Using Model Theory to Specify AI Programs

This paper proposes a method fo r adapting the t r a d i t i o n a l devices of model theory to the task of speci fy ing the input /output behavior of a r t i f i c i a l i n te l l i gence reasoning programs when viewed as inference engines. The method is i l l u s t r a t e d by spec i fy ing two programs, one a toy example and the other a program for r e t r i e v i n g in format ion from a dec lara t ive knowledge base. Close examination shows that many i n t u i t i o n s about the proper t ies of a re t r i eve r can be stated r igorous ly in terms of inference and that the modeltheoret ic spec i f i ca t i o n can then be used to prove that the r e t r i e v e r has these proper t ies . 1. AI Needs Spec i f i ca t ion Techniques The success of a r t i f i c i a l I n te l l i gence as a s c i ence hinges on our a b i l i t y to bu i l d a theory that re la tes a program's s t ruc tu re to i t s behavior. Essential to t h i s enterpr ise are methods for producing r igorous spec i f i ca t ions of programs at a high level of abst rac t ion that can be used to codi fy and communicate our r e s u l t s . There are two spec i f i ca t i on methods predominantly (though c e r t a i n l y not exc lus ive ly ) used: Engl ish prose t y p i c a l l y descr ib ing various s t ruc tu ra l components of the program and t h e i r ro le in the program's performance, and the actual code that is used to implement the program. An English prose spec i f i ca t i on can be h igh ly abstract but , in pract i c e , usual ly at the expense of r i go r and p r e c i s ion . Consider the problem of p red ic t i ng how a program so described in a journa l w i l l behave on examples not considered in the a r t i c l e . Morse ye t , consider the problem of showing that the program has ce r ta in p roper t ies . Though a program's code does not su f fe r from the lack of prec is ion that an English spec i f i ca t i on t y p i c a l l y does, i t i s not s u f f i c i e n t l y abs t rac t . Hence, appropr ia te ly enough, code ra re l y works i t s way in to the l i t e r a t u re . A methodology that has been pursued successfu l ly throughout computer science is that of separat ing the descr ip t ion of what a program computes from how it computes i t . On the one hand there are descr ipThis work has been supported in part by grant GR/D/16062 from the Science and Engineering Research Counci l . t ions of a program's input /output behavior and on the other descr ip t ions of i t s i n te rna l modules, processes, states and data s t ruc tu res . This paper considers the case fo r using model theory to speci fy what a program computes. A method is proposed fo r adapting t r a d i t i o n a l modeltheore t i c techniques and is i l l u s t r a t e d by spec i fy ing two programs. The f i r s t is a toy example used to i l l um ina te the key points of the technique, whi le the second, a knowledge r e t r i e v e r , is used to demonstrate app l i ca t ion of the technique to a r e a l i s t i c AJ program. Beyond what is demonstrated by these examples and those mentioned in the sect ion on re la ted work, l i t t l e is known about the range of a p p l i c a b i l i t y of the technique. 2. Overview of the Proposed Technique A major concern of AI is w i th programs that manipulate representat ions, which 1 take to be data s t ruc tures that denote. This ra ises the p o s s i b i l i t y that many such programs can be viewed as inference engines, de r i v ing conclusions from t h e i r representat ions. This paper is concerned wi th developing techniques fo r using model theory to speci fy the input /output behavior of programs seen from t h i s v iewpoint . For example, a planner can be seen as an i n f e r ence system. The program embodies a theory of ac t ion and i t s input is a pa i r of sentences each pred ica t ing a condi t ion on the wor ld . The planner produces a plan such that i t s theory of act ion l o g i c a l l y impl ies that the second input sentence would be t rue if the plan were performed in any world s a t i s f y i n g the f i r s t input sentence. Unfor tunate ly , many planning systems such as STRIPS (Fikes and Nl lsson, 1971) do not meet such a s p e c i f i c a t i o n . In order to deal w i th the enormously complex problem of f i n d i n g an appropriate plan among the set of a l l p lans, these systems employ problem representat ions that o f ten f i n d plans qu ick ly but do so at the expense of occasiona l l y f a i l i n g to f i n d any at a l l . As an inference engine such a system is incomplete. How then can such an incomplete system be speci f ied? The approach advocated here is to produce another model theory whose l og i ca l imp l i ca t ion r e l a t i o n is weakened in such a way that the program is a sound and complete inference engine w i th respect to I t . Many people i n i t i a l l y f i n d t h i s approach qu i te odd.