On Proof Complexities Relations in Some Systems of Propositional Calculus

The number of linear proofs steps for some sets of formulas is compared in the folowing systems of propositional calculus: PK – seguent system with cut rule, PK— - the same system without cut rule, SPK – the same system with substitution rule, QPK – the same system with quantifier rules. The number of steps of tree-like proofs in the same systems for some considered set of formulas is compared from Alessandra Carbone in [1] and some distinctive property of the system QPK is revealed: QPK has an exponential speed-up over the systems SPK and PK, which, in their turn, have an exponential speed-up over the system PK—. This result drew the heavy interest for the study of the system QPK. In this work for linear proofs steps in the same systems the other relations are received: it is showed that the system QPK has no preference over the system SPK, it is showed also that for the considered formula sets the system PK has no preference over the system PK—, which, in its turn, has no preference over the monotone system PMon. It is proved also, that the same results are reliable for some other sets of formulas and for other systems as well.

Analysis of minimal path and cut vectors in multistate monotone systems and use it for detection of binary type multistate monotone systems

Theorem specifying how all minimal cut (path) vectors to level j can be obtained from all minimal path (cut) vectors to level j in any multistate monotone system is proven. Characterizations of binary type multistate monotone systems and binary type multistate strongly coherent systems by minimal cut and path vectors and corresponding to them, binary type cut and path sets are also demonstrated.

Zero-One Law for the Non-Availability of Multistate Repairable Systems

In a monotone system whose failure depends on a large number of independent components, the non-availability should increase abruptly over a relatively short interval of time. Our first step is a reliability interpretation of a result of Friedgut and Kalai.1 Then we prove a zero-one law which is more adapted in a reliability context. In the last section some multistate extensions of classical models are considered as examples.