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 Function- Behavior-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.. Key words: Formalization of theories of design, first order logic, causal calculus, Function-Behavior-Structure, entropy of a design. 2 1. Introduction The past few decades have seen the concurrent and integrated development of the science of design as well as overarching frameworks of practice of design (e.g. Simon(1996), Papalambros (2015), Pahl and Beitz (1988), Hubka and Eder (1988), Pugh (1991), Suh (1990), Ullrich and Eppinger (1995), Otto and Wood (2001), 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 3 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 4 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. (1990, 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, 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 5 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.. Functions in Gero’s (1990) ontology correspond to functions in the ontologies of Pahl and Beitz (1988), Otto and Wood (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 and Wood (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. 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- 6 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. 7 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 8 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 9 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 10 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 (2003, 2004) provided a causal calculus that integrated causal theories (involving propositions and not actions) with propositional logic. The causal calculus of Bochman (2003, 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 11 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 12 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 (2003, 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 (2003, 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 13 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 (2013, 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 (2013, 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 14 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 non- monotonic setting. 15 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. 2. 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 16 logic. For example, in this work, an SD abbreviated as (S, [t 1, t2]) can represented in classical first order logic as follows: •t ((t ≥ t1 & t ≤ t 2 & t1 ≤ t 2) => A(a, b, .., t)), where A is a predicate with the individuals a, b,..,t as arguments. •t((t ≥ t1) & (t ≤ t 2)) => xy... A(x, y,…, t)), where A is a predicate with the individual variables x, y,..,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. 3. 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 17 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((t ≥ t1 & t ≤ t 2 & t1 ≤ t2) =>(Power(IC Engine) = v & v > 0)) (i.e., 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, t 1 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 18 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 . 3.1 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. FR X1, FRX2, FRX2, …). However, for an FR (FR X) decomposed as above, exactly one of the child FRs (FR X1 or FRX2 or FRX2 or …) should logically imply the parent FR (FR X). 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 FR 1 (Air-fuel mix is sucked into the cylinder) in Table 1, exactly one child FR (FR 13) of FR1 logically implies FR 1. 4. 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. 4.1 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 19 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 t1t2vt((t ≥ t1 & t ≤ t 2 & t1 ≤ t2) =>(Power(IC Engine) = v & v > 0)). Similarly, the behavior of generating 75 hp of power in the power stroke of the IC engine (represented in L as t((t ≥ 20 ms & t ≤ 30 ms) =>(Power(IC Engine) = 75 hp)) logically implies the satisfaction of the PR that the 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)) and also logically implies the satisfaction of the FR to generate power represented in L as t1t2vt((t ≥ t1 & t ≤ t 2 & t1 ≤ t2) 20 =>(Power(IC Engine) = v & v > 0)). 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. 4.2 Formalization of theories of what constitutes a good design 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)). 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: solution- neutrality of FRs of a design; coupling in designs; coupled, uncoupled and decoupled designs; probability of 21 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. 22 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. 23 The formal, axiomatic theory of design is given below. Definitions of terms d ≡ A given design. d’ ≡ A design obtained by modifying a given design d (specific types, if any, of modifications by which d’ is obtained from d are specified within the context of the corresponding theorems in the formal theory below). Q(d) ≡ Quality of the design d from a design-theoretic perspective (see Appendix 1 for the definition of quality of a design from a design-theoretic perspective). E(d) ≡ Entropy of design d (see Appendix 1 for the definition of entropy of a design). O(d)≡ Degree of solution-neutrality of design d (see Appendix 1 for the definition of solution-neutrality of a design). K(d) ≡ Dimensionality of design d (see Appendix 1 for the definition of dimensionality of a design). M(d) ≡ Number of combinations of values of attributes of design d that lie within the specifications of design d (see Appendix 1 for how this quantity is calculated). V(d) ≡ Number of combination of values of attributes of design d that lie within the natural bounds of variation of the attributes (see Appendix 1 for how this quantity is calculated). B(d) ≡ Number of combination of values of attributes of design d that lie between the specified target values and central tendencies (e.g. expected values) of these attributes (see Appendix 1 for how this quantity is calculated). S(d) ≡ Degree of standardization of design d (see Appendix 1 for the definition of degree of standardization of a design). Sym(d) ≡ Degree of symmetry of design d (see Appendix 1 for the definition of degree of symmetry of a design). T(d) ≡ Degree of parametrization of design d (see Appendix 1 for the definition of degree of parametrization of a design). U(d) ≡ Number of non-circular dependencies among sibling sub-designs of d (see Appendix 1 for the definition of non-circular dependency among sibling sub-designs of a design). 24 C(d) ≡ Number of circular dependencies among sibling sub-designs of d (see Appendix 1 for the definition of circular dependency among sibling sub-designs of a design). E(d, i) ≡ Entropy of design d at level i in the hierarchy of FRs of d. Q(d, i) ≡ Quality of design d, from a design-theoretic perspective, at level i in the hierarchy of FRs of d. C(d, i) ≡ Number of circular dependencies among sibling sub-designs of design d, at level i in the hierarchy of FRs of d. U(d, i) ≡ Number of non-circular dependencies among sibling sub-designs of design d, at level i in the hierarchy of FRs of d. ΔX(d’, d) ≡ X(d’) - X(d), where X(d) is one of the following: E(d); Q(d); K(d); O(d); V(d); B(d); S(d); or Sym(d). ΔY((d’, j), (d, i)) ≡ Y(d’, j) – Y(d, i), where Y(d, i) is one of the following: E(d, i); Q(d, i); C(d, i); or U(d, I) (and similarly for Y(d’, j)). 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: (ΔE(d’, d) < 0 => ΔQ(d’, d) > 0) & (ΔE(d’, d) > 0 => ΔQ(d’, d) < 0 & ((ΔE(d’, d) = 0) => (ΔQ(d’, d) = 0)). 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): 25 Theorem 1: The more solution-neutral the FRs of a design, the better the design. That is: ΔO(d’, d) > 0 => ΔE(d’, d) < 0 => ΔQ(d’, d) > 0 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. Consider a design d wherein two or more FRs are not solution neutral. Consider design d’ obtained from design d only by making a non-solution-neutral FR in design d solution-neutral. Similarly, consider design d’’ obtained from design d only by making another non-solution-neutral FR in design d solution-neutral. Let x be the level of the non-solution-neutral FR, in the hierarchy of FRs of d, that was made solution-neutral in design d’. Similarly, let y be the level of the non-solution-neutral FR, in the hierarchy of FRs of d, that was made solution-neutral in design d’’. Then: (x < y) => ΔO(d’, d”) > 0 => ΔE(d’, d’’) < 0 => ΔQ(d’, d”) > 0 (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: ΔK(d’, d) < 0 => ΔE(d’, d) < 0 => ΔQ(d’, d) > 0 (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: ΔM(d’, d) > 0 => ΔE(d’, d) < 0 => ΔQ(d’, d) > 0 26 (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: ΔV(d’, d) < 0 => ΔE(d’, d) < 0 => ΔQ(d’, d) > 0 (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: ΔB(d’, d) < 0 => ΔE(d’, d) < 0 => ΔQ(d’, d) > 0 (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: ΔS(d’, d) > 0 => ΔK(d’, d) < 0 => ΔE(d’, d) < 0 => ΔQ(d’, d) > 0 (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. 27 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: ΔSym(d’, d) > 0 => ΔK(d’, d) < 0 => ΔE(d’, d) < 0 => ΔQ(d’, d) > 0 (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 sub- designs of d. Then: (ΔU(d’, d) < 0 & ΔC(d’, d) ≤ 0) v (ΔU(d’, d) ≤ 0 & ΔC(d’, d) < 0) => ΔE(d’, d) < 0 => ΔQ(d’, d) > 0 (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 non- circular) among sibling sub-designs of design d. Then: (ΔU(d’, d) + ΔC(d’, d) = -(U(d) +C(d)) => ΔQ(d’, d) > 0 Proof: U(d) > 0 v C(d) > 0 (by definition of d)…………………………………………………..…..(1) (U(d) + C(d)) > 0 (by (1))……………………………………………………………….…….(2) (ΔU(d’, d) = -U(d)) <=> (U(d’) = 0) (by definition of d’ and d above)………………………..(3) 28 (ΔC(d’, d) = -C(d)) <=> (C(d’) = 0) (by definition of d’ and d above)………………………..(4) Therefore: ((ΔU(d’, d) = -U(d) & ΔC(d’, d) = -C(d)) => (U(d’) = 0) & (C(d’) = 0) (by (3) and (4))……...(5) (U(d’) = 0) & (C(d’) = 0) <=> Uncoupled(d’) (by definition of uncoupled design)………....(6) U(d) > 0 & C(d) > 0 (by definition of d)………………………………………………….…..(7) (ΔU(d’, d) + ΔC(d’, d) = -(U(d) +C(d)) => (ΔU(d’, d) + ΔC(d’, d)) < 0 (by 7)……………......(8) (ΔU(d’, d) + ΔC(d’, d)) < 0 => ΔE(d’, d) < 0 (by definition of entropy of a design)…………..(9) (ΔU(d’, d) + ΔC(d’, d) = -(U(d) +C(d)) => ΔQ(d’, d) > 0 (by (8), (9) and Axiom 1). Hence proved. 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: (ΔE(d’’, d) < ΔE(d’, d) < 0) & (ΔQ(d’’, d) > ΔQ(d’, d) >0) (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 29 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: (ΔE(d’, d) < ΔE(d’’, d) < 0) & (ΔQ(d’, d) > ΔQ(d’’, d) >0) (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 level z + m (m > 0) in its hierarchy of FRs. Consider a design d’ obtained from design d only by eliminating a non-circular dependency at level z of FRs of d and a design d’’ obtained from design d only by eliminating a non-circular dependency at level z+m of the FRs of d. Then: (ΔE(d’, d) < ΔE(d’’, d) < 0) & (ΔQ(d’, d) > ΔQ(d’’, d) >0) (by definition of entropy of a design and Axiom 1). 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 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 non- circular dependency at level z+m of the FRs of d. Then: (ΔE(d’, d) < ΔE(d’’, d) < 0) & (ΔQ(d’, d) > ΔQ(d’’, d) >0) 30 (by theorems 12 and 13). 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: (ΔE(d’c, d) - ΔE(d’u, d) < 0) => (ΔE(d’, d) < 0) => (ΔQ(d’, d) > 0) (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. Consider a design d’ obtained from design d only by changing the extent of parametrization of d. Then: ΔT(d’, d) > 0 => ΔQ(d’, d) > 0 Proof: ΔT(d’, d) > 0 => ΔK(d’, d) < 0 (since parametrization of parts or components reduces the number of constrained independent attributes of the design)………..(10) Therefore: ΔT(d’, d) > 0 => ΔE(d’, d) < 0 (by (10) and Theorem 3)…..(11) Therefore: ΔT(d’, d) > 0 => ΔQ(d’, d) > 0 (by (11) and Axiom 1). 31 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 32 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. 5. 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) and 1 (b) illustrate t he calculation of the truth values of requirements and constraints of the illustrative IC engine given its structure and the truth values of the input conditions of its behavior. Supplementary Figure 2 illustrates the calculation of the probability of satisfaction of the requirements and constraints of the illustrative IC engine, given the probabilities of its pre-conditions, using a Monte Carlo simulation. The Monte Carlo simulation generates multiple combinations of truth values for the pre-conditions, 33 assuming that each pre-condition follows a binomial distribution with probability of occurrence as specified in Supplementary Figure 2. For each combination of truth values of pre-conditions, the truth values of the intermediate and output SDs are calculated based on the causal relationships specified in Table 3. The probabilities of occurrence of the intermediate and output SDs are then estimated from the truth table simulated as above. --------------Figure 1 (a) goes here------------------ --------------Figure 1 (b) goes here------------------ --------------Table 1 goes here------------------------ --------------Table 2 goes here------------------------ --------------Table 3 goes here------------------------ --------------Table 4 goes here------------------------ Figure 2 shows the coupling (dependency matrix) for the illustrative IC engine (see Appendix 1.2 for a definition of the coupling (dependency) matrix of a design). Figure 3 (a) shows the entropy and components of the entropy of the illustrative IC engine. The relative weight of each component of entropy in Figure 3 (a) is obtained from Figure 3 (b), which illustrates the assignment of relative importance (weights) to various components of entropy of the illustrative IC engine. Since all the FRs in the illustrative IC engine are represented in a solution-neutral manner (see Appendix 2.1 for a definition of solution-neutrality of an FR), its entropy of solution-neutrality is zero. Supplementary Figure 3 (a) shows the calculation of the entropies of dimensionality and conformance of the illustrative IC engine. Supplementary Figure 3 (b) shows the calculation of the entropy of the change manageability of the illustrative IC engine. --------------Figure 2 goes here------------------ --------------Figure 3 (a) goes here-------------- --------------Figure 3 (b) goes here-------------- 34 From the above, it can be seen that the design of the illustrative IC engine, as represented in L, has the following general attributes: a) the FRs of the illustrative IC engine have been stated in a solution-neutral manner (see Tables 1 and 2); b) there exists a proof of satisfaction, in the logic of L, of the requirements and constraints of the design by its structure and behavior (see Table 4); c) the design has a non-zero probability of satisfaction of requirements and constraints by its structure and behavior (see Supplementary Figure 2); and d) the design is decoupled (see Figure 2) (see Appendix 1.2 for a definition of a decoupled design). 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 35 selection of potential improvements in a design has to be guided by the overall impact of the improvements on the entropy of the design. Figures 4 (a) and 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. 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 36 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. 6. 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 37 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 non- circular) 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 38 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. 39 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. 7. Competing Interests: The author declares none. 40 8.