Intuitionistic implication without disjunction

Lavalette, Gerard R. Renardel de; Hendriks, A.; Jongh, Dick de

doi:10.1093/logcom/exq058

Cited by 16 publications

(12 citation statements)

References 18 publications

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“…The equation (2) says that b → c is the maximum of {a ∈ A | a∧b ≤ c}; therefore, a lattice admits at most one "Heyting implication", i.e., an operation → such that it becomes a Heyting algebra. The lattices underlying Heyting algebras are always distributive (in fact, for any a ∈ A, the function b → a ∧ b preserves any join that exists in A, since it is a lower adjoint).…”

Section: ⊓ ⊔mentioning

confidence: 99%

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Duality and Universal Models for the Meet-Implication Fragment of IPC

Bezhanishvili

Coumans

Gool

et al. 2015

Logic, Language, and Computation

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Abstract. In this paper we investigate the fragment of intuitionistic logic which only uses conjunction (meet) and implication, using finite duality for distributive lattices and universal models. We give a description of the finitely generated universal models of this fragment and give a complete characterization of the up-sets of Kripke models of intuitionistic logic which can be defined by meet-implication-formulas. We use these results to derive a new version of subframe formulas for intuitionistic logic and to show that the uniform interpolants of meet-implication-formulas are not necessarily uniform interpolants in the full intuitionistic logic.

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Section: ⊓ ⊔mentioning

confidence: 99%

“…The induction hypothesis is that, for all proper subsets V U , the formula s(ϕ V ) defines V . We distinguish two cases: (1) U has a one minimal point; (2) U has more than one minimal point.…”

Section: ⊓ ⊔mentioning

confidence: 99%

Duality and Universal Models for the Meet-Implication Fragment of IPC

Bezhanishvili

Coumans

Gool

et al. 2015

Logic, Language, and Computation

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“…The fragments of IPC with a restricted number of variables and → as the only connective have finitely many equivalence classes of formulas and models [27,8]. The equivalence class of a given formula can be obtained with the resources of NC 1 , using a technique from Buss [2].…”

Section: Lower and Upper Boundsmentioning

confidence: 99%

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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“…The fragments of intuitionistic logic with double negation were also studied, e.g., in [2,10,14] as well as in [5], where ¬¬ was treated as a modal operator. Our construction works for some of these fragments, e.g., for (→, ¬¬) and (∧, →, ¬¬), where as the equivalent algebraic semantics one can take Hilbert algebras with regularization (HI r ) and Brouwerian semilattices with regularization (B r ), respectively, i.e., Hilbert algebras or Brouwerian semilattices endowed with a retraction r into the set of regular elements that is simultaneously a closure operator with respect to the natural order.…”

Section: Algebraic Semantics For Ipc →¬¬ and Ipc ∧→¬¬mentioning

confidence: 99%