Intuitionistic Frege systems are polynomially equivalent

In a paper by Cook and Reckhow (1979), it is shown that any two classical Frege systems polynomially simulate each other. The same proof does not work for intuitionistic Frege systems, since they can have nonderivable admissible rules. (The rule A/B is derivable if the formula A → B is derivable. The rule A/B is admissible if for all substitutions σ, if σ(A) is derivable, then σ(B) is derivable.) In this paper, we polynomially simulate a single admissible rule. Therefore any two intuitionistic Frege systems polynomially simulate each other. Bibliography: 20 titles.

[1]  P. Pudlák.  Sets and Proofs: On the Complexity of the Propositional Calculus , 1999 .

[2]  Ronald Harrop,et al.  Concerning formulas of the types A→B ν C,A →(Ex)B(x) in intuitionistic formal systems , 1960, Journal of Symbolic Logic.

[3]  Jeffery I. Zucker,et al.  The adequacy problem for inferential logic , 1978, J. Philos. Log..

[4]  G. Mints.  A Short Introduction to Intuitionistic Logic , 2000 .

[5]  Stephen A. Cook,et al.  The Relative Efficiency of Propositional Proof Systems , 1979, Journal of Symbolic Logic.

[6]  G. Kreisel,et al.  Constructivist Approaches to Logic , 1981 .

[7]  Hilary Putnam,et al.  Eine Unableitbarkeitsbeweismethode für den Intuitionistischen Aussagenkalkül , 1957, Arch. Math. Log..

[8]  G. Mints,et al.  Derivability of admissible rules , 1976 .

[9]  Silvio Ghilardi,et al.  Unification in intuitionistic logic , 1999, Journal of Symbolic Logic.

[10]  Peter Schroeder-Heister,et al.  A natural extension of natural deduction , 1984, Journal of Symbolic Logic.

[11]  V. P. Orevkov.  Complexity of Proofs and Their Transformations in Axiomatic Theories , 1993 .

[12]  Harvey M. Friedman,et al.  One hundred and two problems in mathematical logic , 1975, Journal of Symbolic Logic.

[13]  Rosalie Iemhoff,et al.  On the admissible rules of intuitionistic propositional logic , 2001, Journal of Symbolic Logic.

[14]  Samuel R. Buss,et al.  The Complexity of the Disjunction and Existential Properties in Intuitionistic Logic , 1999, Ann. Pure Appl. Log..

[15]  Mauro Ferrari,et al.  On the Complexity of Disjunction and Explicit Definability Properties in Some Intermediate Logics , 2002, LPAR.

[16]  Samuel R. Buss,et al.  On the computational content of intuitionistic propositional proofs , 2001, Ann. Pure Appl. Log..