Temporal patterns and modal structure

J van Benthem
1999 Logic Journal of the IGPL  
Temporal logic arose at the border of philosophy and linguistics. From the seventies onward, it became a major tool also in computer science and artificial intelligence, which have become the most powerful source of new logical developments since. We discuss some recent themes demonstrating new connections with modal logic. In the course of this, we point out some new types of open research questions. Temporal logic as an identifiable research area emerged in the 50s, largely due to the
more » ... ng efforts of Arthur Prior. Born some decades later than modern modal logic, it turned out technically much like it. But unlike in the case of modal logic, from the start, different motivations fed into temporal logic, coming from philosophy (analysis of historical argumentations involving the structure of time), linguistics (description of tenses and other temporal expressions in natural languages), and to some extent also mathematics (theory of linear and branching orders, non-point-based geometries). This breadth reflects, of course, the ubiquity of time in virtually every intellectual endeavour. Van Benthem 1983 ("The Logic of Time") tries to systematize temporal logic across these various motivations, even taking up issues in relativistic physical space-time. In the 80s, temporal logic penetrated into further disciplines, such as cognitive psychology (cf. Jackson & Michon 1985). By far the most spectacular development, however, has taken place at the interface of logic with computer science and artificial intelligence, where temporal logic itself really changed its agenda and its depth of reach. Broad surveys across this more contemporary width are van Benthem 1983 van Benthem , 1989 van Benthem , 1995 . This paper presents some recent trends along this spectrum, without scholarly completeness. There may not even be one unified field of logical studies of time. After all, it is not quite clear that the different 'foster disciplines' are concerned with the same thing. Physicists deal primarily with real-world space-time, linguists and psychologists with 'representation time', and computer scientists with machine-induced 'process time'. At least, one has to be aware that these notions differ, and then, see if or how they relate. E.g., linguistic representation time has to 'fit' space-time to keep us attuned to our physical environment, and computational process-time needs to 'mesh' with space-time to ensure reliable on-line performance. Much of this conceptual interfacing is nontrivial, and remains to be done. Indeed, technically and sociologically, there are now different communities in temporal logic with different styles and agendas, such as those active in pure logic, computer science, or AI -while formal philosophers of time are yet another separate breed. Perhaps the highest status has been achieved by the CSbased variety of temporal logic developed by Amir Pnueli since the mid 70s, studying temporal specification of desired executions of complex distributed programs (cf. De Bakker, de Roever & Rozenberg, eds., 1989). This work was honoured by the 1996 Turing Award, the highest recognition in computer science. Even though this line still shows many traces of the spirit and tools of traditional temporal logic, it is definitely diverging. (E.g., the currently emerging work on specifying 'hybrid temporal systems' is mixing the analysis of 'logical time' with standard physical engineering techniques). Temporal Logic as Modal Logic Prior's temporal logic may be viewed as a two-directional modal logic, with operators F ("at least once in the future") and P ("at least once in the past"). This makes temporal logic and modal logic very close mathematically. Traditionally, however, one stressed a philosophical difference in the 'direction of thought'. Given the relative concreteness of its subject matter, temporal logic constructs description languages for independently given temporal models, while modal logic is in the business of constructing models for given modal languages. (The syntax of modality, alas, is clearer than its semantics.) This particular distinction no longer works these days. Partly under the influence of temporal logic (Gabbay 1981), over the past decade modal logicians, too, have turned with zest to new language construction over Kripke models (or in CS speak: 'labeled transition systems'). Technical themes therefore have tended to converge between the fields. A typical example of this current trend is the theory of the 'process equivalence' of bisimulation in the analysis of expressive power for modal or temporal languages. Simulations: From Matching Single States To Matching Tuples Prior's original propositional F, P language is invariant for two-sided bisimulations, which are like standard modal bisimulations, but now with back-and-forth clauses for
doi:10.1093/jigpal/7.1.7 fatcat:cjkmqmzoyvh3zl7lw2s7jzpclu