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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| Vydáno v: | Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science s. 1 - 14 |
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29.06.2021
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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). |
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| 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 |
| Author_xml | – sequence: 1 givenname: Jean Christoph surname: Jung fullname: Jung, Jean Christoph email: jungj@uni-hildesheim.de organization: University of Hildesheim – sequence: 2 givenname: Frank surname: Wolter fullname: Wolter, Frank 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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