Oleg Grigoriev

Natural deduction calculus for linear-time temporal logic

We present a natural deduction calculus for the propositional linear-time temporal logic and prove its correctness. The system extends the natural deduction construction of the classical propositional logic. This will open the prospect to apply our technique as an automatic reasoning tool in a deliberative decision making framework across various AI applications.

Natural Deduction Calculus for Computation Tree Logic

The authors present a natural deduction calculus for the computation tree logic, CTL, defined with the full set of classical and temporal logic operators. The system extends the natural deduction construction of the linear-time temporal logic. This opens the prospect to apply our technique as an automatic reasoning tool in a deliberative decision making framework across various applications in AI and computer science, where the branching-time setting is required

Automated Natural Deduction for Propositional Linear-Time Temporal Logic

We present a proof searching technique for the natural deduction calculus for the prepositional linear-time temporal logic and prove its correctness. This opens the prospect to apply our technique as an automated reasoning tool in a number of emerging computer science applications and in a deliberative decision making framework across various AI applications.

Relevant generalization starts here (and here = 2)

There is a productive and suggestive approach in philosophical logic based on the idea of generalized truth values. This idea, which stems essentially from the pioneering works by J.M. Dunn, N. Belnap, and which has recently been developed further by Y. Shramko and H. Wansing, is closely connected to the power-setting formation on the base of some initial truth values. Having a set of generalized truth values, one can introduce fundamental logical notions, more specifically, the ones of logical operations and logical entailment. This can be done in two different ways. According to the first one, advanced by M. Dunn, N. Belnap, Y. Shramko and H. Wansing, one defines on the given set of generalized truth values a specific ordering relation (or even several such relations) called the logical order(s), and then interprets logical connectives as well as the entailment relation(s) via this ordering(s). In particular, the negation connective is determined then by the inversion of the logical order. But there is also another method grounded on the notion of a quasi-field of sets, considered by Bialynicki-Birula and Rasiowa. The key point of this approach consists in defining an operation of quasi-complement via the very specific function g and then interpreting entailment just through the relation of set-inclusion between generalized truth values. In this paper, we will give a constructive proof of the claim that, for any finite set V with cardinality greater or equal 2, there exists a representation of a quasi-field of sets isomorphic to de Morgan lattice. In particular, it means that we offer a special procedure, which allows to make our negation de Morgan and our logic relevant.