How to Solve the Problem of Compositionality by Oscillatory Networks

H ow to Solve the Problem of C om positionality by O scillatory N etw orks M arkus W erning (m arkus.w erning@ uni-erfurt.de) D epartm ent of Philosophy, P. O . Box 900221 D -99105 Erfurt, G erm any A bstract Cognitive system s are regarded to be com positional: The sem antic values of com plex representations are deter- m ined by, and dependent on, the sem antic values of prim itive representations. Both classical and connectionist netw orks fail to m odel com positionality in a plausible w ay. The paper introduces oscillatory net- w orks as a third alternative. It provides neurobiological evidence for the adequacy of those netw orks and argues that they are com positional. O scillatory netw orks com - bine the virtues and avoid the shortcom ings of classical and connectionist architectures. C om positionality and System aticity M inds have the capacity to com pose contents. O ther- w ise, they w ould not show a system atic correlation betw een representational capacities: If a m ind is capa- ble of certain intentional states in a certain intentional m ode (perception, thought, im agination, preference, etc.), it m ost probably is also capable of other inten- tional states w ith related contents in the sam e m ode. The capacity to see a red square in a green circle, e.g., is statistically highly correlated w ith the capacity to see a red circle in a green square. To explain this em pirical phenom enon, w hich is closely related to the w ell- know n binding problem , com positional operations are postulated. They enable the system to build com plex representations from prim itive ones so that the sem antic value of the com plex representation is determ ined by, and dependent on, the sem antic values of the prim itives. Several theories have been developed to m eet the re- quirem ent of com positionality. Both classical and connectionist attem pts suffer from severe deficits, though. Fodor and Pylyshyn (1988) for one take recourse to a language of thought, w hich they link to the claim that the brain can be m odeled by a Turing-style com puter. A subject’s having a m ental representation, they believe, consists in the subject’s bearing a com putational rela- tion to a m ental sentence; it is a relation analogous to the relation a Turing m achine’s control head bears to the tape. A ccordingly, the m ind com poses com plex representations from prim itive ones just the w ay a com - puter com poses phrases from w ords: by concatenation. The trouble w ith classical com puter m odels is w ell know n and reaches from the fram e problem , the prob- lem of graceful degradation, and the problem of learn- ing from exam ples (H organ & Tienson, 1996) to prob- lem s that arise from the content sensitivity of logical reasoning (G igerenzer & H ug, 1992). To avoid the pitfalls of classicism , connectionist m odels have been developed. Som e of them attem pt to m eet the com positionality constraint. Sm olensky (1995) m aps the term s and the syntax of a language hom om or- phically onto an algebra of vectors and tensor opera- tions. Each prim itive term of the language is assigned to a vector. Every vector renders a certain distribution of activity w ithin the connectionist netw ork. The syntactic operations of the language have tensor operations as counterparts. Barnden (1991) pursues a related ap- proach. A s far as syntax is concerned, som e connectionist netw orks can com pletely im plem ent com positional languages. The kind of com positionality that is necessary for system aticity, how ever, focuses not on syntactic, but on sem antic features. The capacity to think that a child w ith a red coat is distracted by an old herring is not correlated w ith the capacity to think that a child w ith an old coat is distracted by a red herring. The thoughts ought to be correlated, though, if syntactic com position w as sufficient for system aticity. A lthough both thoughts are syntactically com posed from exactly the sam e prim itives by exactly the sam e operations, they are not correlated because red herring is idiom atic, i.e. because the m apping (red, herring) → red herring is syntacti- cally, but not sem antically com positional. O ne m ay w ell have the capacity to think of red coats and old herrings even though one lacks the capacity to think of red herrings. W e m ay infer, thus, that sem antic com po- sitionality is necessary for system aticity and that syn- tactic com positionality is not sufficient. The strategy to m ap the syntax of a system atic language hom om orphi- cally onto a connectionist netw ork does not suffice to establish that the netw ork itself is system atic. To put the dilem m a in a nutshell, connectionist m od- els seem to be too w eak to explain system aticity, w hereas classical m odels are apparently too strong to be im plem ented by real brains. The rest of the paper w ill explore the option of som ething “in betw een” classical and connectionist architectures. The presented solution differs significantly from other approaches to the di- lem m a (Lange & D yer, 1989; Shastri & A jjanagadde, 1993; H um m el & H olyoak, 1997; Sougne 1999). Espe- cially w ith respect to the representation of relations, the