Superposition Decides the First-Order Logic Fragment Over Ground Theories

Kruglov, Evgeny; Weidenbach, Christoph

doi:10.1007/s11786-012-0135-4

Cited by 15 publications

(28 citation statements)

References 23 publications

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“…What we say there also applies to other developments rooted in that calculus, [1, e. g.]. The specialized version of hierarchic superposition in [20] will be discussed in Sect. 9 below.…”

Section: Introductionmentioning

confidence: 90%

“…It is rather evident, however, that this condition is sometimes stronger than needed. For instance, if the set of input clauses N is ground, then we only have to consider the ground BG-sorted FG terms that actually occur in N [20] (analogously to the Nelson-Oppen combination procedure). A relaxation of sufficient completeness that is also useful for non-ground clauses and that still ensures refutational completeness was given by Kruglov [19]: The example demonstrates that local sufficient completeness is significantly more powerful than sufficient completeness, but this comes at a price.…”

Section: Local Sufficient Completenessmentioning

confidence: 99%

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Hierarchic Superposition Revisited

Baumgartner

Waldmann

2019

Lecture Notes in Computer Science

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Many applications of automated deduction require reasoning in first-order logic modulo background theories, in particular some form of integer arithmetic. A major unsolved research challenge is to design theorem provers that are "reasonably complete" even in the presence of free function symbols ranging into a background theory sort. The hierarchic superposition calculus of Bachmair, Ganzinger, and Waldmann already supports such symbols, but, as we demonstrate, not optimally. This paper aims to rectify the situation by introducing a novel form of clause abstraction, a core component in the hierarchic superposition calculus for transforming clauses into a form needed for internal operation. We argue for the benefits of the resulting calculus and provide two new completeness results: one for the fragment where all background-sorted terms are ground and another one for a special case of linear (integer or rational) arithmetic as a background theory. ⋆ The final authenticated version is available online at https://doi.org/TBD.

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“…What we say there also applies to other developments rooted in that calculus, [1, e. g.]. The specialized version of hierarchic superposition in [20] will be discussed in Sect. 9 below.…”

Section: Introductionmentioning

confidence: 90%

Section: Local Sufficient Completenessmentioning

confidence: 99%

Hierarchic Superposition Revisited

Baumgartner

Waldmann

2019

Lecture Notes in Computer Science

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“…Nevertheless, the SUP(LA) calculus is a decision procedure for the FOL(LA) ground case [21] and for the FOL(LA) fragment resulting from the translation of timed automata [15]. In this paper we extend the latter result to the fragment corresponding to the translation of timed automata extended with unbounded integer variables.…”

Section: Introductionmentioning

confidence: 92%

Automatic Generation of Invariants for Circular Derivations in SUP(LA)

Fietzke

Kruglov

Weidenbach

2012

Logic for Programming, Artificial Intelligence, and Reasoning

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The hierarchic combination of linear arithmetic and first-order logic with free function symbols, FOL(LA), results in a strictly more expressive logic than its two parts. The SUP(LA) calculus can be turned into a decision procedure for interesting fragments of FOL(LA). For example, reachability problems for timed automata can be decided by SUP(LA) using an appropriate translation into FOL(LA). In this paper, we extend the SUP(LA) calculus with an additional inference rule, automatically generating inductive invariants from partial SUP(LA) derivations. The rule enables decidability of more expressive fragments, including reachability for timed automata with unbounded integer variables. We have implemented the rule in the SPASS(LA) theorem prover with promising results, showing that it can considerably speed up proof search and enable termination of saturation for practically relevant problems.

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“…Sufficient completeness of a set of Σ-clauses is a property that is not even recursively enumerable. For certain classes of Σ-clause sets, however, it is possible to establish a variant of sufficient completeness automatically [11,7] See [7] for the corresponding Define inference rule.…”

Section: Hierarchic Theorem Provingmentioning

confidence: 99%