Temporal Logic with Predicate λ-Abstraction

Abstract

Abstract.A predicate linear temporal logic LT L λ= without quantifiers but with predicate abstraction mechanism and equality is considered. The models of LT L λ= can be naturally seen as the systems of pebbles (flexible constants) moving over the elements of some (possibly infinite) domain. This allows to use LT L λ= for the specification of dynamic systems using some resources, such as processes using memory locations, mobile agents occupying some sites, etc. On the other hand we show that LT L λ= is not recursively axiomatizable and, therefore, fully automated verification of LT L λ= specifications is not, in general, possible.

LTL over Integer Periodicity Constraints

“…[CC00] IPC + {x < y, x = y} pspace Σ All undecidability results with {x − y = c, x = c} are consequences of the fact that LTL over the constraint language allowing atomic constraint of the form x = y and x = y + 1 is undecidable by simulation of two-counter machines [CC00]. The recent results from [DLN05,DG05,LP05] answer to the questions left open in [Dem04b] and are evidence that our results are optimal. For instance, PLTL(IPC + ) extended with the freeze operator is expspacecomplete whereas PLTL(IPC ++ ) extended with the freeze operator is already Σ 1 1 -hard.…”

“…For instance, PLTL(IPC + ) extended with the freeze operator is expspacecomplete whereas PLTL(IPC ++ ) extended with the freeze operator is already Σ 1 1 -hard. Indeed, LTL({x = y}) with the freeze operator is shown Σ 1 1 -complete in [DLN05,LP05] The pspace-completeness of PLTL mod leaves open for which constraint system D (not necessarily fragment of Presburger arithmetic), LTL over D is decidable in pspace. Necessary conditions are provided in [DD03] to guarantee the polynomial space upper bound (completion property and frame checking in pspace) and similar conditions are also introduced in [BC02,LM05].…”

“…This is a powerful binder mechanism used in real-time logics [AH93,AH94], in hybrid logics [Gor96,Bla00], in logics with λ-abstraction [Fit02,LP05], and in temporal logics [LMS02,DLN05]. Adding this kind of operator can easily lead to undecidability (see e.g., [Gor96]) when no restriction is required on the Kripke structures.…”

Witness Runs for Counter Machines

“…Freeze operator has been introduced in numerous works, sometimes with different motivations, see e.g. [Hen90,Gor94,Fit02,LP05]. It is also sometimes used implicitly as for the temporal semantics for imperative programs that may use first-order temporal logics, see e.g.…”

Alternation-free modal mu-calculus for data trees

“…An alternative approach to reasoning about data words was considered in [16,23,12,11,22], where expressiveness and algorithmic properties of linear temporal logic extended by freeze quantification (for short, LTL ↓ ) were studied. Freeze quantification was introduced in the context of timed logics (cf., e.g., [1]).…”