Assume ZFC + "There is a weakly compact cardinal" is consistent. Then: (i) For every $S \subseteq \omega, \mathrm{ZFC} +$ "S and the monadic theory of ω 2 are recursive each in the other" is consistent; and (ii) ZFC + "The full second-order theory of ω 2 is interpretable in the monadic theory of ω 2 " is consistent.

LetL be any modal or tense logic with the finite model property. For eachm, definer L (m) to be the smallest numberr such that for any formulaA withm modal operators,A is provable inL if and only ifA is valid in everyL-model with at mostr worlds. Thus, the functionr L determines the size of refutation Kripke models forL. In this paper, we will give an estimation ofr L (m) for some linear modal and tense logicsL.

Temporal logic is gaining recognition as an attractive and versatile formalism for rigorously specifying and reasoning about computer programs, digital circuits and message-passing systems. This book introduces Tempura, a programming language based on temporal logic, Tempura provides a way of directly executing suitable temporal logic specifications of digital circuits, parallel programs and other dynamic systems. Since every Tempura statement is also a temporal formula, the entire temporal logic formalism can be used as the assertion language and semantics. One result is (...) that Tempura has the two seemingly contradictory properties of being a logic programming language and having imperative constructs such as assignment statements. The presentation investigates Interval Temporal Logic, a formalism with conventional temporal operators such as next and always as well as lesser known ones such as chop. This provides the basis for Tempura. The design of an interpreter for Tempura is also included, as are a variety of sample Tempura programs illustrating how to model both hardware and software. (shrink)