Computational Synthesis: Following the Treaded Mathematical Track
Zippora Arzi‐Gonczarowski · 2003
A mathematical approach to intelligent agent modeling is described, where one starts from low level sensoryreactive mechanisms and proceeds, by mathematical synthesis, to scale to high complexity and to model high level, domain independent, intelligent functionalities. The composition of building blocks features regularity, modularity, abstraction, and awareness of hierarchy. The General Approach The essence of mathematical modeling has always been to start from basic, low-level, building blocks which are intuitively convincing and obvious. Then, following a long series of simple steps, to obtain arbitrary high-level constructs. The typical paradigm is the system of natural numbers: The five postulates of Peano capture the pre-theoretical essence of the natural numbers as counters of discrete quantities. Orderly extensions of the natural numbers provide the integers, then the rational numbers, then the real numbers. One starts from fundamental concepts as primitive terms, and asserts certain simple propositions (postulates, axioms) about them. Further terms are then introduced in an orderly manner, using the primitive terms. Theorems express properties of these new terms, applying deductive reasoning to obtain them from the postulates. Another typical paradigm is Euclidean geometry, that models physical space using points and lines as building blocks. 20th century ‘pure’ mathematicians, considering such theories, perceived that they shared domain independent principles and methods (Lawvere & Schanuel 1997). Not the least ones of these abstracted principles and methods are about the identification and the composition of low-level building blocks to scale to high-level constructs, achieving arbitrary complexities. Hilbert coined the term The Genetic Method for the method which is suggestive in the contexts of AI and of biological complexity: Natural intelligent systems started evolving from the earliest nerve cell that was probably a combined receptor (receiving environmental stimuli) and motor unit (producing muscle or gland response). With biological systems as role models, intelligent systems could be Copyright c 2003, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. modeled mathematically by starting from primitive building blocks that capture an abstraction of that, and orderly structured extensions could then be introduced to model higher level funcionalities, applying deduction to obain and to study their properties. Reusage of sub-systems is frequent in the biological domain as well: Evolution theorists use the term exaptations (Gould & Vrba 1982) to refer to minor changes that make use of already existing capabilities to create new behaviors. Exaptation can be readily modeled mathematically by structural abstraction and re-application of relevant constructs and proofs. One ‘no free lunch’ price for generic domain independent models seems to be an extreme care to balance between abstraction that is not detached, and grounding that is not over deterministic. In particular: A bimodality of mathematical modeling is that, on one hand, one treats the concepts as if they were meaningless. That should warrant that all assumptions are stated explicitly as postulates, and no hidden properties of the primitive terms enter from their pretheoretical, commonsense, intuitions. On the other hand, one needs to invariably refer to the domain that is being modeled, making certain that the theory that emerges validly describes the phenomena that gave rise to the formalization. In AI, the open-ended diversity of phenomena that need to be modeled has made it difficult to capture things in a uniform manner and to benefit from mathematization as a powerful modeling tool like other scientific domains. If one were to overcome this last obstacle, then this would probably be by finding a higher, yet suitable, level of abstraction: high enough to absorb a variety of phenomena, but not so high as to conflate all meanings. Category theory (Herrlich & Strecker 1973; MacLane 1972) has been developed precisely for such purposes within mathematics itself. It provides meticulous tools of rigour to capture a structural essence without being over deterministic. Category theory has already been successfully applied to various issues in computer science, like programming language semantics and the design of programs using abstract data types, providing a standard ontology and language of discourse for these areas of research (Barr & Wells 1995). Its advantages for modeling intelligence have been solicited, among others, by (Magnan & Reyes 1994). ISAAC, an Integrated Schema for Affective Artificial Cognition, (Arzi-Gonczarowski & Lehmann 1998b; 1998a; From: AAAI Technical Report SS-03-02. Compilation copyright © 2003, AAAI (www.aaai.org). All rights reserved.