Concurrent Deduction: Classical and Modal

We provide an informal report of work seeking to introduce concurrency into deduction by exploiting modularity. The work arose in the context of a theorem prover for an applied modal action logic, has led to the re-discovery and generalisation of earlier work by Nelson and Oppen and promises a fibered approach to deduction in multi-modal representations of rationality. 1. Background An automated tableau theorem prover was developed for the UK Forest project first order modal action logic in order to demonstrate industrial applications in validating specifictions rather than mathematically interesting theorems. It has been used to prove properties safety-critical specifications for systems of non-trivial size, without being adequate for systems of full industrial scale (see, for example Atkinson and Cunningham 11991]). While the full battery of mechanised deduction methods with sorting, theory unification etc., would undoubtedly have provided further enhancement for this system, there seemed to be an underlying need for a "divide and rule" approach which will reduce deep problems into small shallow parts. Motivated by this we re-explored the salient work of Nelson and Oppen [1979] on co-operating decision processes. As a consequence we were able to discover, in turn, a simple new procedure for co-operating classical tableaux, and on further analysis, a comparable procedure for concurrent action tableau which had previously been elusive. Our interests in mechanising a richer class of multi-modal logics let to the creation of a European Esprit project (Medlar). While we report elsewhere on the work of the Medlar project (see, for example Cunningham, Gabbay and Ohlbach 1991 ), indicate in the final section of this paper the possible development and application of a fibered form of concurrency. 2. Communication between Derivation Processes We address the following question: what has to be communicated when (semi-) decision procedures are used cooperatively to infer joint consequences. For a full answer the detailed processes of derivation must be analysed, but some pre-requisites for success can be derived at a more abstract level. Let Aand B be distinct theories in some logic L and call their common extension A u B the joint theory. For example, A might be the theory of finite lists with membership and B a simple arithmetic. Suitably formulated, the joint theory could be lists with arithmetic elements. Suppose we wish to prove some goal G in the joint theory, but only have the deductive machinery for deriving goals in theories A and B separately. In this case we may wish to separate the goal, so that G = GA u GB, and derive one subgoal G,~ in a subproof using the machinery for theory A and the other subgoal Ga in a subproof using the machinery for theory B. By what means can we (i) separate the goals, (ii) under what conditions is it sufficient to prove the sub-goals, and (iii) if it is not necessary to prove the subgoals in their separate theories is there a complete procedure to infer G from the successful derivation of the subgoals? In other words, we seek an exploitable relation between derivation in the combined theory (a) below, and derivations the separate theories (b): (a) A to B I.-G and (b) A lG,~ & B I(i) First let us address the means of separation. Nelson and Oppen have observed and exploited the separation of joint derivations in a quantifier-free theory in classicallogic with identity. They separate the constituent theories by the introduction of free term variables, leaving the equality symbol and the free term variables as the remaining common parameters. A similar technique for separation of theories can be achieved by the introduction of free propositional variables in classical propositional logic. This is also used in the final 34 From: AAAI Technical Report FS-93-01. Compilation copyright © 1993, AAAI (www.aaai.org). All rights reserved.