A Hierarchical Completeness Proof for Propositional Interval Temporal Logic with Finite Time

Ben Moszkowski
2004 Journal of Applied Non-Classical Logics  
We present a completeness proof for Propositional Interval Temporal Logic (PITL) with finite time which avoids certain difficulties of conventional methods. It is more gradated than previous efforts since we progressively reduce reasoning within the original logic to simpler reasoning in sublogics. Furthermore, our approach benefits from being less constructive since it is able to invoke certain theorems about regular languages over finite words without the need to explicitly describe the
more » ... ated intricate proofs. A modified version of regular expressions called Fusion Expressions is used as part of an intermediate logic called Fusion Logic. Both have the same expressiveness as ÈÁÌÄ but are lower-level notations which play an important role in the hierarchical structure of the overall completeness proof. In particular, showing completeness for ÈÁÌÄ is reduced to showing completeness for Fusion Logic. This in turn is shown to hold relative to completeness for conventional linear-time temporal logic with finite time. Logics based on regular languages over finite words and -words offer a promising but elusive framework for formal specification and verification. A number of such logics and decision procedures have been proposed. In addition, various researchers have obtained complete axiom systems by embedding and expressing the decision procedures directly within the logics. The work described here contributes to this topic by showing how to exploit some interesting links between regular languages and interval-based temporal logics. KEYWORDS: interval temporal logic, completeness proof, fusion product. 56 Issue on Interval Temporal Logics and Duration Calculi Î , . . . are used. They are later formally introduced in Definition 44. Below is the structure of the proof from Ä to ÈÁÌÄ using countably infinite subsets of ÈÁÌÄ formulas denoted as ÈÁÌÄ ¼ , ÈÁÌÄ ½ , . . . and formally introduced later in Definition 80: Here Lemma 84 has a proof by induction. The suffix "a" refers to the base case and the suffix "b" refers to the inductive proof step. Related Work Let us now discuss other work on axiom systems for ÁÌÄ and then some closely related calculi. A proof of completeness for such notations is often based on some kind of decision procedure so we make some mention of this as well. Halpern and Moszkowski [MOS 83a, pages 23-24] prove the decidability of ÈÁÌÄ with quantifiers over finite time by translation to ÉÈÌÄ over finite time which is decidable by an easy modification of an analogous result for conventional ÉÈÌÄ over infinite time by Wolper [WOL 82, SIS 87] (see also [LIC 85] for a direct proof). The satisfiability problem for ÈÁÌÄ has nonelementary complexity and hence is much harder than that for popular logics such as ÈÌÄ. We include statements and proofs of relevant results for ÈÁÌÄ in Appendix A. These difficulties with complexity have also manifested themselves in work on complete axiom systems for ÈÁÌÄ. The topic seems to present more hurdles than in the case for other some related logics. The reader should bear this in mind when attempting to assess progress in this area. Rosner and Pnueli [ROS 86] investigate an axiom system for quantifier-free ÈÁÌÄ with finite and -intervals and the until operator. However it does not contain the operator chop-star which is like Kleene-star for regular expressions. A tableau method serves as the decision procedure underlying the completeness proof and employs an adaptation of Fischer-Ladner closures developed for Propositional Dynamic Logic (È Ä) [FIS 79, HAR 00]. One of the inference rules is quite large and requires constructing an index-table containing indices (including terminal indices) and an accessibility relation for automata transitions connected with tableau construction. Furthermore, the inference rule necessitates deducing three categories of ÈÁÌÄ theorems concerning accessibility between indices in the index-table before an inference can actually be made. Paech [PAE 88] investigates a quantifier-free version of ÈÁÌÄ with -intervals having chop-star limited, like Kleene-star, to a finitely many iterations and including an additional temporal operator unless. Due to a theorem of Thomas [THO 79] A completeness proof for PITL 59 (later more simply proved by Y. Choueka and D. Peleg [CHO 83]), ÈÁÌÄ with such a restricted chop-star is still as expressive as -regular expressions (and hence the logics S1S [BÜC 62] and ÉÈÌÄ) as well as quantifier-free propositional ÈÁÌÄ with unrestricted chop-star (which permits consecutive finite iterations) although with possibly less succinctness. An additional temporal operator unless, which is a variant of the conventional operator until, is also included. Paech presents a complete Gentzen-style proof system which includes some nonconventional axioms which obligate certain ÈÁÌÄ formulas to already be in a form analogous to regular expressions. This can potentially involve complex meta-reasoning about arbitrary ÈÁÌÄ formulas over finite intervals to ensure suitability for these particular axioms. The proof method is adapted from one used by Nishimura [NIS 79] for È Ä and subsequently refined by Valiev [VAL 79]. Consequently, a generalised form of Fischer-Ladner closures is necessary to cope with negation and other aspects of ÈÁÌÄ not found in È Ä programs. Surprisingly, the axioms, unlike those of Rosner and Pnueli, appear to limit intervals to be infinite. Therefore no modular reasoning about finite subintervals is possible. Dutertre [DUT 95] gives two complete proof systems for first-order ÁÌÄ without chop-star for finite time. The first uses a possible-worlds semantics of time and the second considers arbitrary linear orderings of states. Neither is complete for standard discrete-time intervals. Wang Hanpin and Xu Qiwen [Wan 99] generalise Dutertre's results to handle infinite time. Kono [KON 95] presents a tableau-based decision procedure for ÈÁÌÄ with quantifiers and temporal projection over finite time which has been successfully implemented in Prolog 1 . No formal proof is given that the method does not overlook models. Instead, a sketchy argument about termination is presented. Kono suggests that the transformations provide a partial basis for a complete axiom system. Many details are omitted and one of the two proposed axioms for projection is unsound 2 . Moszkowski [MOS 94] presents propositional and first-order axiom systems for ÁÌÄ over finite intervals. This is shown to support proofs involving sequential and parallel aspects of compositionality based on the rely-guarantee paradigm of Jones [JON 83]. The propositional part is claimed to be complete but only a brief outline of a proof is given. This work is extended in [MOS 95] to include a axiomatisation for temporal projection which is complete relative to ÈÁÌÄ without projection. Bowman and Thompson [BOW 98] present a detailed study of a tableau-based decision procedure for quantifier-free ÈÁÌÄ with finite time and temporal projection but do not give an axiom system. They omit considerations about the termination of their method. In [BOW 00, BOW 03] they look at termination and also obtain a completeness proof for an axiomatisation of this version of ÈÁÌÄ. Wolper [WOL 83] presents Extended Temporal Logic ( ÌÄ) which includes operators containing explicit automata descriptions. This make ÌÄ's expressive power equivalent to that of S1S. A decision procedure and complete axiom system are given. Compared with ÈÁÌÄ, ÌÄ's notational reliance on automata makes it less suited for ½. We have extensively used it and never found a bug. ¾. However, Kono has told us that the associated problem in the implemented decision procedure was rectified early on.
doi:10.3166/jancl.14.55-104 fatcat:rfryn6mn6vhidpxpb4mik74gbe