The theory of trees with additional unary predicates and quantification over nodes and branches embraces a rich branching time logic. This theory was reduced in the companion paper to the first-order theory of binary, bounded, well-founded trees with additional unary predicates. Here we prove the decidability of the latter theory.

In Ockhamist branching-time logic [Prior 67], formulas are meant to be evaluated on a specified branch, or history, passing through the moment at hand. The linguistic counterpart of the manifoldness of future is a possibility operator which is read as `at some branch, or history (passing through the moment at hand)'. Both the bundled-trees semantics [Burgess 79] and the $\langle moment, history\rangle$ semantics [Thomason 84] for the possibility operator involve a quantification over sets of moments. The Ockhamist frames are (3-modal) (...) Kripke structures in which this second-order quantification is represented by a first-order quantification. The aim of the present paper is to investigate the notions of modal definability, validity, and axiomatizability concerning 3-modal frames which can be viewed as generalizations of Ockhamist frames. (shrink)

As cause we often specify an event the occurrence of which first guaranteed that of the effect. This notion is explicated in a framework of branching worlds in Sections I to V. VI and VII point out its close relations to the concept of an agent's bringing about an event. The topic of the last two sections is the distinction between causes and necessary circumstances. For this purpose conditionals are used, interpreted with respect to branching worlds without a similarity relation (...) between them. (shrink)

We show that the join of two classical [respectively, regular, normal] modal logics employing distinct modal operators is a conservative extension of each of them.

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)

In this paper we shall introduce a simple temporal logic suitable for reasoning about the temporal aspects of parallel universes, parallel processes, distributed systems, or multiple agents. We will use a variant of the mosaic method to prove decidability of this logic. We also show that the logic does not have the finite model property. This shows that the mosaic method is sometimes a stronger way of establishing decidability.

As cause we often specify an event the occurrence of which first guaranteed that of the effect. This notion is explicated in a framework of branching worlds in Sections I to V. VI and VII point out its close relations to the concept of an agent's bringing about an event. The topic of the last two sections is the distinction between causes and necessary circumstances. For this purpose conditionals are used, interpreted with respect to branching worlds without a similarity relation (...) between them. (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)