353 research outputs found

    A CDCL-style calculus for solving non-linear constraints

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    In this paper we propose a novel approach for checking satisfiability of non-linear constraints over the reals, called ksmt. The procedure is based on conflict resolution in CDCL style calculus, using a composition of symbolical and numerical methods. To deal with the non-linear components in case of conflicts we use numerically constructed restricted linearisations. This approach covers a large number of computable non-linear real functions such as polynomials, rational or trigonometrical functions and beyond. A prototypical implementation has been evaluated on several non-linear SMT-LIB examples and the results have been compared with state-of-the-art SMT solvers.Comment: 17 pages, 3 figures; accepted at FroCoS 2019; software available at <http://informatik.uni-trier.de/~brausse/ksmt/

    Graph Sequence Learning for Premise Selection

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    Premise selection is crucial for large theory reasoning as the sheer size of the problems quickly leads to resource starvation. This paper proposes a premise selection approach inspired by the domain of image captioning, where language models automatically generate a suitable caption for a given image. Likewise, we attempt to generate the sequence of axioms required to construct the proof of a given problem. This is achieved by combining a pre-trained graph neural network with a language model. We evaluated different configurations of our method and experience a 17.7% improvement gain over the baseline.Comment: 17 page

    Heterogeneous Heuristic Optimisation and Scheduling for First-Order Theorem Proving

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    Good heuristics are essential for successful proof search in first-order automated theorem proving. As a result, state-of-the-art theorem provers offer a range of options for tuning the proof search process to specific problems. However, the vast configuration space makes it exceedingly challenging to construct effective heuristics. In this paper we present a new approach called HOS-ML, for automatically discovering new heuristics and mapping problems into optimised local schedules comprising of these heuristics. Our approach is based on interleaving Bayesian hyper-parameter optimisation for discovering promising heuristics and dynamic clustering to make optimisation efficient on heterogeneous problems. HOS-ML also use constraint programming to devise locally optimal schedules and machine learning for mapping unseen problems into such schedules. We evaluated HOS-ML on the theorem prover iProver and demonstrated that it can discover new heuristics that considerably improve performance and can solve problems that have not been solved previously by any other system.<br/

    Scavenger 0.1: A Theorem Prover Based on Conflict Resolution

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    This paper introduces Scavenger, the first theorem prover for pure first-order logic without equality based on the new conflict resolution calculus. Conflict resolution has a restricted resolution inference rule that resembles (a first-order generalization of) unit propagation as well as a rule for assuming decision literals and a rule for deriving new clauses by (a first-order generalization of) conflict-driven clause learning.Comment: Published at CADE 201

    Decidability of the Monadic Shallow Linear First-Order Fragment with Straight Dismatching Constraints

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    The monadic shallow linear Horn fragment is well-known to be decidable and has many application, e.g., in security protocol analysis, tree automata, or abstraction refinement. It was a long standing open problem how to extend the fragment to the non-Horn case, preserving decidability, that would, e.g., enable to express non-determinism in protocols. We prove decidability of the non-Horn monadic shallow linear fragment via ordered resolution further extended with dismatching constraints and discuss some applications of the new decidable fragment.Comment: 29 pages, long version of CADE-26 pape

    Delocalization of Wannier-Stark ladders by phonons: tunneling and stretched polarons

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    We study the coherent dynamics of a Holstein polaron in strong electric fields. A detailed analytical and numerical analysis shows that even for small hopping constant and weak electron-phonon interaction, polaron states can become delocalized if a resonance condition develops between the original Wannier-Stark states and the phonon modes, yielding both tunneling and `stretched' polarons. The unusual stretched polarons are characterized by a phonon cloud that {\em trails} the electron, instead of accompanying it. In general, our novel approach allows us to show that the polaron spectrum has a complex nearly-fractal structure, due to the coherent coupling between states in the Cayley tree which describes the relevant Hilbert space. The eigenstates of a finite ladder are analyzed in terms of the observable tunneling and optical properties of the system.Comment: 7 pages, 4 figure

    First-Order Logic Theorem Proving and Model Building via Approximation and Instantiation

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    In this paper we consider first-order logic theorem proving and model building via approximation and instantiation. Given a clause set we propose its approximation into a simplified clause set where satisfiability is decidable. The approximation extends the signature and preserves unsatisfiability: if the simplified clause set is satisfiable in some model, so is the original clause set in the same model interpreted in the original signature. A refutation generated by a decision procedure on the simplified clause set can then either be lifted to a refutation in the original clause set, or it guides a refinement excluding the previously found unliftable refutation. This way the approach is refutationally complete. We do not step-wise lift refutations but conflicting cores, finite unsatisfiable clause sets representing at least one refutation. The approach is dual to many existing approaches in the literature because our approximation preserves unsatisfiability

    Learning Instantiation in First-Order Logic

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    Contains fulltext : 286055.pdf (Publisher’s version ) (Open Access)AITP 202

    AC-KBO Revisited

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    Equational theories that contain axioms expressing associativity and commutativity (AC) of certain operators are ubiquitous. Theorem proving methods in such theories rely on well-founded orders that are compatible with the AC axioms. In this paper we consider various definitions of AC-compatible Knuth-Bendix orders. The orders of Steinbach and of Korovin and Voronkov are revisited. The former is enhanced to a more powerful version, and we modify the latter to amend its lack of monotonicity on non-ground terms. We further present new complexity results. An extension reflecting the recent proposal of subterm coefficients in standard Knuth-Bendix orders is also given. The various orders are compared on problems in termination and completion.Comment: 31 pages, To appear in Theory and Practice of Logic Programming (TPLP) special issue for the 12th International Symposium on Functional and Logic Programming (FLOPS 2014

    Finding Finite Models in Multi-Sorted First-Order Logic

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    This work extends the existing MACE-style finite model finding approach to multi-sorted first order logic. This existing approach iteratively assumes increasing domain sizes and encodes the related ground problem as a SAT problem. When moving to the multi-sorted setting each sort may have a different domain size, leading to an explosion in the search space. This paper focusses on methods to tame that search space. The key approach adds additional information to the SAT encoding to suggest which domains should be grown. Evaluation of an implementation of techniques in the Vampire theorem prover shows that they dramatically reduce the search space and that this is an effective approach to find finite models in multi-sorted first order logic.Comment: SAT 201
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