The activity of designing can be seen as a form of planning (Simon(1996)). The situation calculus (McCarthy and Hayes (1981)) was introduced to formally represent and reason about plans in an artificial intelligence context. A situation is defined as "the complete state of the universe at an instant of time". Each situation is associated with an instant of time. Changes in situations over time are represented using fluents, which are functions of situations ranging over situations (situational fluents) or truth values (propositional fluents). Fluents are also used to represent the results of actions. Causality (i.e. one situation causing another) is represented, using situational fluents, as a proposition A in a situation x causing a situation y to come about in which the proposition B is true. Attempts to formalize theories of action and change (including the situation calculus), engendered problems in maintaining the consistency the theory in the face of incomplete knowledge of situations. Reasoning under conditions of incomplete knowledge is characteristic of common sense reasoning (Mueller (2014)). Formalization of common sense reasoning about actions and change requires ensuring the consistency of the theory as new knowledge about the world becomes available. This sometimes necessitates revising the premises or conclusions of the theory as new knowledge about the world becomes available. In contrast to classical logic which is monotonic in nature (i.e., the truth values of sentences in a theory are not subject to change), formal reasoning under conditions of incomplete knowledge is non-monotonic in nature (i.e., the truth values of sentences in a theory could change as new knowledge is incorporated as premises into the theory) (McDermott and Doyle (1980), Reiter (1980)). Non-monotonic reasoning engenders problems in maintaining the consistency of theories of action and change as new knowledge becomes available. These include the following: the Frame Problem (i.e., the problem of having to specify a potentially infinite number of fluents whose values do not change over situations or time) (McCarthy and Hayes (1981)); Qualification Problem (i.e. the problem of having to specify a potentially infinite number of preconditions for a change) (McCarthy (1977)); and the Ramification Problem (i.e. the problem of having to specify a potentially infinite number of consequences of a change) (Finger (1987)). Methods such as default reasoning with the closed world assumption (Reiter (1980)), circumscription (McCarthy (1980)) and explicit representation of cause-effect relationsips (e.g. McCain and Turner (1997)) are used to address the above problems. Default reasoning uses rules of inference that are known mostly to be true (e.g if x is a bird, then x can fly), unless there is specific information based on which to conclude otherwise (e.g. x is a penguin and penguins cannot fly). The closed world assumption (Reiter (1978)) is the assumption that if a certain sentence is not explicitly stated to be true of a given object, then it is assumed that the sentence is not true of the object (e.g. if it is not explicitly stated that x is a penguin, then it is assumed that x is not a penguin). In a similar vein, circumscription involves apriori identifying all the arguments of a predicate for which sentences containing the predicate are true in a given theory (e.g apriori identifying all the x's for which "x is a bird", and "if x is a bird, then x can fly" are true). The explicit representation of cause-effect relationships (e.g. as causal laws (McCain and Turner (1997)) mitigates the Ramification problem by distinguishing between the following: inferences of effects from causes; and inferences of causes given evidence of effects. The event calculus (Kowalski and Sergot (1989), Shanahan(1999)) represents and reasons about change in terms of events pertaining to specific objects, predicates and times of interest, in contrast to the situations of situation calculus that pertain to the state of the entire universe at a given point in time. The event calculus (Shanahan (1999)) represents temporal changes as events that occur at points in time or propositions that hold over points or durations of time. Causality is represented in the event calculus as a material implication of events or propositions that hold over time by events or actions causing them. Pearl (2000) organized causal knowledge within the framework of structural causal models, consisting of exogenous variables, endogenous variables and functions causally relating exogenous to endogenous variables, to make predictions, identify potential interventions (actions to achieve a desired effect) and hypothesize about counterfactuals. In structural causal models, the structure of causal relationships is represented as a directed acyclic graph (DAG) whose nodes represent parameters of interest (e.g. fluents) and whose arcs represent causal relationships among the parameters of interest. The value of a node in the DAG of causal relationships is determined as a function of the values of its immediate parents in the DAG. Pearl (1988) pointed out the importance of explicitly taking into account causality in default reasoning to reflect differences between causal and evidential support for inferences (a proposition inferred as the effect of a cause known to have occurred weakens the basis for abducting the occurrence of other potential causes for the same effect). He categorized default rules (inferences) as causal (e.g. expectation of smoke from fire) or evidential (e.g. inference of fire from smoke). Geffner (1990) formalized causal theories in propositional modal logic. McCain and Turner (1997) formalized causal theories, in terms of propositions and actions, as sets of causal laws (of the form A => B to be read as "If A is true, then B has a causal explanation", which includes the case of A causing B, and where A can be a logical combination of propositions or actions, and B is a proposition). Bochman (2003Bochman ( , 2004 provided a causal calculus that integrated causal theories (involving propositions and not actions) with propositional logic. The causal calculus of Bochman (2003Bochman ( , 2004 was generalized to enable causal explanation of formulas expressible in first order logic (Lifschitz (1997), Lifschitz and Yang(2013)), which also enabled the logical representation of structural causal models of Pearl (2000) (Bochman and Lifschitz (2015)).
The logical representation of the FBS ontology of a design requires the representation of the following: objects (e.g. parts, sub-assemblies, etc.); attributes of objects (e.g. dimensions, materials, position, etc.); relationships among objects (e.g. a piston being a part of the piston sub-assembly); behavior of objects (e.g. rotation of crank from 0 degrees to 180 degrees causes the piston to move from top dead center position to bottom dead center position and inlet valve to open, etc.); FRs as abstractions of behavior (e.g. crank is rotated, piston is retracted, etc.); constraints on FRs (e.g. ranges of power generated within corresponding ranges of frequencies of crank rotation), etc.. Quantification (universal or existential) over objects such as parts or instants of time will be an important property to have in the representation of theories of design (e.g. "for all cylinders in the IC engine, the length of the cylinder is 90 mm" can be used to standardize the specifications of the cylinders in the IC engine; and "there exists time instants t1 and t2 such that in the duration [t1, t2], the power generated by the IC engine is greater than 0", is a statement that can be used to logically represent the FR of "power is generated by the IC engine"). Further, any scientific theory progresses from observing correlations and explaining them using causal theories (e.g. correlation between an attribute A (e.g. proportion of unburnt fuel in exhaust gas of an IC engine) and attribute B (e.g. air-fuel ratio at the inlet of the IC engine) of a design could be discovered to be due to a causal relationship between A and B (a lower air-fuel ratio causes a higher proportion of unburnt fuel to be present in the exhaust gases of the IC engine due to reduction in the quantity of oxygen that is required to burn the fuel) or due to a common cause C affecting both A and B (e.g. the correlation between the speeds of the piston and valves being can explained by the causal relationship between the speed of the crank shaft and the speeds of the piston and valves, etc.). Thus a formal theory of why a design displays a certain behavior or how a design satisfies its requirements would also include the representation of causation of states or events by other states or events.
The situation calculus (McCarthy and Hayes (1981) treats both physical things (e.g. parts) as well as situations as objects (subject to universal and existential quantification), while defining a situation as "the complete state of the universe at an instant of time". The concept of situation in the situation calculus lacks intuitiveness in the context of designing, which is done in very specific contexts and not taking into account the state of the world as a whole. Further, intuitively, from the standpoint of classical first order logic, a situation could be seen as the conjunct of all the predicates that hold true for the objects of the universe at a given point in time. Therefore, quantification over situations could intuitively seen be as quantification over predicates. Thus, from the standpoint of classical first order logic, the situation calculus has the implicit structure of a higher (second) order logic. The event calculus (Shanahan (1999)) does not quantify over things (e.g parts) in its ontology, and therefore is also not an intuitive choice to represent the FBS ontology of a design. On the other hand, classical first order logic can intuitively represent the following: parts and sub-assemblies as objects; the attributes and relationships among parts as predicates; behavior as logic programs involving first order formulas, FRs as existentially quantified first order formulas (obtained by replacing individuals in first order formulas representing states constituting behavior with existentially quantified individual variables), constraints as formulas in classical first order logic, etc.. Higher order logics, such as second order logic (Boolos and Jeffrey (1989)), provide more expressive power by allowing quantification over predicates, functions and sentences, which is not allowed in classical first order logic. However, higher order logics also have important limitations (e.g. it is not always possible to assess whether a theory in second order logic is consistent (i.e., free of contradictions) (Boolos and Jeffrey (1989)), and perhaps consequently not as widely used as classical first order logic. A limitation of classical first order logic, relevant to the representation of theories of design, is that it does not have the means to explicitly infer effect from cause in a manner that is distinct from other kinds of implication (e.g. inference based on correlation). The causal formalisms of Geffner (1990), McCain and Turner (1997) and Bochman (2003Bochman ( , 2004 do not use the language of classical first order logic to represent causal theories. The formalisms of Lifschitz (1997) and Lifschitz and Yang (2013) can represent causal theories in terms of first order formulas. However, the logical representation of causal rules (e.g. A => B) in the formalisms of Geffner (1990), McCain and Turner (1997), Lifschitz (1997), Bochman (2003Bochman ( , 2004 and Lifschitz and Yang (2013) pertain to inferring from A whether B has a causal explanation, rather than stating that A causes B (possibly originating in Pearl's (1988) idea of distinguishing statements with causal explanations from statements with non-causal (e.g. evidential) explanations in non-monotonic reasoning). However, the simplest representation of causality in behavior would be statements such as "A causes B" rather than statements such as "A implies that B has been caused". Simon (1996) described a complex system as one having a large number of parts with many interactions, and also described the importance and pervasiveness of hierarchy in complex systems. The Information Axiom of ADT (Suh (1990)) expressed the probability of a design satisfying its FRs in terms of information entropy (Shannon (1948)). Braha and Maimon (1998) analyze the complexity of a design into structural (in terms of the number and variety of symbols used to represent the design) and functional (in terms of the probability of satisfaction of requirements and constraints) complexity. They quantify the level of abstraction of a design, in terms of its information content, relative to that at the highest level of abstraction of the design. The definition of information content by Braha and Maimon (1998) allows for the explicit characterization of the impact of standardization and symmetry of parts on the information content of a design. El-Haik and Yang (1999) expressed the complexity of a design in terms of Boltzmann entropy having the following components: variability (pertaining to the probability of satisfaction of requirements and constraints); vulnerability (reflective of the size of the design problem in terms of number of FRs, sensitivity of FRs to design parameters, and dependencies among design parameters); and correlation (among design parameters). Frey, et al. (2000) provided approaches to compute the information content of decoupled designs, and showed that summing information content without regard to the dependencies among design parameters (DPs) could substantially overestimate the information content of a design. Guenov (2002) provided a measure of entropy, associated with coupling in the design matrix of ADT, for use in high level design. Khan, et al. (2007) used the concept of entropy to measure the diversity of components to satisfy functions, i.e. they measured, to an extent, the standardization of components used in a design. Modrak, et al. (2015) applied the measure of entropy of coupling developed by Guenov (2002) to characterize the complexity associated with mass customization of products. Krus (2013Krus ( , 2015 characterized the evolution of a design through various phases (such as: the generation of design space, i.e. the set of all possible designs; generation of concepts; screening and optimization of concepts) in terms of reducing entropy (i.e. increasing information) corresponding to each stage of the design process. Krus (2013Krus ( , 2015 also enabled the explicit characterization of the impact of standardization and parametric design of parts on the information content of the design space. Taking into account the considerations discussed above pertaining to a suitable formal logic for the science of design, this work provides a symbolic logical framework (L), consisting of classical first order logic augmented with a calculus of causal processes (to represent statements such as "A causes B", where A and B are sentences in the language of classical first order logic) to formalize, axiomatize and integrate the following types of theories of design: a) What is a design (formally represented as a system of requirements (including FRs), constraints, structure and behavior); b) Whether and how a design satisfies its requirements and constraints (represented as a formal theory, based on monotonic reasoning, of satisfaction of requirements (including FRs) and constraints by the structure and behavior of the design); and c) What is a good design (represented as a formal, axiomatic theory of general attributes of a good design). To formalize and axiomatize the theory of what constitutes a good design (i.e., the general attributes of a good design), this work does the following: a) defines general attributes of a good design (e.g. those pertaining to the probability of satisfaction of requirements and constraints and change manageability of the design) in terms of the representation of a design in L; b) provides a measure of entropy (Shannon (1948), Lebowitz (1993)) of a design, which generalizes the concept of information content of Suh (1990) to include other general attributes of a design (e.g. dimensionality, solution-neutrality of FRs, change manageability, etc.); and c) posits a utility function of a design that serves as an index of the quality of the design from a design-theoretic perspective (i.e., the extent to which the design exhibits general attributes of a good design such as solution-neutrality of requirements, high probability of satisfaction of requirements and constraints, change mangeability, etc.) that is defined to vary monotonically and inversely with the entropy of the design (from here onwards where quality of the design is mentioned, it means the quality of the design from a design-theoretic perspective, as described above). This utility function representing the quality of a design from a design theoretic perspective enables the re-framing of imperative statements (e.g. ensure the solution-neutrality of FRs) as declarative statements in predicate logic (e.g. improving the solution-neutrality of FRs reduces the entropy of the design, and therefore increases the quality of the design), as suggested by Simon (1998).
Since the emphasis of this work is on representing theories of design in a suitable logical framework rather than automating the generation or improvement of designs, the type of reasoning focused on in this work is monotonic reasoning. However, the use of a causal calculus in this work naturally mitigates the Qualification and Ramification problems (McCain and Turner (1997)). This could further be strengthened within the logical framework used in this work by methods of circumscription involving inclusion of statements such as the following: a) A v B v C <=> D if only A or B or C are explicitly stated in the theory as being the direct causes of D; A & B & C <=> D if only A, B and C in conjunction are explicitly stated in the theory as being the direct causes of D (analogous to the literal completion approach of McCain and Turner (1997), to limit the immediate causes of an effect to only those that are explicitly stated in the theory); b) If for all values of an individual variables x, y,.. within the corresponding ranges of values R1, R2, ... , a given sentence A is true, then include a sentence in the theory that states that the sentence A is true only for the values of x, y,... within their respective ranges R1, R2,…(an example of predicate completion (McCarthy (1980)). The above two approaches of circumscription have been illustrated in this work (see footnote for Table 3), and can be used to apply L in a nonmonotonic setting.
Section 2 describes the symbolic logical framework (L) for the science of design. Section 3 discusses the representation of designs (as systems of requirements (including FRs), constraints, structure and behavior) in L.
Section 4 presents the following types of formal theories of design: a) Whether and how the structure and behavior of a design satisfy its requirements and constraints (expressed as a formal proof of satisfaction of requirements and constraints by the structure and behavior of the design); and b) An axiomatic theory of what constitutes a good design (i.e. the general attributes that characterize a good system of requirements, constraints and structure and behavior). Section 5 illustrates the representation and analysis of designs in the symbolic logical framework L using the example of a schematic IC engine. Section 6 provides a summary and discussion of this work. Appendix 1 provides definitions of important concepts used through the remaining sections of this work. Appendix 2 provides the calculus of causal processes.

Symbolic Logical Framework (L)
The symbolic logical framework (L) described in this work consists of a language and logic to describe and reason about objects and processes. L builds on classical first order (i.e. predicate) logic (Boolos and Jeffrey (1988)) by incorporating a calculus of causal processes. The aspect of L that is predicate logic enables the description and reasoning about states of objects. The calculus of causal processes of L enables the description and reasoning about causal relationships among states of objects over time.
In this work: objects having certain attributes or being in certain relationships with other objects is called a state; a state that holds for a duration of time is called a state-duration (SD); a process is defined as a description of states of objects over time; and a causal process is a process that includes descriptions of causal relationships among states or SDs. An SD is a sentence in classical first order logic that describes a state holding over a specified duration of time. An SD is abbreviated in L as (S, D), wherein S denotes a state and D a duration of time. The representation of an SD in the form (S, D) in L is an abbreviation for a sentence in classical first order logic. For example, in this work, an SD abbreviated as (S, [t1, t2]) can represented in classical first order logic as follows: , where A is a predicate with the individuals a, b,..,t as arguments.
Processes (as defined in Appendix 1.1) are represented in L in the following ways: a) SDs; b) sentences formed by conjunction or disjunction of states and at least one SD; and c) sentences containing one or more causal implication ( C =>) signs. Sentences containing one or more causal implication ( C =>) signs represent causal processes in L. Each causal implication ( C =>) sign corresponds to a causal process. The sentence, formed from states or SDs only using the operations of negation, conjunction or disjunction, which is immediately to the left of the causal implication ( C =>) sign is called the input sentence of the corresponding causal process, and that immediately to the right of the causal implication ( C =>) sign is called the output sentence of the causal process.
The use of L to represent recursive processes is described in Appendix 1.1. Definitions of objects, attributes, states, SDs, processes, etc. are given and their representation in L described in Appendix 1.1. The language of L is defined in Supplementary Tables 1 and 2. The calculus of causal processes is given in Appendix 2.

Representation of the design of an artifact in the symbolic logical framework of L
A design is represented in the language of L as a set of sentences stating the requirements (including FRs), constraints, structure and behavior of the design. Requirements (including FRs), constraints and structure are represented in L as sentences that are states, SDs, or logical combinations of states or SDs using negation, conjunction or disjunction. In addition, every sentence representing an FR in L contains one or more existentially quantified individual (object) variables, as described further in the following paragraph. Behavior is represented in L as a temporal or causal process. Sentences in L that represent behavior as a causal process have one or more instances of the causal implication ( C =>) sign.
In this work functional requirements (FRs) are expressed as SDs to be achieved by the design (e.g. similar to the definition of function as an effect in Chandrasekaran and Josephson(1997)). FRs are represented in L as SDs with existential quantifiers. For example, the FR to generate power is representable in L as: t1t2vt , there exists a time at which the power of the IC engine is greater than zero). Constraints that characterize FRs, e.g. specified range of power of an IC engine (also called performance requirements (PRs) in this work), can be obtained from FRs with values specified for one or more existentially quantified variables. For example, the performance requirement that an IC engine generate 75 hp of power (representable in L as: t1t2t((t ≥ t1 & t ≤ t 2 & t1 ≤ t2) =>(Power(IC Engine) = 75 hp)) can be obtained from the representation of the corresponding FR to generate power by specifying the value of 75 hp for the existentially quantified variable v in the representation of the FR given above. Similarly, expected behavior of the design can be obtained from an FR or PR by specifying values for one or more existentially quantified variables in the FR or PR. For example the expected behavior of generating 75 hp of power in the power stroke of the IC engine (representable in L as: t((t ≥ 30 ms & t ≤ 40 ms) =>(Power(IC Engine) = 75 hp)) can be obtained from the representation of the corresponding FR above by specifying values for the existentially quantified variables v, t1 and t2, or can be obtained from the representation of the corresponding PR above by specifying values for the existentially quantified variables t1 and t2. The structure of a design is represented as a set of states in L. The behavior of a design is represented as a process (temporal or causal) in L, i.e. as a set of sentences involving SDs and, if behavior is represented as a causal process, one or more causal implication ( C =>) signs. L allows the representation of designs at various levels of abstraction (including at the conceptual design stage as well as stages involving progressively more detail) Appendix 1.2 provides definitions pertaining to designs, their representation in L, and the explication of their attributes in terms of their representation in L. The representation of the requirements, including FRs, structure and behavior of a design in the language of L is illustrated in Tables 1 to 3 for the design of a schematic IC engine shown in Figure 1 of Section 5.

Functional decomposition in the framework of L
An FR (FRX) represented (as an existentially quantified sentence) in the language of L can be decomposed into (i.e., elaborated at the next level of detail as) a temporal or causal process involving child FRs (e.g. FRX1, FRX2, FRX2, …). However, for an FR (FRX) decomposed as above, exactly one of the child FRs (FRX1 or FRX2 or FRX2 or …) should logically imply the parent FR (FRX). For the behavior of the design to satisfy the FRs (and PRs) of the design, each lowest level FR (or PR) of the design should be logically entailed by one or more sentences describing the behavior of the design. For example, in the functional decomposition of FR1 (Air-fuel mix is sucked into the cylinder) in Table 1, exactly one child FR (FR13) of FR1 logically implies FR1.

Formalization of Theories of Design in L
The following types of theories of design are formalized using L: a) Theories of a whether and how the structure and behavior of a design satisfy the requirements and constraints of the design; and b) Theory of what constitutes a good design (in the sense of a system of requirements, constraints, structure and behavior). The formalization of the above kinds of theories is described below.

Formalization of theories of whether and how the structure and behavior of a design satisfy the requirements and constraints of the design
A formal, axiomatic theory, describing how the structure and behavior of a design satisfy its requirements and constraints, can be represented in L. Such a theory (henceforth called the theory of satisfaction of the requirements and constraints of a design) is illustrated (see Section 5) in Table 5 for the schematic IC engine shown in Figure 1, whose requirements, constraints, structure and behavior are shown in Tables 1-3. In a theory of satisfaction of the requirements and constraints of a design, the specified structure and behavior of the design constitute the axioms (or premises), and the sentences derived from these axioms constitute the theorems (or conclusions based on the premises). The structure and behavior of a design can be said to satisfy a given requirement of the design if and only if the sentence representing the requirement in L occurs as a theorem (i.e., the satisfaction of the requirement is entailed by the specified structure and behavior constituting the premises) in the above formal, axiomatic theory of satisfaction of requirements and constraints of the design. Similarly, a given constraint is said to be satisfied (i.e., not violated) by the structure and behavior of a design if and only if the negation of the constraint is not entailed by the specified structure and behavior of the design. Consequently, if the sentence in L representing a constraint is entailed by the specified structure and behavior of the design, then also the constraint is said to be satisfied by the specified structure and behavior of the design.
If the structure and behavior do satisfy the requirements and constraints of the design, then how the structure and behavior of a design satisfy the design's requirements and constraints is expressed as a formal proof in L. It can be seen from Section 3 above that FRs, PRs and behavior can be represented in L in a hierarchy of existential quantification. A PR that is obtained by specifying the values of one or more existentially quantified variables in an FR logically implies the FR. Similarly a description of behavior that is obtained by specifying the values of one or more existentially quantified variables in a PR or an FR logically implies the PR and FR, respectively. For example, the PR that an IC engine generate 75 hp of power (represented in L as t1t2t((t ≥ t1 & t ≤ t 2 & t1 ≤ t2) =>(Power(IC Engine) = 75 hp)) logically implies the satisfaction of the FR to generate power represented in L as ). This possibility of logical implication of FRs and PRs by behavior, and the logic of L including its causal calculus, together enable the proof of satisfaction of FRs and constraints by the structure and behavior of the design. Table 4 illustrates the proof of satisfaction of FRs and constraints by the structure and behavior of an illustrative IC engine. Garvin (1987) proposed eight dimensions of quality of a product, viz.: performance, features, reliability, durability, conformance, serviceability (maintainability), aesthetics and perceived quality. Bralla (1996) proposed upgradability as an additional dimension of quality. The above nine dimensions of quality can be distilled into two generally desirable attributes of a design: a) satisfaction of requirements (including functionality, performance, features, conformance, aesthetics, etc.); and b) change manageability (including maintainabilty and upgradability). The information axiom of ADT (Suh(1990)) and the consequent desirability of wide specifications, low variability and low bias pertain to the attribute of requirements satisfaction. The independence axiom of ADT (Suh(1990)) and its implications pertaining to the desirability of uncoupled or decoupled designs over coupled designs, and the concept of modularity of a design (Ullrich and Eppinger (1995)) pertain to the attribute of change manageability of a design. Other generally desirable attributes of a design (Pahl and Beitz(1988), Ullrich and Eppinger(1995), Suh(1990)) include FRs stated in a solution-neutral manner so as to allow a wide number and variety of solutions to satisfy them, standardization of and symmetry in the components of a product, and a parametric design (Woodbury (2010), Oxman and Gu (2015)).

Formalization of theories of what constitutes a good design
The formal representation of a design in terms of its requirements (including FRs), constraints, structure and behavior in the language of L enables the definition of the above desirable properties of a design in terms of the representation of the design in the language of L. Appendix 1 provides definitions of various general attributes of a design, in terms of the representation of the design in the framework of L, including the following: solutionneutrality of FRs of a design; coupling in designs; coupled, uncoupled and decoupled designs; probability of success (and consequently, information content) of a design; standardization of and symmetry in the components of a design; and parametric designs.
The formalization of theories of what constitutes a good design (in terms of requirements, constraints, structure and behavior) can be achieved by representing the above generally desirable attributes of a design as sentences in a formal theory of design. Further, axiomatization of such a theory requires that one or more sentences in the theory be axioms from which the remaining of the above generally desirable attributes of a design be derived as theorems.
The approach taken in this work for the creation of such a formal, axiomatic theory of design, is now described.
First, the concept of entropy (Shannon (1948), Lebowitz (1993)) underlying the information axiom of ADT (Suh (1991)) is generalized to include the following pertaining to a design: a) solution-neutrality of FRs (at various levels in the hierarchy of FRs); b) dimensionality (the number of mutually independent, constraned attributes) of the design; c) the probability of satisfaction of requirements and constraints by the structure and behavior of the design (calculated using a Bayesian framework (Pearl (2000)); and d) change manageability of the design. The entropy associated with the probability of satisfaction of requirements and constraints of the design is further analyzed into the components of width of specified ranges for the attributes of the design, variability of attributes of the design around their respective central tendencies, and bias in the attributes of the design (i.e., shift of the central tendencies of the attributes of the design from their respective specified target values). The entropy of change manageability of a design takes into account the nature (e.g. circular or non-circular), extent (number of) and relative importance of dependencies among sibling sub-designs at various levels in the hierarchy of the design. The entropy of a design as defined in this work, in addition to taking into account the factors mentioned above, also enables the explicit characterization of the impact of standardization, symmetry and the extent of parametrization of a design, on the overall entropy of the design. In this work we posit that lesser the entropy of a design the better is the design. The entropy of a design in terms of its components mentioned above is defined in Appendix 1.
Having defined the entropy of a design as above, a utility function "Q(d)" of a design (d) representing the quality of a design from a design-theoretic perspective is posited. The quality of a design from design-theoretic perspective (Q(d)) is the degree to which the design (d) exhibits the general attributes of a good design such as those discussed above (e.g. solution-neutrality, high probability of satisfaction of requirements and constraints, change-manageability, etc.). The definitions of generally desirable attributes of a design in terms of its representation in L, the definition of the entropy of a design, and the concept of the quality of a design from a design-theoretic perspective together enable the creation of a formal, axiomatic theory of what constitutes a good design (in the sense of a system of requirements, constraints, structure and behavior).
The first (and only) axiom of the theory states that lower the entropy of a design, the higher its quality (i.e., better the design) from a design-theoretic perspective. From this axiom and the definition of the entropy of a design are derived various theorems that formally state the desirability of the following: solution-neutrality of FRs, a small number of independent attributes of a design; wide specifications; low variability and bias in the attributes of the design; standardization of parts and components; symmetry of parts and components; uncoupled designs over coupled designs; of circular over non-circular dependencies among sibling sub-designs; avoiding dependencies among sibling sub-designs at higher levels in the hierarchy of sub-designs than at lower levels; and avoiding non-circular dependencies at higher levels in the hierarchy of sub-designs than at lower levels. As an example of how such a theory could help identify attributes of a good design over and above the set of attributes discussed above, the desirability of parametric designs (Woodbury (2010), Oxman and Gu (2015)) is also derived as part of the theory discussed above. For example, the practice of following rules of thumb to specify a relatively large number of attributes (e.g. various parameters of the inlet and exhaust valves of an IC engine, such as valve stem diameter, thickness of the valve head, etc.) in terms of the specifications of a small set of attributes of the design (e.g. the specified diameter of the valve head (Heywood (1988)), could be seen as a type of parametric design.
The formal, axiomatic theory of design is given below.

Definitions of terms
Axiom 1: The quality of a design from a design-theoretic perspective varies inversely with the entropy of the design. That is, lower (higher) the entropy of a design the better (worse) is its quality from a design-theoretic perspective. (Here onwards, the phrase "better the design" is taken to mean "better the design from a design theoretic perspective").
That is: The following theorems follow from the definition of entropy (see Appendix 1) and Axiom 1 above (the statements of the theorems in words are by default qualified by trade-offs that might exist among various components of entropy of a design): Theorem 1: The more solution-neutral the FRs of a design, the better the design.
That is: Theorem 2: The solution-neutrality of an FR higher in the hierarchy of FRs of a design is more important than the solution-neutrality of an FR lower in the hierarchy of FRs of the design.
(by definitions of solution-neutrality of a design, entropy of a design and Axiom 1).
Theorem 3: Lower the dimensionality of a design (k), the better the design.
Consider a design d' obtained from design d only by reducing the dimensionality (number of constrained independent attributes) of d. Then: (by the following: definition of design d' with respect to design d given above; definition of dimensionality of a design; definition of entropy of a design; and Axiom 1).
Theorem 4: Wider the specified ranges of attributes of a design, the better the design.
Consider a design d' obtained from design d only by changing the specified range of values, i.e. widening or narrowing the specifications for one or more attributes of design d. Then: (by definition of entropy of a design and Axiom 1).
Theorem 5: Lower the variability in the attributes of the structure and behavior of a design about their mean, the better the design.
Consider a design d' that is obtained from design d only by reducing the variation of one or more attributes of the structure or behavior of d about their respective means. Then: (by definition of entropy of a design and Axiom 1).
Theorem 6: Lower the bias in the values of attributes of the structure and behavior of a design from the specified target value of the attribute, the better the design.
Consider a design d' that is obtained from design d only by changing the bias in the values of the attributes of the structure or behavior of d from the specified target values for the attributes. Then: (by definition of entropy of bias of a design and Axiom 1).
Theorem 7: More the standardization of parts or components of a design, the better the design.
Consider a design d' obtained from design d only by changing the extent of standardization of parts or components of d. Then: (since standardization of parts or components reduces the number of constrained independent attributes of the design; and by definition of dimensionality of a design, definition of entropy of a design and Axiom 1).
Theorem 8: More the symmetry in the parts or components of a design, the better the design.
Consider a design d' obtained from design d only changing the extent of symmetry in the form of the parts or components of d. Then: (since the value of one attribute in a pair of attributes that are mirror images of each other can be obtained from the value of the other attribute in the pair, thereby reducing the number of constrained independent attributes of the design; by definitions of dimensionality and entropy of a design; and Axiom 1).
Theorem 9: Fewer the dependencies among sibling sub-designs of a design, the better the design.
Consider a design d' obtained from design d only by changing the number of dependencies among sibling subdesigns of d. Then: (by definition of entropy of a design and Axiom 1).
Theorem 10: An uncoupled design is better than a coupled design.
Consider a design d that has one or more circular or non-circular dependencies among its sibling sub-designs.
Consider a design d' obtained from design d only by changing the number dependencies (circular or noncircular) among sibling sub-designs of design d. Then: Therefore: (3) and (4) (8), (9) and Axiom 1).
Theorem 11: Within a given level of FRs, eliminating a single circular dependency is more important than eliminating a non-circular dependency in a design.
Consider a design d that has one or more circular dependencies and one or more non-circular dependencies among its sibling sub-designs at level z in its hierarchy of FRs. Consider a design d' obtained from design d only by eliminating a circular dependency and a design d'' obtained from design d only by eliminating a non-circular dependency, each at level z in the hierarchy of FRs of d. Then: (by definition of entropy of a design and Axiom 1).
Theorem 12: Eliminating a circular dependency at a given level in the hierarchy of FRs of a design is more important than eliminating a circular dependency at a lower level in the hierarchy of FRs of a design.
Consider a design d that has one or more circular dependencies among its sibling sub-designs at level z and level z + m (m > 0) in its hierarchy of FRs. Consider a design d' obtained from design d only by eliminating a circular dependency at level z of FRs of d and a design d'' obtained from design d only by eliminating a circular dependency at level z+m of the FRs of d. Then: (by definition of entropy of a design and Axiom 1).
Theorem 13: Eliminating a non-circular dependency at a given level in the hierarchy of FRs of a design is more important than eliminating a non-circular dependency at a lower level in the hierarchy of FRs of a design.
Consider a design d that has one or more non-circular dependencies among its sibling sub-designs at level z and Theorem 14: Eliminating a circular dependency at a given level in the hierarchy of FRs of a design is more important than eliminating a non-circular dependency at a lower level in the hierarchy of FRs of a design.
Consider a design d that has one or more non-circular dependencies among its sibling sub-designs at level z and Theorem 15: A decoupled design is better than a coupled design that is not decoupled (but within limits determined by the number of non-circular dependencies that are equivalent, in terms of their contribution to the entropy of a design, to a circular dependency, at various levels of FRs of a design).
Consider a design d that is coupled but not decoupled. Let design d' be obtained from design d only by eliminating all the circular dependencies in d and adding zero or more non-circular dependencies. Let d'c be the design obtained from design d only by eliminating all the circular dependencies in d. d'u be the design obtained from design d only by adding zero or more non-circular dependencies Then: (By definition of entropy of a design and Axiom 1).
Theorem 16: More parametric the design (i.e., higher the proportion of dependent to independent attributes of a design), the better it is. Thus it can be seen from the above formalization that notions of what constitutes a good design, such as a high probability of satisfaction of requirements and constraints, change manageability (enabled by modularity), low dimensionality, high standardization, high symmetry, high parametrization, etc., are often not mutually independent, but have to be traded off against one another in accordance with the needs of stakeholders of the design, including the users of the design and the organization designing a product or solution. For example, a design that is highly optimized for performance might entail a reduced degree of standardization than otherwise might have been possible (e.g. an IC engine re-optimized for weight might use plastic components where previously aluminum components might have been used, thereby resulting in reduced use of standard parts). It can also be seen that formalization of the concept of coupling in design necessitates the recognition (especially in the light of the trade-off among attributes of a good design discussed above) of the following nuances of coupling in design (e.g. over and above whether a design is uncoupled, coupled or decoupled): a) the extent of coupling (e.g. number of non-circular and circular dependencies); b) the relative importance of circular compared to non-circular coupling; and c) the relative importance of coupling in higher levels of FRs compared to that in lower levels of FRs. Also, it can be seen that the above formalization enables the explicit identification of the desirability of parametric designs and provides a formal explanation of why parametrization in designs is desirable (i.e. parametrization tends to reduce the entropy of a design by reducing its dimensionality).
As can be seen from the description of the entropy of a design and the consequent theory of of what constitutes a good design given above, the entropy of a design could potentially be reduced, and therefore the design potentially improved, by one or more of the following ways: a) making FRs solution-neutral; b) reducing the dimensionality of the design (k); c) increasing the probability of success of the design; and d) reducing coupling in the design. The dimensionality of the design could be reduced by multiple ways including: eliminating one or more FRs; eliminating components or parts; increasing standardization of parts or components; increasing symmetry of parts or components; making the design more parametric (e.g. adopting empirically validated rules of thumb whereby certain attributes of a part or component are defined as functions of other attributes of the part or component). The probability of success of a design could be improved by widening the specifications and/or reducing the variability and bias of attributes of the design. Coupling in the design could primarily be reduced by eliminating circular dependencies (i.e., trying to make the design decoupled), and secondarily by reducing or eliminating the number of non-circular dependencies (i.e., trying to make the design uncoupled, once it has been made decoupled) among sibling sub-designs of the design. However, reduction in one component of the entropy of the design could potentially be accompanied by an increase in another component of the entropy of the design. For example, reducing the dimensionality of a design by integrating parts could potentially be accompanied by increased coupling, and standardization of parts could potentially impact the probability of success of the design. Thus, it can be seen that the structure of entropy of a design could be used to guide the search for improvements in the design, while the values of the parameters determining the impact of dimensionality, solution-neutrality, circular or non-circular couplings, etc. on the entropy of the design (see Appendix 1 for definition of entropy of design in terms of the above) could be used to guide trade-offs among various considerations such as solution-neutrality, dimensionality, probability of success, change manageability, etc. of a design.

Illustrative example of a schematic IC engine
A schematic of a spark ignition IC engine shown in Figure 1 (a) and 1(b) is used to illustrate the representation and analysis of designs using the symbolic logical framework L discussed above. For the illustrative IC engine, the representation of the following in the language of L is shown in Tables 1, 2 and 3, respectively: a) requirements and constraints; b) structure; and c) behavior. Table 4 shows the proof (and expressions to calculate the probability) of satisfaction of the requirements and constraints of the illustrative IC engine by its structure and behavior. Supplementary Figures 1 (a)
The structure of entropy (design-theoretic (as defined in Appendix 1.2), and not the thermodynamic entropy of heat engines) of the illustrative IC engine enables the search for improvements in its design. For example, the entropy of the design of the illustrative IC engine could potentially be reduced by: a) reducing the dimensionality of the design; b) increasing the probability of satisfaction of the requirements and constraints of the design; and c) improving the change-manageability of the design. The dimensionality of the design could potentially be reduced, for example, by making the inlet valve unidirectional (thus eliminating the complex valve train used to control the timing of opening and closing the valve), standardizing the inlet and exhaust valves, eliminating the components for spark generation (e.g. by converting the IC engine into a compression ignition engine), etc.. The probability of satisfaction of requirements and constraints of the design could potentially be improved by appropriately modifying the specifications of and controlling the variation of statistically significant factors that determine the performance of the design. Since the illustrative IC engine is a decoupled design, improving its change manageability might not take priority over the other sources of entropy of the design.
The reduction in one component of entropy of the design obtained by each of the above potential improvements to the design of the illustrative IC engine might also have a tendency to increase the entropy of another component of entropy of the design, unless further adjustments are made to the design. For example, making the inlet valve unidirectional, while reducing the entropy of dimensionality, could increase the entropy of satisfaction of requirements and constraints by reducing the power output of the engine. Thus, since it is likely that any given design improvement could reduce one component of entropy while increasing another, the selection of potential improvements in a design has to be guided by the overall impact of the improvements on the entropy of the design. Figure 4 (b) illustrate the trade-offs among various components of entropy of a design associated with potential improvements in the design. Figure 4 (a) illustrates the overall impact of standardizing the inlet and exhaust values on the entropy of the design of the schematic IC engine, assuming that the above improvement reduces the power of the engine marginally. Figure 4 (b) illustrates the overall impact of changing the design of the inlet valve from a cam-operated poppet valve to a unidirectional valve, assuming that this change, while eliminating the complex mechanism (valve train) required to control operation of the valve, causes a substantial reduction in the power of the engine. Such analyses could help prioritize ideas for improving a design by analyzing the impact of the potential improvement on the various components of entropy, as well as the overall entropy, of the design.

Figures 4 (a) and
Designers could assign different relative weights to the components of entropy depending on the purpose and context of the design, as shown in Figure 3(b). For example, the relative weight of entropy of satisfaction of requirements and constraints could be an order of magnitude higher than the other components of entropy if the product is used in a context where the cost of non-conformance is really high (e.g. surgical instruments), and the relative weight for change manageability could be relatively high in contexts where new versions of the product are released frequently (e.g. software products or automobiles). Figure 4(c) illustrates the sensitivity of the illustrative IC engine to the relative importance of various components of entropy shown in Figure 3 (b) (in this case, the effect of +/-10% change in the relative importance of each component of entropy on that component of entropy of the illustrative IC engine). For example, it can be seen that the design of the illustrative IC engine, as is, is not impacted by the relative importance (weight) assigned to solution neutrality or circular coupling in the design. This is because the design, as is, is solution-neutral and does not contain any circular coupling. Also, as expected from the weights given in Figure 3 (b), the design of the illustrative IC engine is most sensitive to conformance to requirements and constraints, relatively less sensitive to dimensionality and marginally sensitive to non-circular coupling. Thus the weights assigned to each component of entropy could potentially play a role in guiding the search for improvements of the design by emphasizing (or de-emphasizing) the components of entropy that are important in the context of the design.

Summary and Discussion
This work provides a symbolic logical framework (L), consisting of first order logic augmented with a causal calculus, to formalize, axiomatize and integrate the following types of theories of design: a) What is a design; b) How does a design satisfy its requirements and constraints (represented as a formal theory of satisfaction of requirements (including FRs) and constraints by the structure and behavior of the design); and c) What is a good design (represented as a formal, axiomatic theory of general attributes of a good design). A design is formally represented in L, in keeping with the FBS ontology (Gero (1990), Vescovi, et al. (1994, Umeda, et al. (1990)), as a system of requirements (including FRs), temporal and causal relationships among FRs, constraints, structure and behavior. The wide applicability of L enables the following: use of a single language to represent functions, behavior and structure, thereby enabling their seamless integration; and generality in the formalization of theories of design.
A central question pertaining to any design (in the sense of an artifact) is: "How will the design satisfy its requirements and constraints?". L enables the formal proof of satisfaction of FRs by the structure and behavior of the product through the use of logical, material and causal implication. FRs are represented as existentially quantified forms of behavior using L, which enables the logical implication of FRs (e.g. power is generated) by behavior (e.g. 75hp of power is generated). The formal representation of a design in terms of its requirements (including FRs), constraints, structure and behavior in the language of L enables the definition of generally desirable properties of designs (e.g. solution-neutrality of requirements, high probability of satisfaction of requirements and constraints, change manageability enabled by a modular architecture, standardization, symmetry, etc. (Pahl and Beitz (1988), Ullrich and Eppinger (1995), Suh (1990)) in terms of the representation of the design in the language of L. Solution-neutrality of an FR is defined as the absence (non-usage), in the representation of the FR in the language of L, of individuals or predicates specific to a solution (structure and behavior) for the FR. Coupling in the design of an artifact represented in L is a dependency of one sub-design (structure and behavior intended to satisfy a specific FR) on a sibling sub-design in the hierarchy of the design and its sub-designs. An uncoupled design is one wherein there are no couplings. A coupled design is one where there are one or more couplings. A decoupled design is a coupled design where the couplings are not circular.
The probability of success of a design is defined as the probability of the structure and behavior of the design satisfying all of its requirements and constraints. The probability of success of a design is computed in a Bayesian framework (Pearl (2000)). The entropy of a design is then defined in terms of general attributes of a design represented in the language of L. The definition of entropy of a design used in this work is a generalization of the concept of entropy (Shannon (1948)) underlying the information axiom of ADT (Suh (1990)), to encompass the following: solution-neutrality of FRs; dimensionality (i.e., the number of independent constrained attributes) of the design; satisfaction of requirements and constraints; and change-manageability. The entropy of satisfaction of requirements and constraints is further analyzed into the entropies of the following pertaining to the attributes of the design: width of specifications; variability about the central tendency of the attribute; and bias (shift of the central tendency of the attribute from the specified target value for the attribute).
The entropy of change manageability of a design takes into account the number and types (circular and noncircular) of dependencies (couplings) among sibling sub-designs in the hierarchy of sub-designs of the design.
The quality of a design from a design-theoretic perspective (i.e., the degree to which a design exhibits desirable attributes such as those discussed above) is then defined as a utility function of the design that increases (or decreases) with decreasing (or increasing) entropy of the design.
A formal, axiomatic theory of what constitutes a good design is then constructed based on the following: a) definitions of general attributes of a good design as represented in the language of L; b) definition of the entropy of a design in terms of or related to the definitions of general attributes of a good design as represented in the language of L; and c) the utility function Q representing the quality of a design from a design-theoretic perspective. In the formal, axiomatic theory of design thus constructed, it is taken as an axiom that lower the entropy of a design, the higher its quality (i.e., the better it is) from a design-theoretic perspective. Based on the above axiom, and the definition of entropy of a design, the desirability of general attributes of a design such as solution-neutrality of FRs, wide specifications, low variability and bias, less coupling (i.e. a more modular architecture), higher standardization and symmetry, etc. follow as theorems.
The use of L is illustrated in the context of a schematic IC engine. The requirements (including FRs), constraints, structure and behavior of the IC engine are represented in the language of L. A formal proof of satisfaction of the requirements and constraints of the illustrative IC engine by its structure and behavior is provided using the logic of L. The probability of satisfaction of the requirements and constraints of the illustrative IC engine by its structure and behavior is demonstrated in a Bayesian framework using a Monte-Carlo simulation. It is shown that the FRs of the illustrative IC engine are represented in L in a solution-neutral manner. It is also shown that the design of the illustrative IC engine, as represented in the language of L, is decoupled. The computation of the entropy of the illustrative IC engine is demonstrated. The use of the entropy of the design to guide the search for and assess trade-offs among potential improvements in the design is illustrated. The specification of the relative importance of various components of entropy of a design in consonance with the purpose and context of the design activity, to guide the search for solutions, is described and illustrated.
Although monotonic reasoning has primarily been used in this work in keeping with the emphasis of this work on formalizing theories of design rather than automating the generation of designs, the causal calculus of L naturally mitigates the Qualification and Ramification problems encountered in non-monotonic reasoning.
Further, approaches of circumscription (analogous to literal completion of McCain and Turner (1997)) and predicate subscription (McCarthy (1980) have been illustrated (see footnote of Table 3), and can be used to apply L in a non-monotonic setting.
L, being a first order logic, is undecidable (Boolos and Jeffrey (1989)), i.e., there is no method by which it can always be determined whether any given sentence in L is derivable (or not derivable) from any given set of premises in L. An implication of the undecidability of L in the context of the representation and reasoning about designs is that it cannot always be determined, using the logic of L, whether or not a given structure and behavior of a design, as represented in L, will satisfy its requirements and constraints. This is not to say that whether the structure and behavior of a design satisfy its requirements and constraints cannot be determined at all. For most practical cases, especially at the conceptual design stage, it should be possible to logically ascertain whether or not the structure and behavior imply the satisfaction of requirements and constraints. In fact, if the requirements and constraints of a design are indeed implied, in the logic of L, by the structure and behavior of a design, it can eventually be shown that this is so, through repeated application of the rules of logic of L (by virtue of the fact that first order logic is semi-decidable (Boolos and Jeffrey (1989)). However, the time required to demonstrate the soundness of a sound design could, in the worst case, grow exponentially with the size (e.g. number of symbols used) of description of the design in terms of its structure, behavior, requirements and constraints (by virtue of the fact that first order logic has exponential worst-case complexity (Papadimitriou (1994), Vardi (1982).
An important aspect of design is the joint consideration of form and function (e.g. Palmer and Shapiro (1993)).
This work focuses on the function and not the form or geometry of a design. This work does not, as yet, include in its scope approaches for the automatic synthesis of designs (e.g. using a search for design prototypes as in Gero(1990), Umeda, et al. (1990) or using grammar based approaches such as those described by Stiny and Gips (1971), Schmidt and Cagan (1995), Sridharan and Campbell (2005), etc.). The above limitations could be the subject matter for future work. In particular, the following could be explored in the future: automated search or synthesis of designs or design improvements guided by a formal, axiomatic, integrated theory of what constitutes a good design; and use or extension of the symbolic logical framework L to formalize, axiomatize and integrate theories of what constitutes a good design process.

Object
An object is a thing such as an IC engine, sub-assembly of the IC engine (e.g. piton sub-assembly), a part of an IC engine (e.g. piston, crank shaft, etc.), a feature of a part (e.g. cylinder bore, cavity in the cylinder head, etc.).

Domain
A domain is a set of objects (e.g. domain of IC engine consisting of the IC engine, its sub-assemblies and its parts). A domain is represented in L as a set of individual letters or names, e.g. (a, b, c, etc.).

Simple (or atomic) Object
A simple object is one that is not composed from other objects (e.g. a piston is a simple object in the domain consisting of the parts and sub-assemblies of an IC engine).

Compound Object
A compound object is one that is composed from other objects (e.g. a piston sub-assembly, consisting of piston, piston pin, etc. is a compound object in the domain consisting of the parts and sub-assemblies of an IC engine).

Attribute or Property of an Object
An attribute or property of an object is a descriptor of the object. For example, being of a cylindrical shape, having a diameter of 90 mm, having a length of 50 mm, being made of aluminum, etc. are attributes or properties of a piston. An attribute of an object is represented in L using a function symbol whose argument is the individual letter representing the object and whose range is the set of individual letters that represent the values that the attribute can take.

State of a simple object
The state of a simple object is the set of properties of the object and the relationships in which the object exists with other objects. For example, the state of a piston can be described in terms of its position in the engine cylinder, its temperature, etc..

State of a compound object
The state of a compound object is the set of properties that the object, and its constituent objects, have and the relationships that the object, and its constituent objects, have with other objects. For example, the state of a piston assembly can be described in terms of the attributes (e.g. position, velocity) and relationships of the piston assembly (e.g. type of motion relative to cylinder bore) and its constituent parts (e.g. temperature of piston, type of fit between piston and piston ring, etc.). The state of a simple or compound object is represented as a sentence in L (excluding the use of the causal implication ( C =>) sign in the sentence representing the state of the object).

Atomic State
An atomic state is a state that is not composed from other states using conjunction or disjunction. For e.g., the rotational speed of a crank shaft being 3000 rpm is an atomic state of the crank shaft. Atomic states are represented as atomic sentences (sentences that are not formed form other sentences using conjunction or disjunction) in L. For example, the state of a piston being in the top dead center position in the cylinder of the IC engine is an atomic state, and is represented in L using the atomic sentence: (Position (Piston) = Top Dead Center).

Compound State
A compound state is a state that is composed from other states using conjunction or disjunction. Compound

Duration
A duration is continuous period of time with a start time and end time, including the start and end times. When the end time is the same as the start time, the duration reduces to an instant of time.

State-Duration [SD]
An SD is a state that persists over a duration of time. In this work, an SD abbreviated as (S, [t1, t2]) can represented in classical first order logic as follows: • t ((t ≥ t1) & (t ≤ t 2)) => A(a, b, ..)), where A is a predicate with the individuals a, b,..(but not t) as arguments.

(but not t) as arguments.
For example, the SD of the inlet valve of an IC engine being open from 0 ms to 10 ms can be abbreviated in L as: (Open (Inlet Valve), [0 ms, 10 ms]) and fully represented as a sentence in classical first order logic as: t ((t ≥ 0 ms) & (t ≤ 10 ms)) => Open(Inlet Valve)).

Atomic SD
An atomic SD is an SD whose state is an atomic state. For example, the SD of the inlet valve of an IC engine being open from 0 ms to 10 ms, and represented in L as t ((t ≥ 0 ms) & (t ≤ 10 ms)) => Open(Inlet Valve)), is an atomic SD.

Compound SD
A compound SD is an SD whose state is a compound state. For example, the SD of the inlet valve of an IC engine being open from 0 ms to 10 ms, and the exhaust valve being closed in the same duration, and represented in L as t ((t ≥ 0 ms) & (t ≤ 10 ms)) => Open(Inlet Valve) & Closed (Exhaust Valve)) is a compound SD.

Process
A process is the evolution of states of objects over time. A process can be represented in L as a sentence formed as follows: • By the conjunction or disjunction of one or more states and at least one SD; • Containing at least two distinct instants of time in the union of the durations in all the SDs constituting the process.

Causal Process
A causal process is a process that includes descriptions of causal relationships among states or SDs (i.e., which combinations of states or SDs cause which other combinations of states or SDs).
A causal process is represented in L as a sentence containing one or more causal implication ( C =>) signs. Each instance of a causal implication ( C =>) sign in a sentence corresponds to a causal process. Each causal implication ( C =>) sign in a sentence should be immediately preceded by a sentence formed from states or SDs, using only the operations of conjunction and disjunction, called the input sentence of the causal process. Each causal implication ( C =>) sign in a sentence should be immediately succeeded by a sentence formed from states or SDs, using only the operations of conjunction and disjunction, called the output sentence of the causal process. In a causal process containing multiple causal implication ( c ) signs, causal implication ( c ) sign in the process denotes a causal sub-process in the given causal process.
The first example of a process above contains a single c  sign and the second contains two c  signs.
Each state or SD in the input sentence of the causal process corresponding to a given causal implication ( c ) sign is called an input state or SD of the causal process. Similarly, each state or SD in the output sentence of the causal process corresponding to a given causal implication ( c ) sign is called an input state or SD of the causal process. In a causal process (P) if the output sentence of a causal sub-process (P1) is also an input sentence or logically implies or contained in the input sentence of another causal sub-process (P2), then the output sentence of P1 is called an intermediate sentence of the given causal process P. The states or SDs of an intermediate sentence of causal process are called intermediate states or SDs. All the input and output sentences of the causal sub-processes of a given causal process that are not intermediate sentences of the given causal process are also input and output sentences, respectively, of the given causal process.
A causal process represented in L has to satisfy the following conditions:  Not flowing backward in time: If the causal process consists of only one causal implication ( c ) sign, then no output SD of the causal process should have the start time of its duration earlier than the start time of the duration of any input SD of the causal process. If the causal process consists of multiple causal implication ( c ) signs, then for any causal sub-process corresponding to a causal implication ( c ) sign in the given causal process, no output SD of the causal sub-process should have the start time of its duration earlier than the start time of the duration of any input SD of the causal sub-process. This condition subsumes within it the condition of non-circularity of causal processes, i.e., the output SD of a causal process cannot have an effect on the input SD of the causal process.
 Non-circularity: The output sentence of a causal process cannot have an effect on the input sentence of the causal process. This implies the condition of non-reflexivity, i.e., a state or SD cannot cause itself.  Non-redundancy: Neither the input nor the output sentence of a causal process should not be a tautology.
The input sentence of a causal process should not logically imply its output sentence.
More rules pertaining to the usage of the causal implication ( c ) sign are given in Section 2.1.

Recursive Process
A recursive process is represented in L by specifying the following: a sentence that is a starting criterion for the process; a sentence that is a stopping criterion, if required, for the process; a causal process that executes when its starting condition is satisfied and so long as its stopping condition is not true; and with each iteration of the causal process causing the starting condition for the next iteration of the process to be true. For example, the recursive process of keeping the inlet valve open in an IC engine for the first 10 ms of each cycle of duration 40ms while the ignition is on can be represented as follows: t(t = 0 ms <=> T(t))……(initialization of time as a starting criterion; T is a predicate used to enable the assignment of a specific instant of time to the individual variable t)………………………………..………...

Deterministic and Non-deterministic Processes
Consider a process (causal or non-causal) executed n times. The process is deterministic if and only if, in each of the n executions of the process, the truth value of each state or SD constituting the process remains the same. A process that is not deterministic is non-deterministic (or stochastic).

Probability of success of a causal process
The probability of success of a causal process is defined as the conditional probability of the output sentence of the causal process being true given the truth of the input sentence of the causal process.
Let C be a causal process whose input sentence is A and output sentence is B. Then the probability of success of a causal process C (P(C)) ≡ P(B/A).

Requirement
A sentence that has to be made true by the structure and behavior of a design. A requirement is represented in L as a sentence that does not include a causal implication sign.

Functional Requirement (FR)
A requirement that is an abstraction of the behavior expected from a design. An FR is represented in L as a sentence (not including a causal implication sign) containing existentially quantified individual variables whose instantiation as individual constants (representing specific objects) can be used to obtain a representation of behavior that satisfies the FR by logical implication. For example, t1t2vt , the IC engine generates powe) is a representation of an FR, such that assigning specific values to the variables t1, t2 and v can be used to obtain a representation of behavior such as t((t ≥ 30 ms & t ≤ 40 ms) =>(Power(IC Engine) = 75 hp)) (i.e. the IC engine generates power of 75 hp during 30 ms to 40 ms of its cycle of operation) that satisfies the FR by logical implication.

Constraint
A sentence that should not be negated by the structure and behavior of a design. constraint is represented in L as a sentence that does not include a causal implication sign.

Structure of a design
The attributes of and relationships among the objects that constitute that constitute the design. A structure is represented in L as a set of sentences that do not include SDs, and do not include the causal implication sign.

Behavior of a design
A process (causal or non-causal) involving the objects that constitute the design. Behavior is represented in L as a set of sentences that could (and typically would) include the causal implication sign.

Solution Concept
The structure and behavior of a design that are intended to satisfy the requirements and constraints of the design.

Design
A system of requirements (including FRs), constraints, structure and behavior.

Sub-Design
A sub-system (d) of a design (D) such that: d is also a design; and the FRs of d are contained in the hierarchy of FRs of D.

Sibling Sub-Designs
Sub-designs (d1 and d2) of a design D are sibling sub-designs of design D if and only if the highest level FRs of d1 and d2, are siblings in the hierarchy of FRs of D.

Solution-neutrality of an FR
An FR in the hierarchy of FRs of a design is solution-neutral if and only if it is not defined in terms of any object that is not already contained in a solution concept selected for a higher level FR of which the given FR is a descendant. For example, let S1 (valve train linking crank shaft motion to inlet valve opening and closing) be a solution concept for (the highest level FR in the hierarchy of FRs) FR1 (the inlet valve is opened) and S11 (valve train with overhead cam and follower) be a solution concept for FR11 (motion is transferred from crank shaft to inlet valve), which is a child of FR1. Since FR11 does not refer to objects (e.g. overhead cam) in S11 that are not already contained in the higher level solution concept (S1), FR11 is stated in a solution-neutral manner. However, if FR11 were stated as "inlet valve is pushed by overhead cam" then FR11 would not be solution-neutral, since the object overhead cam is contained in the solution concept (S11) for FR11 , but not contained in the solution concept (S1) of FR1. Similarly, if FR11...1 is the child of a child of (and so on) of FR1 then, for FR11...1 to be solution-neutral, FR11...1 can refer to any object (e.g. crank shaft) explicitly contained in the representation of the solution concepts of the parent of, the parent of parent of (and so on) of FR11...1, but cannot refer to any object (e.g. overhead cam) that is contained in the solution concept for FR11...1 but not in the solution concepts of the parent of, the parent of parent of (and so on) of FR11...1.

A fragment B of a design depends on a fragment A of a design if and only if:
• A is contained in B; or • A causes B; or • A implies B, where A is a sentence that describes structure or behavior and B is an FR or constraint. A special case of this condition is when A is a sentence that implies B by virtue of B being an an existentially quantified form of A (e.g. behavior A logically implies FR B, since the instantiation of one or more existentially quantified variables in B with individual constants yields A).

Dependency (coupling) among designs
A design D2 depends on design D1 if and only if there is a term in D2 that depends on a term in D1.

Dependency (coupling) matrix of a design
Consider the square matrix shown in Supplementary Figure 4 A matrix, as shown in Supplementary Figure 1, satisfying the description given above is called the dependency matrix of design D.

Coupled Design
A design is coupled if and only if there is a dependency among any two sibling sub-designs of the design. That is, a design is coupled if and only if there is at least one non-zero element in its dependency matrix.

Uncoupled Design
A design that is not a coupled design is an uncoupled design. That is, a design is uncoupled if and only if all the elements in its dependency matrix are zero.

Decoupled design
A design is decoupled if and only if all the elements in the lower triangular portion of its dependency matrix are zero and there exists at least one non-zero element in the upper triangular portion of the dependency matrix.

Probability of Success of a Design
The probability that all the requirements and constraints of a design are satisfied.

Information Content of a Design
The logarithm to the base 2 of the reciprocal of the probability of success of a design (Suh(1990)).
That is, information content of design D ≡ log2(P(D)); where P(D) is the probability of success of design D.

Valid Design
A design whose representation in L does not contain any inconsistencies within the logic of L. A design that is not valid will contain contradications in its representation (e.g. in structure, behavior, FRs, constraints, their relationships, etc.), which will make the design impossible to realize and/or render unreliable assessments of the satisfaction of FRs and constraints by the structure and behavior of the design.

Sound Design
A design whose structure and behavior satisfy its requirements and constraints.

Dimensionality of a design
The dimensionality of a design is the minimum number of independent attributes (parameters) required to completely specify the design. The dimensionality of a design is specific to the level of abstraction of a design.
For example, the dimensionality of a design at the conceptual design stage is likely to be lower than its dimensionality at the detailed design stage.

Degree of parametrization of a design
The degree of parametrization of a design is defined as the ratio of the total number of attributes used to completely specify a design to the dimensionality of the design. That is: Degree of parametrization of design D (g(D)) ≡ j/k; where: • j is the number of parameters used to completely specify the design D • k is the dimensionality of design D For example, let attributes required to completely specify the geometry of an inlet valve be as follows: base diameter, base thickness, stem diameter and stem length. Thus the total number of attributes required to completely specify the geometry of the inlet valve (j) = 4. However, let the practice of designing of inlet valves follow the following rules of thumb: • Base thickness = 1/10th of base diameter • Stem diameter = 1/4th of base diameter Then the minimum number of independent attributes required to completely specify the geometry of the inlet valve are base diameter and stem length. Thus the dimensionality of the geometry of the inlet valve (k) = 2.
Thus the degree of parametrization of the inlet valve = j/k = 4/2 = 2.

The entropy of dimensionality of a design
The entropy of dimensionality of a design D (E Dim (D)) is defined as the natural log of the dimensionality of D.
That is: where K(D) is the dimensionality of design D.

Degree of solution-neutrality of an FR
The degree of solution-neutrality of an FR in the hierarchy of FRs of a design is defined as the total number of potential solutions for the FR (aggregated over the potential solutions for all its descendant FRs, if any, in the hierarchy of FRs). For an FR that is at the lowest level in the hierarchy of FRs of a design, the degree of solution-neutrality is simply the number of solution concepts enumerated for that FR. For an FR that is not at the lowest level in the hierarchy of FRs, the degree of solution-neutrality is recursively enumerated over its descendant FRs in the hierarchy of FRs, as follows: For example, let FR1 be "Allow air-fuel mix into the IC engine". Solution concepts at the level of abstraction of FR1 could be enumerated as follows: S1 ≡ Poppet valve mechanism; S2 ≡ Butterfly valve mechanism; S3 ≡ Unidirectional valve mechanism. Consider S1 (Poppet valve mechanism). This solution concept could entail the following next level FRs: FR11 ≡ Open poppet valve; FR12 ≡ Close poppet valve. The solution concepts for FR11 at the level of abstraction of FR11 could be enumerated as follows: S11 ≡ Overhead cam mechanism; S2 ≡ Cam, pushrod and tappet mechanism. Similarly, the solution concepts for FR12 at the level of abstraction of FR12 could be enumerated as follows: S21 ≡ Cam slot mechanism; S22 ≡ Spring. Similarly let 2 second level FRs be entailed for each of the solution concepts S2 and S3 for FR1, and let each of the second level FRs entailed by S2 and S3, respectively, have 2 solution concepts enumerated at the corresponding level of abstraction of FRs. Thus the degree of solution-neutrality of FR11 = 2 (number of elements in the set {S11, S12}, since FR11 is at lowest level of FRs considered in this example), and similarly, that of FR12 = 2. Therefore the solution-neutrality of FR1 considering S1 as the selected solution concept =2*2 =4. Similarly, the degree of solution-nuetrality of FR1 considering S1 and S2, respectively as the solution-concept selected for FR1, is 4 each. Therefore the solutionneutrality of FR1 (obtained as the sum of its solution-neutrality considering each of S1, S2 and S3 individually as the selected solution concept) = 4+4+4 = 12.

Degree of solution-neutrality of the totality of FRs of a design
The degree of solution-neutrality of the totality of FRs of a design is the product of the degrees of solutionneutrality of the highest level FRs of the design. Thus:

Alternative definition of the degree of solution-neutrality of the FRs of a design
An alternative definition of the degree of solution-neutrality of the totality of FRs of a design is given below: ; where:  ND is the degree of solution-neutrality of the totality of FRs of D; This alternative definition of the degree of solution-neutrality of a design considers solution-neutrality of an FR as a binary variable (using the term g(i, j) above), emphasizes the relative importance of solution-neutrality at the higher levels in the hierarchy of FRs (using the term fi above) and does not require an explicit enumeration of all possible solution concepts over all possible hierarchies of FRs of a design.

Entropy of solution-neutrality of the FRs of a design
The entropy of solution-neutrality associated with an FRi is defined as the natural log of the reciprocal of the

The degree of flexibility of specifications of a design (M)
The degree of flexibility of specifications of a design is the number of combinations of discrete values, of the independent attributes required to completely specify the design, which are consistent with (i.e. do not violate) the specifications of the design.
The number of discrete values that an attribute (X) can take is defined as follows:  Case 1: X is a discrete variable, i.e., X can take only discrete values. Then the number of discrete values that X can take is the number of values of X that lie within the range of X that is specified in the requirements or constraints of D. For example, if X is the state of an on-off switch, then X can take only two discrete values (on and off).
 Case 2: X is a continuous variable. Let the specified range of X be [x1, x2]. The range of discrete values of x is defined as follows: (x1, x1 + r, x1 + 2r, x1 + 3r,….x1 + nr, x2), where r is the precision of the measurement system specified to measure x, and n is the maximum whole number such that x1 + nr ≤ x2.
For example, if X is a continuous variable with specified range of [1, 3.5] units, and the precision of the measuring instrument specified to measure X is 0.5 units, then the range of discrete values of X is given by [1, 1.5, 2.0, 2.5, 3.0, 3.5].
Thus if a design D is completely specified by 2 attributes X1 (with m discrete values in its specified range) and X2 (with n discrete values in its specified range), then the degree of flexibility of specifications of D is given by m*n.

Entropy of specifications of a design (E S (D))
The entropy of specifications (E S (D)) of a design D is defined as follows: E S (D) = ln(1/M), where M is the degree of flexibility of specifications of the design D.

Entropy of variability of a design (E V (D))
The entropy of variability of a design D, denoted by E V (D), is defined as follows: The approach to defining the entropy of variability of a design described above is partly analogous to that defined in Krus (2015) to estimate design information entropy for a the range of a continuous variable divided into equal regions, each with an identical probability of occurrence.

Entropy of bias of a design (E B (D))
The entropy of bias of a design D, denoted by E B (D), is defined as follows:

Degree of coupling of a design
The degree of coupling of a design is defined as follows: gi ≡ 1/10 (i-1) ; J1 ≡ A number greater than 1 and sufficiently large (e.g. 10 (L-1) ) to reflect the importance accorded to a circular coupling among sibling FRs of the design; J2 ≡ A number greater than 1 and sufficiently smaller than J1 (e.g. J1/1000) to reflect the much lower importance accorded to a non-circular coupling compared to a circular coupling in a design; ci ≡ Number of circular couplings among sibling sub-designs in level i of FRs; ui ≡ Number of non-circular couplings among sibling sub-designs in level i of FRs.
The above definition of degree of change manageability of a design gives a higher importance (weight, using the term gi above) to couplings among higher level FRs than to couplings among lower level FRs.

Entropy of change manageability of a design (E CM (D))
The entropy of change manageability of a design D, denoted by E CM (D), is defined as follows: Where H is the degree of coupling of the design D.

Entropy of requirements and constraints (including specifications) of a design (E R (D))
The entropy of the requirements and constraints (including specifications) of a design D, denoted by E R (D), is defined as follows:  L is a sufficiently large constant real number to ensure that E R (D) is positive for the design D;  k is the dimensionality of D;  O is the degree of solution-neutrality of the totality of FRs of D;  M is the degree of flexibility of specifications of D.

Entropy of Satisfaction of Requirements and Constraints of a Design(E RS (D))
The entropy of satisfaction of requirements and constraints of a design is defined as the sum of the following types of entropies of the design: entropy of requirements and constraints; entropy of variability; and entropy of bias. That is:

Entropy of conformance of a design to its specifications
The entropy of conformance to specifications of a design D is defined as:

Entropy of a Design
The entropy of a design (E(D)) is defined as the sum of the following: entropy of satisfaction of requirements and constraints; and entropy of change manageability. That is:

Alternative definition of entropy of a design
An alternative definition of the entropy of a design (E(D)) is as the sum of the following: entropy of solution neutrality, entropy of dimensionality, entropy of conformance and entropy of change manageability. That is: Situations in which this definition of entropy might be preferred include the situations where the use of entropy of conformance to specifications (as opposed to entropy of satisfaction of requirements and constraints) might be preferred (as described above in definition of entropy of conformance to specifications).

Quality of a design from a design theoretic perspective (Q(D))
The quality of a design (D) from a design theoretic perspective, denoted by Q(D), is an index of the extent to which a design exhibits the attributes of a good design as discussed in Section 4.2. In this work Q(D) is a real number in the range [0, ∞]. Q(D) varies with the entropy of the design (E(D)) as described in Section 4.2. For the sake of simplicity, in this work the scope of consideration of standardization of parts or components is restricted to the product and does not span across products (i.e., the definition of standardization in this work does not include reuse of a part or component across products, or industry-wide standards (e.g. of nuts or bolts).

Standardization of parts or components of a design
However, the scope of consideration of standardization could be easily broadened as follows: S(D) ≡ N(D)*R(D)/U(D); where S(D), N(D) and U(D) are as defined above, and R(D) ≡ the number of unique parts or components that are specified as per industry standards or are re-used across the products of the organization

Symmetry of parts or components of a design
The degree of symmetry of parts or components of design D is defined as:  Z is the set of unique parts or components of design D  U(D) is the total number of unique parts or components of design D.
 ki = the total number of planes of bilateral symmetry or axes of radial symmetry that are used in specifying the part or component i.         Piston is moved to its lowest point in the cylinder(bottom dead center(BDC)).

Expanded(Combustion Gases)) FR43
Piston is moved to BDC t431t((t = t431) => Position(Piston) = BDC)) FR5 Combustion gases   1.31 liters S14 Four bar mechanism described by the degrees of freedom and types of joints among the parts of the IC engine as described in sentences S8 to S13. S14  S8 &S9 &…S19 S15 Structure (mechanism involving parts such as cam shaft, cam, etc.) converting the motion of the crank shaft into the motion of the inlet and exhaust valves Not represented here in terms of its component parts for the sake of simplicity, but used in Table 6 as an input to the causal process transforming the movement of the crank shaft to the movements of the inlet and exhaust valves. S16 Structure (mechanism involving parts such as crankshaft position sensor, high tension coil, spark plug, etc.) generating a spark as the piston reaches TDC in the power stroke.
Not represented here in terms of its component parts for the sake of simplicity, but used in Table 6 as an input to the causal process transforming the movement of the crank shaft to the movements of the inlet and exhaust valves.

Alias Description Representation in L Input conditions (sentences) I10
Crank shaft is at 180 degrees (pointing towards the piston) t((t = 0 ms) =>

I12
Exhaust valve is closed until the start of the exhaust process t((t ≥ 0 ms & t ≤ 30 ms) => Closed(Exhaust Valve)) I13 Piston is at its highest point in the cylinder(Top Dead Center  Piston is moved to BDC SD41 c => SD42 P43 The crank shaft is rotated to 270 degrees S14 & SD42 c => SD43 P50 The crank shaft moves from 270 degrees to 720 degrees under its own momentum SD43 c => SD50

P51
The exhaust valve remains open during the exhaust process S43 & S15 c => SD51 P52 The piston moves from BDC to TDC S14 & SD50 c => SD52 P53 Combustion gases are exhausted from cylinder SD51 & SD52 c => SD53 P54 The exhaust valve closes at the end of the exhaust process S15 & SD50 c => SD54 Intermediate SDs SD10 The crank shaft is oriented at 180° at the end of the suction stroke. t((t = 10 ms) =>Orientation The inlet valve is open during the suctions stroke. t((t > 0 ms & t ≤ 30 ms) =>Open(Inlet Valve)) SD12 The piston at BDC at the end of the suction stroke. t((t = 10 ms) =>Position(Piston) = BDC) SD13 The air-fuel mixture is in the cylinder during the suction stroke. t((t = 10 ms) =>InCylinder(Air-Fuel Mix)) SD20 The orientation of the crank shaft is at 360° at the end of the compression stroke. t((t = 10 ms) => Orientation (Crank Shaft) = 360°) SD21 The inlet valve is closed immediately after the suction stroke and remains so for the rest of the strokes. t((t > 10 ms & t ≤ 40 ms) =>Closed(Inlet Valve)) SD22 The piston at TDC at the end of the compression stroke. t((t = 20 ms) => Position(Piston) = TDC) SD23 The air-fuel mix is compressed during the compression stroke. t((t = 20 ms) =>Compressed(Air-Fuel Mix)) SD30 The spark is generated at the end of the compression stroke. t((t = 20 ms) =>Generated(Spark)) SD31 The air-fuel mixture is ignited at the beginning of the power stroke. Orientation of the crank shaft is 540° at the end of the power stroke. t((t = 30 ms) =>Orientation (Crank Shaft) = 540°) SD50 Orientation of the crank shaft is 720° at the end of the exhaust stroke. t((t = 40 ms) =>(Orientation (Crank Shaft)) SD51 Exhaust valve is open during the exhaust stroke. t((t > 30 ms & t ≤ 40 ms) => Open(Exhaust Valve)) SD52 The piston is at TDC at the end of the exhaust stroke. t((t = 40 ms) => Position(Piston) = TDC) Output SDs SD53 Combustion gases are exhausted during the exhaust stroke. t((t = 40 ms) => Exhausted (Combustion Gases)) SD54 The exhaust valve is closed immediately after the exhaust stroke. t((t > 40 ms =>

Closed(Exhaust Valve))
Footnotes: • L provides flexibility to specify attributes or parameters to be equal to some value (e.g. density of air at inlet to engine = 1.125 kg/m 3 ) or to be within a range of values (E.g. power of IC engine = 75 hp +/-5 hp), as appropriate to the context (e.g. conceptual stage of design or later stages of design involving more detail, whether the attribute is a categorical (e.g. material type) or numerical attribute (e.g. power), etc.).
• Examples of circumscription that could be applied to the above definition of behavior are as follows: • S15 & I11 & SD10 <=> SD11 (corresponding to the causal process P11 defined above as: S15 & I11 & SD10 c => SD11) to limit the immediate cause of SD11 to only that explicitly stated in the theory, i.e. S15 & I11 & SD10. (this is analogous to the literal completion approach of McCain and Turner (1997)).