Algebras of general non-deterministic predicates

M.S. Nikitchenko, O.S. Shkilniak, S.S. Shkilniak

Abstract


Logics of general nondeterministic qua-siary predicates, called GND-predicates, are defined and investigated. These logics are program-oriented logical for-malisms that reflect such properties of programs as partiality, nondeterminism, and non-fixed arity. GND-predicates generalize partial predicates of the rela-tional type. The main attention is paid to the construction of composition algebras of GND-predicates. Compositions of GND-predicates are described, their properties are formulated. For these predicates, such important laws of tradi-tional logic as the law of absorption and the law of distributivity for for  and & are not valid. Various types of GND-predicates are identified. GND-predicates can be modeled as 7-value total deter-ministic predicates (TD7-predicates). A 7-element algebra of truth values of TD7-predicates is defined and all of its subalgebras are described. Each such subalgebra induces a corresponding al-gebra of TD7-predicates, which then in-duces the algebra of GND-predicates. This makes possible to identify a number of important composition algebras of general nondeterministic predicates. The languages of pure first-order logics of GND-predicates and their interpretations are described. The relations of a logical G-consequence and a logical G-equivalence are introduced. The relation of the logical G-consequence is mono-tonic, reflexive, and transitive; for it the properties of the decomposition of for-mulas are satisfied. On the basis of these properties, it is planned to construct cal-culi of sequential type for the logic of GND-predicates.

Problems in programming 2018; 1: 05-21


Keywords


logic; algebra; non-deterministic predicate; logical consequence

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References


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DOI: https://doi.org/10.15407/pp2018.01.005

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