Intuitionistic implication makes model checking hard

Mundhenk, Martin; Weiss, Felix

doi:10.2168/lmcs-8(2:3)2012

Cited by 3 publications

(8 citation statements)

References 24 publications

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“…The model checking problem-i.e. the problem to determine whether a given formula is satisfied by a given intuitionistic Kripke model-for IPC is P-complete [13], even for the fragment with two variables only [14]. More surprisingly, for the fragment with one variable IPC 1 we show the model checking problem to be AC 1 -complete.…”

Section: Introductionmentioning

confidence: 71%

The model checking problem for intuitionistic propositional logic with one variable is AC1-complete

Mundhenk¹,

Weiss²

2010

Preprint

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Section: Introductionmentioning

confidence: 71%

The model checking problem for intuitionistic propositional logic with one variable is AC1-complete

Mundhenk¹,

Weiss²

2010

Preprint

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“…It is interesting to notice that the complexity results for IPC and for KC with at least two variables are the same for the model checking problem [18]. But for the fragments with one variable, the complexity of IPC 1 is higher than that of KC 1 .…”

Section: Lower and Upper Boundsmentioning

confidence: 91%

“…The model checking problem-i.e. the problem to determine whether a given formula is satisfied by a given intuitionistic Kripke model-for IPC is P-complete [17], even for the fragment with two variables only [18]. More surprisingly, for the fragment with one variable IPC 1 we show the model checking problem to be AC 1 -complete.…”

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confidence: 83%

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An AC 1-complete model checking problem for intuitionistic logic

Mundhenk

Wei

2013

comput. complex.

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Abstract. We show that the model checking problem for intuitionistic propositional logic with one variable is complete for logspace-uniform AC 1 . The basic tool we use is the connection between intuitionistic logic and Heyting algebras, investigating its complexity theoretical aspects. For superintuitionistic logics with one variable, we obtain NC 1 -completeness for the model checking problem.Key words. complexity, intuitionistic logic, model checking, AC 1 , Heyting algebra AMS subject classifications. 03B20, 06D20, 68Q171. Introduction. Intuitionistic logic (see e.g. [10,28]) is a part of classical logic that can be proven using constructive proofs-e.g. by proofs that do not use reductio ad absurdum. For example, the law of the excluded middle a ∨ ¬a and the weak law of the excluded middle ¬a ∨ ¬¬a do not have constructive proofs and are not valid in intuitionistic logic. Not surprisingly, constructivism has its costs. Whereas the validity problem is coNP-complete for classical propositional logic [6], for intuitionistic propositional logic it is PSPACE-complete [24,25]. The computational hardness of intuitionistic logic is already reached with the fragment that has only formulas with two variables: the validity problem for this fragment is already PSPACE-complete [22]. Recall that every fragment of classical propositional logic with a fixed number of variables has an NC 1 -complete validity problem (follows from [2]).The most common semantics for intuitionistic logic are Heyting semantics [11] and Kripke semantics [14, 13]-see also [23, Chap. 2]. The Heyting semantics bases on Heyting algebras, and Kripke semantics bases on directed graphs that can straightforwardly be adapted to model state-transition systems. Therefore it is used as the standard semantics for modal and hybrid logics. Model checking as we do with Kripke models is not suited for Heyting algebras. Therefore we also use Kripke semantics for intuitionistic logic. All our and all mentioned complexity results below refer to Kripke semantics.In this paper, we consider the complexity of intuitionistic propositional logic IPC with one variable. The model checking problem-i.e. the problem to determine whether a given formula is satisfied by a given intuitionistic Kripke model-for IPC is P-complete [17], even for the fragment with two variables only [18]. More surprisingly, for the fragment with one variable IPC 1 we show the model checking problem to be AC 1 -complete. To our knowledge, this is the first "natural" AC 1 -complete problem, whereas formerly known AC 1 -complete problems (see e.g. [1]) have some explicit logarithmic bound in the problem definition. A basic ingredient for the AC 1 -completeness lies in normal forms for models and formulas as found by Nishimura [20], that we

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“…Whereas the complexity of validity and evaluation for primal logic is maximal already for atom-free formulas, for intuitionistic logic the maximal complexity is reached with two atoms, and the cases without atoms resp. with one atom have lower complexity [4][5][6].…”

Section: Introductionmentioning

confidence: 99%