Herbrand’s Theorem for Prenex Gödel Logic and Its Consequences for Theorem Proving

Baaz, Matthias; Ciabattoni, Agata; Fermüller, Christian G.

doi:10.1007/3-540-45653-8_14

Cited by 26 publications

(31 citation statements)

References 22 publications

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“…Monadic prenex G [0,1] was shown to be undecidable in [2]. This result is generalized below, where we show that in fact for all infinite V , monadic G V is undecidable, even when restricted to prenex formulas.…”

Section: Undecidability Of Infinite-valued Gödel Logicsmentioning

confidence: 51%

Monadic Fragments of Gödel Logics: Decidability and Undecidability Results

Baaz

Ciabattoni

Fermüller

Logic for Programming, Artificial Intelligence, and Reasoning

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Abstract. The monadic fragments of first-order Gödel logics are investigated. It is shown that all finite-valued monadic Gödel logics are decidable; whereas, with the possible exception of one (G ↑ ), all infinitevalued monadic Gödel logics are undecidable. For the missing case G ↑ the decidability of an important sub-case, that is well motivated also from an application oriented point of view, is proven. A tight bound for the cardinality of finite models that have to be checked to guarantee validity is extracted from the proof. Moreover, monadic G ↑ , like all other infinite-valued logics, is shown to be undecidable if the projection operator is added, while all finite-valued monadic Gödel logics remain decidable with .

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Section: Undecidability Of Infinite-valued Gödel Logicsmentioning

confidence: 51%

Monadic Fragments of Gödel Logics: Decidability and Undecidability Results

Baaz

Ciabattoni

Fermüller

Logic for Programming, Artificial Intelligence, and Reasoning

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“…The literature contains various analytic calculi for G, see, e.g., [58,1,36,17,43,7,12,39]. Among them, several calculi are better suited for proof search than hypersequent calculi.…”

Section: Resultsmentioning

confidence: 99%

“…Among them, several calculi are better suited for proof search than hypersequent calculi. This holds in particular for sequent of relations calculi [17,13], goal-oriented proof procedures [43], the systems recently defined in [7,8,39] or the resolution-style chaining calculi used in [12]. However, the mentioned calculi cannot be modified in a simple way to include quantifiers, modalities or to formalize related logics.…”

Section: Resultsmentioning

confidence: 99%

Hypersequent Calculi for Godel Logics -- a Survey

Baaz¹,

Ciabattoni²,

Fermüller³

2003

Journal of Logic and Computation

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Hypersequent calculi arise by generalizing standard sequent calculi to refer to whole contexts of sequents instead of single sequents. We present a number of results using hypersequents to obtain a Gentzen-style characterization for the family of Gödel logics. We first describe analytic calculi for propositional finite and infinite-valued Gödel logics. We then show that the framework of hypersequents allows one to move straightforwardly from the propositional level to first-order as well as propositional quantification. A certain type of modalities, enhancing the expressive power of Gödel logic, is also considered.

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“…3 The end-hypersequent H σ of the HG-proof σ that forms the input of hyperCERES can be of two forms: either it contains only weak quantifier occurrences or it consists of prenex formulas only. 4 In the latter case we have to Skolemize the proof first (step 1) and de-Skolemize it after cut elimination (step 7): 5. apply θ to the reduced proofs R 1 (σ ), .…”

Section: Overview Of Hyperceresmentioning

confidence: 99%