Abstract
We study some structural and numerical properties of varieties for determinative disjunctive normal forms, introduced here for given propositional formula. We consider for classical and non-classical propositional logics some proof systems, which are constructed on the base of determinative disjunctive normal forms. We investigate also the relation between the proof complexities in some well-known classical and non-classical proof systems (Resolution, Cut-free sequent, Gentzen refutation, Cutting planes etc.) and numerical properties of varieties for determinative disjunctive normal forms for classical and non-classical tautologies.
Highlights
Many of the proof systems of classical propositional logic use presentation of tautologies or contradictions in disjunctive normal forms (DNF) or in conjuctive normal forms (CNF)
If we consider a tautology in the capacity of the Boolean function, it has a unique prime implicant, which corresponds to empty conjunct, but it is well known that there are “hard” and “simple” tautologies, the representations of classical tautologies, of non-classical tautologies, in some above varieties of DNF are not entirely correct
Using some notions given in Chubaryan, An. and Chubaryan, Arm. (2007) for tautologies, in this paper for given propositional formula we introduce the notions of determinative conjunct, minimal determinative conjunct, to which corresponds determinative prime interval, determinative disjunctive normal form, perfect dDNF, abridged dDNF, dead-end dDNF, shortest dDNF and minimal dDNF
Summary
Many of the proof systems of classical propositional logic use presentation of tautologies or contradictions in disjunctive normal forms (DNF) or in conjuctive normal forms (CNF). (2007) for tautologies, in this paper for given propositional formula we introduce the notions of determinative conjunct, minimal determinative conjunct, to which corresponds determinative prime interval, determinative disjunctive normal form (dDNF), perfect dDNF, abridged dDNF, dead-end dDNF, shortest dDNF and minimal dDNF. We show that there are some essential difference between the numerical and structural properties of the same varieties of DNF and dDNF for a given formula. We investigate the relations between numerical properties for the varieties of dDNF and proof complexities for classical and non-classical tautologies in the systems, based on dDNF, and in some other proof systems. An extended abstract of some parts of this paper appeared as Chubaryan, An., Chubaryan, Arm. and Abajyan (2011)
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