Living without Beth and Craig: Definitions and Interpolants in the Guarded and Two-Variable Fragments

In logics with the Craig interpolation property (CIP) the existence of an interpolant for an implication follows from the validity of the implication. In logics with the projective Beth definability property (PBDP), the existence of an explicit definition of a relation follows from the validity of a...

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Vydané v:Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science s. 1 - 14
Hlavní autori: Jung, Jean Christoph, Wolter, Frank
Médium: Konferenčný príspevok..
Jazyk:English
Vydavateľské údaje: IEEE 29.06.2021
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Complexity theory
Computer science
Interpolation
Upper bound
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Abstract In logics with the Craig interpolation property (CIP) the existence of an interpolant for an implication follows from the validity of the implication. In logics with the projective Beth definability property (PBDP), the existence of an explicit definition of a relation follows from the validity of a formula expressing its implicit definability. The two-variable fragment, FO 2 , and the guarded fragment, GF, of first-order logic both fail to have the CIP and the PBDP. We show that nevertheless in both fragments the existence of interpolants and explicit definitions is decidable. In GF, both problems are 3EXPTIME-complete in general, and 2EXPTIME-complete if the arity of relation symbols is bounded by a constant c ≥ 3. In FO 2 , we prove a CON2EXPTIME upper bound and a 2EXPTIME lower bound for both problems. Thus, both for GF and FO 2 existence of interpolants and explicit definitions are decidable but harder than validity (in case of FO 2 under standard complexity assumptions).
AbstractList In logics with the Craig interpolation property (CIP) the existence of an interpolant for an implication follows from the validity of the implication. In logics with the projective Beth definability property (PBDP), the existence of an explicit definition of a relation follows from the validity of a formula expressing its implicit definability. The two-variable fragment, FO 2 , and the guarded fragment, GF, of first-order logic both fail to have the CIP and the PBDP. We show that nevertheless in both fragments the existence of interpolants and explicit definitions is decidable. In GF, both problems are 3EXPTIME-complete in general, and 2EXPTIME-complete if the arity of relation symbols is bounded by a constant c ≥ 3. In FO 2 , we prove a CON2EXPTIME upper bound and a 2EXPTIME lower bound for both problems. Thus, both for GF and FO 2 existence of interpolants and explicit definitions are decidable but harder than validity (in case of FO 2 under standard complexity assumptions).
Author Jung, Jean Christoph
Wolter, Frank
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  surname: Wolter
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  email: wolter@liverpool.ac.uk
  organization: University of Hildesheim
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Snippet In logics with the Craig interpolation property (CIP) the existence of an interpolant for an implication follows from the validity of the implication. In...
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SubjectTerms Complexity theory
Computer science
Interpolation
Upper bound
Title Living without Beth and Craig: Definitions and Interpolants in the Guarded and Two-Variable Fragments
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