The decidability of one-variable propositional calculi.

Gladstone, M. D.

doi:10.1305/ndjfl/1093882551

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“…A natural and interesting question arises with respect to this generalization: there is an enumerable set of propositional formulas M for which the condition [P 0 ] ∩ M = ∅ holds if and only if Axm and Cmpl are undecidable for P 0 . Since Gladstone in [7] proved that Drv is decidable for every one-variable propositional calculus, it seems to be interesting to consider only formulas containing two or more variables. Theorem 5.1 shows that twovariables formulas are sufficient.…”

Section: Conclusion and Further Researchmentioning

confidence: 99%

Undecidable problems for propositional calculi with implication

Bokov¹

2016

Logic Jnl IGPL

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In this article, we deal with propositional calculi over a signature containing the classical implication → with the rules of modus ponens and substitution. For these calculi we consider few recognizing problems such as recognizing derivations, extensions, completeness, and axiomatizations. The main result of this paper is to prove that the problem of recognizing extensions is undecidable for every propositional calculus, and the problems of recognizing axiomatizations and completeness are undecidable for propositional calculi containing the formula x → (y → x). As a corollary, the problem of derivability of a fixed formula A is also undecidable for all A. Moreover, we give a historical survey of related results.

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Section: Conclusion and Further Researchmentioning

confidence: 99%

Undecidable problems for propositional calculi with implication

Bokov¹

2016

Logic Jnl IGPL

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“…In 1976, Hughes [8] constructed an undecidable implicational propositional calculus using axioms in 2 variables. Finally, Gladstone in 1979 [7] proved that every 1-variable propositional calculus is decidable.…”

Section: Introductionmentioning

confidence: 99%

On the number of variables in undecidable superintuitionistic propositional calculi

Bokov

2016

Logic Jnl IGPL

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In this paper, we construct an undecidable 3-variable superintuitionistic propositional calculus, i.e., a finitely axiomatizable extension of the intuitionistic propositional calculus with axioms containing only 3 variables. Since there are no 2-variable superintuitionistic propositional calculi, this is the minimal possible number of variables.transforms s, m, n into t, m + 1, n , and the instruction s → t, −1, 0 / u, 0, 0 transforms s, m, n into t, m − 1, n if m > 0 and into (u, m, n) if m = 0. The meaning of the others is defined analogously.Let M be a Minsky machine, then the notation s, m, n M −→ t, k, l means that the configuration t, k, l is obtained from s, m, n by applying an instruction of machine M once. We write s, m, n M =⇒ t, k, l if the configuration t, k, l is obtained from s, m, n by applying instructions of machine M in finitely many steps (possibly, in 0 steps). Particularly, we always have s, m, n M =⇒ s, m, n . The configuration problem for a Minsky machine M and a configuration s, m, n is, given a configuration t, k, l , to determine whether s, m, n M =⇒ t, k, l .Theorem 3.2 (Minsky, [11]). There exist a Minsky machine M and a configuration s, m, n for which the configuration problem is undecidable.Let M be a Minsky machine and s 0 , m 0 , n 0 a configuration for which the configuration problem is undecidable.

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The decidability of one-variable propositional calculi.

Cited by 2 publications

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Undecidable problems for propositional calculi with implication

Undecidable problems for propositional calculi with implication

On the number of variables in undecidable superintuitionistic propositional calculi

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