LTL over Integer Periodicity Constraints

Abstract

Abstract. Periodicity constraints are used in many logical formalisms, in fragments of Presburger LTL, in calendar logics, and in logics for access control, to quote a few examples. In the paper, we introduce the logic PLTL mod , an extension of Linear-Time Temporal Logic LTL with past-time operators whose atomic formulae are defined from a first-order constraint language dealing with periodicity. Although the underlying constraint language is a fragment of Presburger arithmetic shown to admit a pspace-complete satisfiability problem, we establish that PLTL mod model-checking and satisfiability problems remain in pspace as plain LTL (full Presburger LTL is known to be highly undecidable). This is particularly interesting for dealing with periodicity constraints since the language of PLTL mod has a language more concise than existing languages and the temporalization of our first-order language of periodicity constraints has the same worst case complexity as the underlying constraint language. Finally, we show examples of introduction the quantification in the logical language that provide to PLTL mod , expspacecomplete problems. As another application, we establish that the equivalence problem for extended single-string automata, known to express the equality of time granularities, is pspace-complete by designing a reduction from QBF and by using our results for PLTL mod .

Impartial Anticipation in Runtime-Verification

“…We have shown that this task is simple in the setting of Pnueli's LTL. However, for some relevant richer logics, like real-time logics [11] or logics interpreted over the integers [5], this task is considerably more involved [4]. Nevertheless, as the approach given for LTL suggests, there is a typical pattern for deriving those functions, which we describe in this section.…”

“…In the setting of the real-time logic TLTL [11], an event-clock automaton is constructed accepting precisely the models of the formula at hand. In the setting of Demri's PLTL mod [5], words over an infinite alphabet (representing integer valuations) are abstracted to words of a finite alphabet and a suitable construction of a Büchi automaton accepting these symbolic evaluations is provided.…”

“…Furthermore, a monitor synthesis algorithm is also obtained for the LTL enriched by past operators or forgettable past operators [12] with a corresponding satisfiability check by automata abstraction. PLTL mod [5]: PLTL mod is a decidable fragment of Presburger LTL, which extends LTL with first-order integer arithmetic constraints. The alphabet of a PLTL mod formula is infinite because it includes all valuations of variables over Z.…”

“…We define precisely the automata-theoretic approach and we elaborate certain criteria (forgettable past and faithfulness of abstraction) under which a monitor is directly obtained from automata accepting the models of a formula at hand. We show that automata constructions existing in the literature for several linear-time logics satisfy the introduced criteria such that we derive easily impartial and anticipating monitors, e.g., for the linear-time µ-calculus [6,7], monadic second-order logic (over words) [8], RLTL [9], and PLTL mod [5] which is interpreted over sequences of integer valuations.…”

“…In this paper, a uniform approach for synthesizing monitors checking correctness properties specified in linear-time logics at runtime is provided, which is based on our approach in [3]. To this end, we define linear-time logics as logics interpreted over infinite words, for example, LTL, PLTL mod [5], linear µ-calculus, timed LTL etc. Uniformly, we give an impartial and anticipatory semantics for linear-time logics suitable for runtime verification.…”