Labelled Unit Superposition Calculi for Instantiation-Based Reasoning

Korovin, Konstantin; Sticksel, Christoph

doi:10.1007/978-3-642-16242-8_33

Cited by 3 publications

(5 citation statements)

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“…We considered the Inst-Gen calculus, semantic selection, hyper-inferences, redundancy elimination, dismatching constraints, simplifications by propositional reasoning, saturation strategies and finally implementation issues and evaluation. There are number of further extensions, that were not considered in this paper, such as integration of equational [19,33] and theory reasoning in the black-box style [20]. These extensions open novel opportunities to utilise efficient solvers modulo theories, SMT solvers, which have recently gained great popularity due to demand in applications such as software and hardware verification.…”

Section: Discussionmentioning

confidence: 99%

“…Our experiments show that even this naive approach of equality integration works reasonably well in the instantiation-based setting, most likely due to the semantic literal selection and absence of recombination of clauses with equality axioms. For more advanced treatment of equality based on combination of ordered unit superposition with Inst-Gen and the corresponding system iProver-Eq we refer to [32,33].…”

Section: Implementation Of Inst-gen In Iprovermentioning

confidence: 99%

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Inst-Gen – A Modular Approach to Instantiation-Based Automated Reasoning

Korovin

2013

Programming Logics

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Abstract. Inst-Gen is an instantiation-based reasoning method for first-order logic introduced in [18]. One of the distinctive features of Inst-Gen is a modular combination of first-order reasoning with efficient ground reasoning. Thus, Inst-Gen provides a framework for utilising efficient off-the-shelf propositional SAT and SMT solvers as part of general first-order reasoning. In this paper we present a unified view on the developments of the Inst-Gen method: (i) completeness proofs; (ii) abstract and concrete criteria for redundancy elimination, including dismatching constraints and global subsumption; (iii) implementation details and evaluation.

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Section: Discussionmentioning

confidence: 99%

Section: Implementation Of Inst-gen In Iprovermentioning

confidence: 99%

Inst-Gen – A Modular Approach to Instantiation-Based Automated Reasoning

Korovin

2013

Programming Logics

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“…In a similar way as for the EPR fragment it is easy to see that instance based methods are also decision procedures for the non-cyclic fragment. We formulate this as a theorem for Inst-Gen [15,20] and Inst-Gen-Eq [16,22] but it also holds for other instantiation based methods such as Model Evolution. InstGen is an instantiation-based method, complete for first-order logic and Inst-Gen-Eq is its extension with superposition-based equational reasoning.…”

Section: Definitionmentioning

confidence: 99%

Non-cyclic Sorts for First-Order Satisfiability

Korovin

2013

Frontiers of Combining Systems

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Abstract. In this paper we investigate the finite satisfiability problem for firstorder logic. We show that the finite satisfiability problem can be represented as a sequence of satisfiability problems in a fragment of many-sorted logic, which we call the non-cyclic fragment. The non-cyclic fragment can be seen as a generalisation of the effectively propositional fragment (EPR) in the many-sorted setting. We show that the non-cyclic fragment is decidable by instantiation-based methods and present a linear time algorithm for checking whether a given clause set is in this fragment. One of the distinctive features of our finite satisfiability translation is that it avoids unnecessary flattening of terms, which can be crucial for efficiency. We implemented our finite model finding translation in iProver and evaluated it over the TPTP library. Using our translation it was possible solve a large class of problems which could not be solved by other systems.

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“…Definition clauses derived in the sufficient completeness transform are also labelled by the assumption they originate from. (This idea has much in common with Labelled superposition [14] and Constrained Resolution [7]). …”

Section: Definition 32 (Labelled Clause)mentioning

confidence: 99%

“…The Inst-Gen calculus [9] uses unification to produce a set of possibly conflicting instances to pass to a Sat solver (or SMT solver). In a relevant variation, the InstGen-Eq calculus [14] uses an SMT solver to select literals of the clause set which may combine to form a model, then a version of unit superposition with labels is used to extract contradictory instances that follow from this candidate model. The Model-Evolution calculus [3] maintains a set of literals to represent the current candidate model of the clause set and also uses unification to produce possibly conflicting clauses in order to refine this model.…”

Section: Related Workmentioning

confidence: 99%

A Model Guided Instantiation Heuristic for the Superposition Calculus with Theories

Bax

EPiC Series in Computing

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Generalised Model Finding (GMF) is a quantifier instantiation heuristic for the superposition calculus in the presence of interpreted theories with finitely quantified free function symbols ranging into theory sorts. The free function symbols are approximated by finite partial function graphs along with simplifying assumptions which are iteratively refined. Here we present an outline of the GMF approach, give an improvement that addresses some efficiency issues and then present some ideas for extending it with concepts from instantiation based theorem proving. OverviewThe inclusion of interpreted theories in a theorem proving context naturally leads to completeness issues. In the classical first-order logic satisfiability problem we are at least guaranteed refutational completeness, but reasoning modulo theories and including uninterpreted function symbols is often not even semi-decidable. Many applications of theorem proving reduce to the satisfiability problem, from ranking function and loop invariant synthesis in software verification to counter-example finding. In addition to reasoning modulo theories, these applications also require the introduction of uninterpreted function symbols and quantifier reasoning.We aim to recover some completeness in the specific case where there are only finitely many ground instances of (uninterpreted) terms ranging into the interpreted theory. This is guaranteed with the assumption that all variables inside uninterpreted terms of certain sorts are quantified over finite ranges.In the following the interpreted theory will be Linear Integer Arithmetic (LIA), which we view as a particular class of models that satisfy the LIA axioms. Concretely, clauses over the LIA signature (+ 2 , − 1 , 0, s 1 , < 2 ) are evaluated using Cooper's Quantifier Elimination algorithm which decides ∃∀ quantified LIA formulas. Sorts, operators, terms and literals from the LIA theory are called background. New operators which extend the signature of the interpreted theory are called foreground or free.Define background-sorted foreground (BSFG) terms as those which have at their head an uninterpreted function symbol with a background sort.The running example will use the usual array theory with integer indices and elements, this has the operators read : Array × Z → Z, write : Array × Z × Z → Array (these are foreground operators; read is a BSFG operator) and axioms T Array = (1) read(write(A, I, X), I) ≈ X (2) read(write(A, I, X), J) ≈ read(A, J) ∨ I ≈ J * NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence Program.

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Labelled Unit Superposition Calculi for Instantiation-Based Reasoning

Cited by 3 publications

References 11 publications

Inst-Gen – A Modular Approach to Instantiation-Based Automated Reasoning

Inst-Gen – A Modular Approach to Instantiation-Based Automated Reasoning

Non-cyclic Sorts for First-Order Satisfiability

A Model Guided Instantiation Heuristic for the Superposition Calculus with Theories

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