This paper investigates modular combinations of temporal logic systems. Four combination methods are described and studied with respect to the transfer of logical properties from the component one-dimensional temporal logics to the resulting combined two-dimensional temporal logic. Three basic logical properties are analyzed, namely soundness, completeness, and decidability. Each combination method comprises three submethods that combine the languages, the inference systems, and the semantics of two one-dimensional temporal logic systems, generating families of two-dimensional temporal languages with varying expressivity and varying (...) degrees of transfer of logical properties. The temporalization method and the independent combination method are shown to transfer all three basic logical properties. The method of full join of logic systems generates a considerably more expressive language but fails to transfer completeness and decidability in several cases. So a weaker method of restricted join is proposed and shown to transfer all three basic logical properties. (shrink)

We introduce a methodology whereby an arbitrary logic system L can be enriched with temporal features to create a new system T(L). The new system is constructed by combining L with a pure propositional temporal logic T (such as linear temporal logic with Since and Until) in a special way. We refer to this method as adding a temporal dimension to L or just temporalising L. We show that the logic system T(L) preserves several properties of the original temporal logic (...) like soundness, completeness, decidability, conservativeness and separation over linear flows of time. We then focus on the temporalisation of first-order logic, and a comparison is make with other first-order approaches to the handling of time. (shrink)

We introduce a methodology whereby an arbitrary logic system L can be enriched with temporal features to create a new system T(L). The new system is constructed by combining L with a pure propositional temporal logic T (such as linear temporal logic with “Since” and “Until”) in a special way. We refer to this method as “adding a temporal dimension to L” or just “temporalising L”. We show that the logic system T(L) preserves several properties of the original temporal logic (...) like soundness, completeness, decidability, conservativeness and separation over linear flows of time. We then focus on the temporalisation of first-order logic, and a comparison is make with other first-order approaches to the handling of time. (shrink)

A way is found to add axioms to sequent calculi that maintains the eliminability of cut, through the representation of axioms as rules of inference of a suitable form. By this method, the structural analysis of proofs is extended from pure logic to free-variable theories, covering all classical theories, and a wide class of constructive theories. All results are proved for systems in which also the rules of weakening and contraction can be eliminated. Applications include a system of predicate logic (...) with equality in which also cuts on the equality axioms are eliminated. (shrink)

We present a complete and cut-free proof-system for a fragment of MTL, where modal operators are only labelled by bounded intervals with rational endpoints.

We introduce a complete and cut-free proof system for a sufficiently expressive fragment of Metric Temporal Logic over dense time domains in which a schema of induction is provable. So doing we extend results previously obtained by Montagna et al. to unbounded temporal operators.