Cognitive Architecture and Second-Order

Cognitive architecture and second-order systematicity: categorical compositionality and a (co)recursion model of systematic learning 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 each other. An example that is pertinent to the classical explanation is the learning (or memorization) of arbitrary asso- Systematicity commonly means that having certain cognitive capacities entails having certain other cognitive capacities. ciations. For instance, if one has the capacity to learn (mem- Learning is a cognitive capacity central to cognitive science, orize) that the first day of the Japanese financial year is April but systematic learning of cognitive capacities—second-order 1st, then one also has the capacity to learn (memorize) that systematicity—has received little investigation. We proposed associative learning as an instance of second-order systematic- the atomic number of carbon is 6. This example is a legiti- ity that poses a paradox for classical theory, because this form mate instance of systematicity (at the second level) given that of systematicity involves the kinds of associative constructions systematicity has been characterized as equivalence classes that were explicitly rejected by the classical explanation. In fact, both first and second-order forms of systematicity can of cognitive capacities (McLaughlin, 2009). be derived from the formal, category-theoretic concept of uni- Elsewhere (Phillips & Wilson, submitted), we argue that versal morphisms to address this problem. In this paper, we associative learning is problematic for classical theory, be- derived a model of systematic associative learning based on (co)recursion, which is another kind of universal construction. cause it involves the kinds of associative constructions that This result increases the extent to which category theory pro- were explicitly rejected by the classical explanation. Our vides a foundation for cognitive architecture. category theoretic explanation of systematicity resolves this Keywords: cognitive architecture; systematicity; composi- problem, because both first and second-order forms of sys- tionality; learning; category theory; coalgebra tematicity are derived from the same categorical construction: universal morphisms. Here, we derive a model of systematic Introduction associative learning based on (co)recursion, which is another The problem of systematicity for theories of cognitive sci- kind of universal construction. ence is to explain why certain cognitive capacities typically co-exist (Fodor & Pylyshyn, 1988); why, for example, having Outline of paper the ability to identify square as the top object in a scene con- The remaining four sections of this paper are outlined as fol- sisting of a square above a triangle implies having the ability lows. The second section provides the basic category theory to identify triangle as the top object in a scene consisting of definitions and motivating examples needed for our model. a triangle above a square. More formally, an instance of sys- The third section provides concrete examples of anamor- tematicity occurs when one has cognitive capacity c1 if and phisms (corecursion) as a conceptual guide to the model given only if c2 (McLaughlin, 2009): thus, systematicity is the par- in the fourth section. Discussion of this work is in the last titioning of cognitive capacities into equivalence classes. The section. Readers already familiar with the category theory ap- problem is to provide an explanation for systematicity that proach to corecursion may prefer to skip straight to the fourth does not rely on ad hoc assumptions: i.e., auxiliary assump- section, since the category theory employed here is already tions that are motivated only to fit the data, cannot be verified well known, with the possible exception of the recently de- independently of verifying the theory, and are unconnected to veloped adjoint extensions (Hinze, 2013). Readers not famil- the theory’s core principles (Aizawa, 2003). iar with a category theory approach may prefer to skip to the Learning is another cognitive capacity. Hence, using the third section for simple concrete examples as a way of orient- characterization of systematicity as equivalence classes of ing themselves for the model that follows. The second sec- cognitive capacities (McLaughlin, 2009), we have another tion, then, provides points of reference for technical details form of systematicity: i.e., the capacity to learn cognitive ca- and motivating examples when needed. pacity c1 if and only if the capacity to learn cognitive capacity c2, which is sometimes referred to as second-order sys- Basic category theory tematicity (Aizawa, 2003). Aizawa (2003), citing Chomsky In this section, we provide the basic category theory needed (1980), provides an example from language: a person has the for our model. In the interests of brevity and clarity, we only capacity to learn one natural language (say, Chinese) if they provide definitions and examples directly pertaining to the have the capacity to learn another (say, German). Importantly, model, omitting the (albeit, well known) theorems and lem- the learned capacities need not be systematically related to mas that justify statements. Deeper and broader introductions 1877 to category theory and categorical treatments of (co)recursion Example 2 (Right product functor). The right product func- can be found in many textbooks on the topic (e.g., Mac Lane, tor PB : Set ! Set sends each set X to the Cartesian product 1998; Bird & de Moor, 1997). In the context of systematic- X ×B and each function f : X ! Y to the product of functions ity, this paper builds upon our earlier work (see Phillips & f × 1B : X × B ! Y × B;(x;b) 7! ( f (x);b), i.e., f × 1B maps Wilson, 2010), and particularly in the context of recursive ca- (x;b) to ( f (x);b). pacities (Phillips & Wilson, 2012), where the theoretical principles of first-order systematicity and other technical details Example 3 (Right exponential functor). The right exponen- L ! can be found. tial functor B : Set Set sends each set X to the function set XB, which is the set of functions f f : B ! Xg, and each Definition 1 (Category). A category C consists of: function g : X ! Y to the function L(g) : XB ! Y B; f 7! g◦ f . • a collection of objects (A, B, ...); Example 4 (List). List-related constructions built from a set • a collection of morphisms ( f , g, ...), written f : A ! B of elements A are obtained from an endofunctor on the cate- 7! × 7! × to indicate that A and B are respectively the domain and gory Set: i.e., FA : X 1 + A X; f 11 + 1A f , where 1 × codomain of f , including an identity morphism, denoted corresponds to the empty list, and + and are (respectively) the disjoint union and Cartesian product of sets or functions. 1A : A ! A, for each object A in C; and • a composition operation that sends a pair of compatible Definition 5 (Natural transformation). A natural transforma- morphisms f : A ! B and g : B ! C, i.e., where the tion h from a functor F : C ! D to a functor G : C ! D, writ- h !: fh ! codomain of f equals the domain of g, to their composite ten : F G, is a family of D-morphisms A : F(A) morphism, denoted g ◦ f : A ! C, G(A)jA is an object in Cg such that for each morphism f : A ! B in C we have G( f )◦hA = hB ◦F( f ), i.e., the following that satisfy the axioms of: associativity—h◦(g◦ f ) = (h◦g)◦ diagram is commutative (equational): f ; and identity— f ◦ 1A = f = 1B ◦ f for each f in C. h Example 1 Set . Set A ( ) The category has sets for objects, func- F(A) / G(A) (1) tions between sets for morphisms, and composition is composition of functions. F( f ) G( f ) Definition 2 (Terminal object). In a category C, a terminal / F(B) h G(B) object is an object, denoted 1, such that for every object Z B there exists a unique morphism u : Z ! 1. Remark. In Set, a singleton set is a terminal object, whose Definition 6 (Final morphism). A final morphism from a only element is denoted ∗ when its identity is not required. functor F : C ! D to an object X in D is a pair (A;f) con- Other categories may have terminal objects with further inter- sisting of an object A in C and an morphism f : F(A) ! X nal structure, as we shall see for categories of (co)algebras. in D such that for every object Z in Z and every morphism f : F(Z) ! X in C there exists a unique morphism u : Z ! A Definition 3 (Isomorphism). An isomorphism is a morphism such that f = f◦F(u), as indicated by the following commu- f : A ! B such that their exists a morphism g : B ! A sat- tative diagram (dashed arrows indicate unique existence): isfying f ◦ g = 1B and g ◦ f = 1A. Morphism g is called the inverse of f , and denoted f −1. Z F(Z) (2) Remark. In a category C, the collection of morphisms with BB B domain object A and codomain object B is called a hom-set, BBf u F(u) BB denoted Hom (A;B). As we shall see, hom-sets play an im- B C / portant role in category theory. A F(A) f X Definition 4 (Functor). A functor F from a category C to a ! category D is a structure-preserving map, written F : C D, Remark. The dual of final morphism is initial morphism, that maps each object A in C to the object F(A) in D and each whose definition is obtained by reversing the directions of the ! ! morphism f : A B in C to the arrow F( f ) : F(A) F(B) morphisms in the definition of final morphism.

Load more