MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

68 Robert Goldblatt[Hennessy and Liu, 1995]. They provide modalities that formalise complex structuralassertions, for example the formula 〈c!x〉α expressing “it is possible to outputsome value v on channel c and thereby evolve to a state in which α[v/x] is true”.Axiomatisations of various modal process logics may be found, inter alia, in[Stirling, 1987] and [Larsen, 1990]. Other work on modal aspects of process algebrais collected in [Ponse et al., 1995].7.3 Temporal Logic for ConcurrencyIn 1977 Amir Pnueli, motivated by a reading of [Rescher and Urquhart, 1971], 61proposed to use temporal logic to formalising reasoning about the behaviour of concurrentprograms involving a number of processors acting in parallel and sharinga memory environment, so that each can alter the values of variables used by theothers (see Pnueli [1977; 1981]). This is particularly relevant to the specificationand analysis of reactive programs, like operating systems and systems for airlinereservation or process control, that repeatedly interact with their environmentand are not expected to terminate. As such a program runs, each success stateis obtained by one processor being chosen to execute one instruction. Thus froman initial state x 0 , many different sequences x 0 , x 1 ,. . . of states may be generateddepending on which processors get chosen to act at each step.Pnueli observed that temporal modalities could be used to formulate computationallysignificant properties of execution sequences, such as fair scheduling (noprocessor is delayed forever), freedom from deadlock (when none can act), andmany others. He used Prior’s future-tense modality G (and its dual F ), but withthe Diodorean reading of “at all future states including the present”, as well asa connective X with the reading “at the next state”. The latter had first beenintroduced to tense logic for discrete time by Dana Scott (see [Prior, 1967, p. 66]).Programs do not appear in the syntax in this approach. Instead, temporal formulasdescribe properties of a particular execution sequence of a single (concurrent)program.The paper of Gabbay, Pnueli, Shelah and Stavi [1980] added a binary connectiveU to this formalism, with α Uβ meaning “α until β”, i.e. “β will be true, and αwill be true at all times until β is”. This connective and its past-tense versionα since β had been studied by Hans Kamp [1968] who showed that they forman expressively complete set of connectives in the sense that for models in whichtime is a complete linear ordering, all tense-logical connectives can be defined interms of them. Gabbay et al. adapted this to show that U by itself plays a similarrole for the future-tense logic of state sequences. They gave an axiomatisation forthis extended logic, which they called DUX, and proved that it is decidable. Byway of illustration of the expressive completeness of U, they noted that F α can bedefined as ⊤ Uα, and then Gα as ¬F ¬α, while Xα can be defined as ⊥ Uα. DUXis now more commonly known as PLTL (propositional linear temporal logic).61 See [Hasle and Øhrstrøm, 2004, p. 222].