﻿ Relations of logical consequence in logics of partial predicates with composition of predicate complement | Shkilniak | PROBLEMS IN PROGRAMMING

DOI: https://doi.org/10.15407/pp2019.03.011

### Relations of logical consequence in logics of partial predicates with composition of predicate complement

O.S. Shkilniak

#### Abstract

In this paper we study software-oriented logics of partial predicates with new special non-monotonic operation (composition) of the predicate complement. We denote these logics by LC and composition of the predicate complement by . Such operations are used in various versions of the Floyd-Hoare program logic with partial pre- and post-conditions. We describe first order composition algebras and LC languages. For LC, a number of logical consequence relations (Pc|=T, Pc|=F, Rc|=T, Rc|=F, Pc|=TF, Rc|=TF, Pс|=IR) and logical consequence relations under the conditions of undefinedness (P|=T^, P|=F^, R|=T^, R|=F^, P|=TF^, R|=TF^) are specified. Properties of the defined relations are investigated, differences and the relationship between them are given. For the introduced relations, we describe the conditions for their guaranteed presence, the decomposition conditions for formulas and the properties of quantifier elimination. The theorem of elimination of the conditions of undefinedness for the relations |=T^ and |=F is proved. Thus, the relations P|=T^, P|=F^, R|=T^ and R|=F^ can be expressed by Pc|=T, Pc|=F, Rc|=T and Rc|=F respectively. However, it is shown that |=IR^ cannot be expressed by Pс|=IR. Moreover, it is impossible to define correctly the decomposition conditions for  formulas for Pс|=IR. Properties of decomposition conditions for  formulas are different for the relations |=T and |=F, therefore properties of decomposition and equivalent transformations must be specified indirectly through the corresponding properties of |=T and |=F. First order sequent calculi for the introduced logical consequence relations for LC and logical consequence relations under the conditions of undefinedness will be constructed in in the forthcoming articles.

Problems in programming 2019; 3: 11-27

#### Keywords

logic; partial predicate; composition al­gebra; logical consequence

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#### References

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DOI: https://doi.org/10.15407/pp2019.03.011

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