Hierarchies of Partially Ordered Connectives and Quantifiers

Krynicki, Michał

doi:10.1002/malq.19930390134

Cited by 13 publications

(7 citation statements)

References 8 publications

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“…, n}, we have that Q i = ∃; all other quantifiers are universal quantifiers. Using a result of Krynicki's [16] it is not so hard to see that Σ 1 1 ♣ = NP on finite structures. Krynicki showed, namely, that first-order logic prefixed by the quantifier below (with unbound k) coincides with full Σ …”

Section: Theorem 3 ([7 11]) For Every Integer M the Following Are mentioning

confidence: 99%

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Partially Ordered Connectives and ∑1 1 on Finite Models

Sevenster

Tulenheimo

2006

Logical Approaches to Computational Barriers

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Abstract. In this paper we take up the study of Henkin quantifiers with boolean variables [4] also known as partially ordered connectives [19]. We consider first-order formulae prefixed by partially ordered connectives, denoted D, on finite structures. We characterize D as a fragment of second-order existential logic Σ 1 1 ♥ whose formulae do not allow for existential variables being argument of predicate variables. We show that Σ 1 1 ♥ harbors a strict hierarchy induced by the arity of predicate variables and that it is not closed under complementation, by means of a game-theoretical argument. Admitting for at most one existential variable to appear as the argument of a predicate variable already yields a logic coinciding with full Σ 1 1 , thus we show.

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Section: Theorem 3 ([7 11]) For Every Integer M the Following Are mentioning

confidence: 99%

“…Other publications on Henkin quantifiers and partially ordered connectives in relation with complexity theory include [13,14,[16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Partially Ordered Connectives and ∑1 1 on Finite Models

Sevenster

Tulenheimo

2006

Logical Approaches to Computational Barriers

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“…In this section we show that the prenex action recall fragment IF p,r AR(∃) has the same expressive power as the full IF logic. In the proof of this result we exploit the fact that existential second-order logic is captured by Henkin quantifiers with two rows: Theorem 1 ( [14]). For any ESO sentence there is an equivalent sentence of the form…”

Section: Explicit Definition Of Henkin Quantifiers By Signallingmentioning

confidence: 99%

“…In Section 4 we will show that the Henkin prefixes H n 2 are explicitly definable in the prenex, regular fragment of action recall (therefore, by means of signalling). The H n 2 prefixes, taken together, are known to capture all ESO definable concepts ( [14]); therefore, the prenex, regular fragment of action recall suffices for full IF expressive power.…”

Section: Introductionmentioning

confidence: 99%

Independence-Friendly Logic Without Henkin Quantification

Barbero

Hella

Rönnholm

2017

Logic, Language, Information, and Computation

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We analyze from a global point of view the expressive resources of IF logic that do not stem from Henkin (partially-ordered) quantification. When one restricts attention to regular IF sentences, this amounts to the study of the fragment of IF logic which is individuated by the game-theoretical property of Action Recall. We prove that the fragment of Action Recall can express all existential second-order (ESO) properties. This can be accomplished already by the prenex fragment of Action Recall, whose only second-order source of expressiveness are the so-called signalling patterns. The proof shows that a complete set of Henkin prefixes is explicitly definable in the fragment of Action Recall. In the more general case, in which also irregular IF sentences are allowed, we show that full ESO expressive power can be achieved using neither Henkin nor signalling patterns.

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“…We will show that the Henkin prefixes H n 2 are explicitly definable in the prenex, regular fragment of AR (therefore, by means of signalling). The H n 2 prefixes, taken together, are known to capture all ESO definable concepts [20]; thus, the prenex, regular fragment of action recall suffices for full IF expressive power. We also present a simpler translation of the H n 2 prefixes into the non-prenex fragment of AR without signalling patterns; in this case, the relevant source of expressiveness is a form of "signalling by disjunction".…”

Section: Introductionmentioning

confidence: 99%

Independence-friendly logic without Henkin quantification

2021

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We analyze the expressive resources of $$\mathrm {IF}$$ IF logic that do not stem from Henkin (partially-ordered) quantification. When one restricts attention to regular $$\mathrm {IF}$$ IF sentences, this amounts to the study of the fragment of $$\mathrm {IF}$$ IF logic which is individuated by the game-theoretical property of action recall (AR). We prove that the fragment of prenex AR sentences can express all existential second-order properties. We then show that the same can be achieved in the non-prenex fragment of AR, by using “signalling by disjunction” instead of Henkin or signalling patterns. We also study irregular IF logic (in which requantification of variables is allowed) and analyze its correspondence to regular IF logic. By using new methods, we prove that the game-theoretical property of knowledge memory is a first-order syntactical constraint also for irregular sentences, and we identify another new first-order fragment. Finally we discover that irregular prefixes behave quite differently in finite and infinite models. In particular, we show that, over infinite structures, every irregular prefix is equivalent to a regular one; and we present an irregular prefix which is second order on finite models but collapses to a first-order prefix on infinite models.

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Hierarchies of Partially Ordered Connectives and Quantifiers

Cited by 13 publications

References 8 publications

Partially Ordered Connectives and ∑1 1 on Finite Models

Partially Ordered Connectives and ∑1 1 on Finite Models

Independence-Friendly Logic Without Henkin Quantification

Independence-friendly logic without Henkin quantification

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