ON NON-MONOTONOUS PROPERTIES OF SOME CLASSICAL AND NONCLASSICAL PROPOSITIONAL PROOF SYSTEMS

2020 ◽  
Vol 54 (3 (253)) ◽  
pp. 127-136
Author(s):  
Anahit A. Chubaryan ◽  
Arsen A. Hambardzumyan
Keyword(s):  
Cut Rule ◽  
Proof Systems ◽  
Nonclassical Logics ◽  
Frege Systems ◽  
Sequent Systems ◽  
Propositional Proof Systems ◽  
Propositional Proof

We investigate the relations between the proof lines of non-minimal tautologies and its minimal tautologies for the Frege systems, the sequent systems with cut rule and the systems of natural deductions of classical and nonclassical logics. We show that for these systems there are sequences of tautologies ψn, every one of which has unique minimal tautologies φn such that for each n the minimal proof lines of φn are an order more than the minimal proof lines of ψn.

Disjoint NP-Pairs and Propositional Proof Systems

2014 ◽  
Vol 45 (4) ◽  
pp. 59-75 ◽  
Author(s):  
C. Glaßer ◽  
A. Hughes ◽  
A. L. Selman ◽  
N. Wisiol
Keyword(s):  
Proof Systems ◽  
Propositional Proof Systems ◽  
Propositional Proof

The relative efficiency of propositional proof systems

1979 ◽  
Vol 44 (1) ◽  
pp. 36-50 ◽  
Author(s):  
Stephen A. Cook ◽  
Robert A. Reckhow
Keyword(s):  
Relative Efficiency ◽  
Natural Deduction ◽  
Type Systems ◽  
Linear Increase ◽  
Proof System ◽  
Proof Systems ◽  
System A ◽  
Frege Systems ◽  
Propositional Proof Systems ◽  
Hilbert Type

We are interested in studying the length of the shortest proof of a propositional tautology in various proof systems as a function of the length of the tautology. The smallest upper bound known for this function is exponential, no matter what the proof system. A question we would like to answer (but have not been able to) is whether this function has a polynomial bound for some proof system. (This question is motivated below.) Our results here are relative results.In §§2 and 3 we indicate that all standard Hilbert type systems (or Frege systems, as we call them) and natural deduction systems are equivalent, up to application of a polynomial, as far as minimum proof length goes. In §4 we introduce extended Frege systems, which allow introduction of abbreviations for formulas. Since these abbreviations can be iterated, they eliminate the need for a possible exponential growth in formula length in a proof, as is illustrated by an example (the pigeonhole principle). In fact, Theorem 4.6 (which is a variation of a theorem of Statman) states that with a penalty of at most a linear increase in the number of lines of a proof in an extended Frege system, no line in the proof need be more than a constant times the length of the formula proved.

Propositional proof systems, the consistency of first order theories and the complexity of computations

1989 ◽  
Vol 54 (3) ◽  
pp. 1063-1079 ◽  
Author(s):  
Jan Krajíček ◽  
Pavel Pudlák
Keyword(s):  
First Order ◽  
Proof Systems ◽  
Propositional Proof Systems ◽  
Propositional Proof

AbstractWe consider the problem about the length of proofs of the sentences saying that there is no proof of contradiction in S whose length is < n. We show the relation of this problem to some problems about propositional proof systems.

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