Decidable fragments of first-order temporal logics

Abstract

Accepted versio

Monodic temporal logic with quantified propositional variables

“…The decidability of monodic fragments of first-order temporal logic is usually established using 'quasimodels' [12]. Quasimodels for a formula φ have, at each moment in time n, a 'state candidate' which is a collection of possible sets, indexed by elements t in some finite set T , of the formulae ψ in sub x φ such that ψ is true at moment in time n, for some element a (substituting for x) of the domain of a first-order model J(n) that 'realizes' the state candidate.…”

“…A test for satisfiability of φ is constructed by encoding the existence of a quasimodel into one of the (decidable) second-order theories of natural numbers with successor functions and monadic predicates S1S or S2S. The exact nature of quasimodels depends on the linear [12] or branching [13], [3] temporal logic of interest. Our aim in this section is to determine when an EMTL pos formula φ has a model for all (temporal) assignments υ in some set of assignments Υ where the assignments to P i ∈ Π vary.…”

“…For the latter to hold, the underlying firstorder temporal logic being extended has to be decidable/recursively enumerable, and so we work with the monodic fragment [12]. We show that unsatisfiability is recursively enumerable for the monodic fragment where propositional quantification occurs outside of quantification of individual variables and grammar operators, inside of which externally quantified propositions are within positive occurrences of existential quantifiers for individual variables in the overall formula.…”

“…In section 2, we define the monodic fragment of first-order temporal logic and monodic fragments with grammar operators and propositional quantification. The structures used for proving decidability of monodic temporal logics are 'quasimodels' [12]. In section 3, we give an expanded form of quasimodel corresponding to a collection of models of a monodic formula differing on the assignments of propositions.…”

Specifying and Reasoning About Dynamic Access-Control Policies

“…Given that our model involves both relational terms and a transition system, our analyses require logics that integrate predicate logic and temporal operators. Hodkinson et al have shown [19,20] that such logics have very bad decidability properties, even when the first-order components are restricted to decidable fragments. For example, the monadic fragment of first-order linear temporal logic is undecidable, even restricted to the 2-variable case.…”