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A Category Theory Explanation for the Systematicity of Recursive Cognitive

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A Category Theory Explanation for the Systematicity of Recursive Cognitive Categorial compositionality continued (further): A category theory explanation for the systematicity of recursive cognitive capacities Steven Phillips ([email protected]) Mathematical Neuroinformatics Group, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8568 JAPAN William H. Wilson ([email protected]) School of Computer Science and Engineering, The University of New South Wales, Sydney, New South Wales, 2052 AUSTRALIA Abstract together afford cognition) whereby cognitive capacity is or- ganized around groups of related abilities. A standard exam- Human cognitive capacity includes recursively definable con- cepts, which are prevalent in domains involving lists, numbers, ple since the original formulation of the problem (Fodor & and languages. Cognitive science currently lacks a satisfactory Pylyshyn, 1988) has been that you don’t find people with the explanation for the systematic nature of recursive cognitive ca- capacity to infer John as the lover from the statement John pacities. The category-theoretic constructs of initial F-algebra, catamorphism, and their duals, final coalgebra and anamor- loves Mary without having the capacity to infer Mary as the phism provide a formal, systematic treatment of recursion in lover from the related statement Mary loves John. An in- computer science. Here, we use this formalism to explain the stance of systematicity is when a cognizer has cognitive ca- systematicity of recursive cognitive capacities without ad hoc assumptions (i.e., why the capacity for some recursive cogni- pacity c1 if and only if the cognizer has cognitive capacity tive abilities implies the capacity for certain others, to the same c2 (see McLaughlin, 2009). So, e.g., systematicity is evident explanatory standard used in our account of systematicity for where one has the capacity to remove the jokers if and only if non-recursive capacities). The presence/absence of an initial algebra/final coalgebra implies the presence/absence of all sys- one has the capacity to remove the aces (assuming, of course, tematically related recursive capacities in that domain. This one has the capacity to identify jokers and aces). formulation also clarifies the theoretical relationship between recursive cognitive capacities. In particular, the link between The classical explanation for systematicity has two compo- number and language does not depend on recursion, as such, nents: (1) combinatorial syntactically structured representa- but on the underlying functor on which the group of recursive capacities is based. Thus, many species (and infants) can em- tions; and (2) processes that are sensitive to (i.e., compatible ploy recursive processes without having a full-blown capacity with) those syntactic structures. In a classical cognitive ar- for number and language.1 chitecture, mental representations of constituent entities (e.g., Keywords: systematicity; category theory; endofunctor, F- John, Mary) are tokened (instantiated) whenever the mental (co)algebra; (final) initial algebra; universal construction; cata- representations of their complex hosts (e.g., John loves Mary) morphism; anamorphism; classicism; connectionism are tokened, with the meaning of a complex host represen- tation obtained (recursively) from the meaning assigned to Introduction its constituent mental representations and their syntactic rela- Many cognitive domains include recursively definable con- tionships. By analogy to language, this form of mental repre- cepts (i.e., concepts defined with reference to themselves), sentation is called a language of thought (LoT) (Fodor, 1975). such as domains involving lists, or languages. In card games, for example, a deck of cards can be defined (recursively) as a The three aspects of systematicity, i.e., systematicity of rep- top card (perhaps turned face up to reveal its value) and a (re- resentation, systematicity of inference, and compositionality maining) deck of cards. To include finite decks, the definition of representation (Fodor & Pylyshyn, 1988), can often be de- has an alternative clause specifying an empty deck; that is, a rived from classical cognitive architectures, because the same deck is either empty, or contains a top card and a (smaller) component processes are often used for each and every mem- deck. Operations on recursively defined concepts may also ber of a group of systematically-related capacities. For in- be defined recursively. For example, removing jokers from stance, a classical system has the capacity to represent John a deck of cards can be defined (recursively) as removing the loves Mary if and only if the system has a capacity to repre- top card if it is a joker and then removing jokers from the sent Mary loves John when the common component process ! remaining deck of cards. Given that you don’t find people is something like a production rule: S Agent loves Patient who can remove the jokers from a hand of seven cards with- (where John and Mary are produced from Agent and Patient out being able to remove jokers from a deck of fifty-three, by other production rules)—systematicity of representation. recursion-related capacities are further instances (see below) Likewise, a classical system has the capacity to infer John of the systematic nature of human cognition. as the lover in John loves Mary if and only if it has the ca- Systematicity is a property of human cognitive architec- pacity to infer Mary as the lover in Mary loves John given ture (i.e., the basic processes and modes of composition that a common process that is sensitive to the syntactic struc- ture whereby the lover constituent is represented by the first 1This paper is a short version of Phillips and Wilson (2012). token—systematicity of inference. And, again, the capacity 869 to assign the semantic content of John being the lover of Mary any component processes, and so do not provide a basis to the representation John loves Mary if and only if there is for systematicity, even though systematicity could be sup- the capacity to assign the semantic content of Mary being the ported when both tasks are implemented in either the first lover of John to the representation Mary loves John derives style only, or the second style only. Notice that we are not from the tokening principle (above) mediating classical repre- unfairly stressing classical theory by apportioning capacity sentations and processes: the process for juxtaposing tokens at the level of constituents—systematicity concerns “molec- (symbols) John, loves, and Mary to form John loves Mary ular”, not “atomic” capacities (Fodor & Pylyshyn, 1988). with corresponding semantic content is the same process that Rather, given constituent capacities lower and higher, clas- is used to form Mary loves John with corresponding content. sical theory admits two independent compositional forms, as Classical compositionality would seem to provide an el- the example illustrates. egant explanation for systematicity with regard to recursive In this paper, we extend our category theory explanation capacities, even though it fails to provide a full account of to recursive capacities using universal constructions called an systematicity generally (Aizawa, 2003).2 For recursive defi- initial F-algebra and a final F-coalgebra, which have been nitions, like the deck of cards, one self-referencing rule typ- extensively developed in computer science as a theoretical ba- ically covers all cases (bar the terminating case, such as the sis for recursive computations (Manes & Arbib, 1986). Our empty deck). For example, removing jokers from a single previous work (Phillips & Wilson, 2010, 2011) dealt with hand, or an entire deck invokes the same component process. non-recursive domains using a kind of universal construc- The two tasks only differ in the number of recursive steps. tion called adjoint functors—a functor is a way relating cate- gories, which can be viewed as a way of constructing objects Classical, but not systematically recursive and morphisms from one category based on those in another. However, the classical explanation with regard to recursive The current work uses endofunctors, which relate categories capacities still suffers the same general problem that it suffers to themselves, hence their relevance to recursion. for non-recursive capacities. To illustrate, suppose one card game requires removing the lowest value card in the hand Category theory: F-(co)algebras dealt, while another card game requires removing the highest A category theory treatment of recursion starts with the con- value card. Suppose the following recursive procedure, low- cept of an F-algebra constructed on an endofunctor F.3 est, for identifying the lowest valued card in a deck of cards Definition (F-algebra). F C ! C F- (containing at least one card): For an endofunctor : , an algebra is a pair (A;a), where A is an object and a : F(A) ! A lowest(c : cs) = lower(c;lowest(cs)) is a morphism in C. ( ) = × lowest(c : []) = c For example, if F A A A, then the addition operator + : A × A 7! A is an F-algebra. where a deck of cards c : cs is represented by a recursively Definition (F-algebra homomorphism). An F-algebra ho- defined list with c as the top card and cs as the remaining momorphism h : (A;a) ! (B;b) is a morphism h : A ! B (in deck, [] is the empty deck, and lower returns the lower of two C) such that the following diagram commutes: cards. Suppose, also, the following classical non-recursive a / procedure, highest, for identifying the highest valued card: F(A) A (1) highest(cs) = (i;high) (0;undefined) F(h) h while i < n do F(B) / b B (i;high) (i + 1;higher(high;csi)) return high That is, h ◦ a = b ◦ F(h). ! where deck cs is represented by an array of n cards with po- Definition (F-algebra category). For endofunctor F : C a sition indexed by i (i.e., cs is the ith card), high maintains C, an F-algebra category Alg(F) has F-algebras (A; ) for i a ! b a representation of the (currently) highest card, higher re- objects, and F-algebra homomorphisms h : (A; ) (B; ) turns the higher of two cards (undefined is some value guar- for morphisms.
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