Andrei Voronkov

Automated Deduction—CADE-18

Description Logics are a family of class based knowledge representation formalisms characterised by the use of various constructors to build complex classes from simpler ones, and by an emphasis on the provision of sound, complete and (empirically) tractable reasoning services. They have a wide range of applications, but their use as ontology languages has been highlighted by the recent explosion of interest in the “Semantic Web”, where ontologies are set to play a key role. DAML+OIL is a description logic based ontology language specifically designed for use on the web. The logical basis of the language means that reasoning services can be provided, both to support ontology design and to make DAML+OIL described web resources more accessible to automated processes.

Path Feasibility Analysis for String-Manipulating Programs

We discuss the problem of path feasibility for programs manipulating strings using a collection of standard string library functions. We prove results on the complexity of this problem, including its undecidability in the general case and decidability of some special cases. In the context of test-case generation, we are interested in an efficient finite model finding method for string constraints. To this end we develop a two-tier finite model finding procedure. First, an integer abstraction of string constraints are passed to an SMT (Satisfiability Modulo Theories) solver. The abstraction is either unsatisfiable, or the solver produces a model that fixes lengths of enough strings to reduce the entire problem to be finite domain. The resulting fixed-length string constraints are then solved in a second phase. We implemented the procedure in a symbolic execution framework, report on the encouraging results and discuss directions for improving the method further.

First-Order Theorem Proving and Vampire

In this paper we give a short introduction in first-order theorem proving and the use of the theorem prover Vampire. We discuss the superposition calculus and explain the key concepts of saturation and redundancy elimination, present saturation algorithms and preprocessing, and demonstrate how these concepts are implemented in Vampire. Further, we also cover more recent topics and features of Vampire designed for advanced applications, including satisfiability checking, theory reasoning, interpolation, consequence elimination, and program analysis.

Finding Loop Invariants for Programs over Arrays Using a Theorem Prover

Invariants with quantifiers are important for verification and static analysis of programsover arrays due to the unbounded nature of arrays. Such invariants can expressrelationships among array elements and properties involving array and scalar variablesof the loop.This talk presents how quantified loop invariants of programs over arrayscan be automatically inferred using a first order theorem prover,reducing the burden of annotating loops with complete invariants.Unlike all previously known methods, our method is ableto generate loop invariants containing quantifier alternations

The anatomy of vampire - Implementing bottom-up procedures with code trees

We present an implementation technique for a class of bottom-up logic procedures. The technique is based oncode trees. It is intended to speed up most important and costly operations, such as subsumption and resolution. As an example, we consider the forward subsumption problem which is the bottleneck of many systems implementing first-order logic.To efficiently implement subsumption, we specialize subsumption algorithms at run time, using theabstract subsumption machine. The abstract subsumption machine makes subsumption-check using sequences of instructions that are similar to the WAM instructions. It gives an efficient implementation of the “clause at a time” subsumption problem. To implement subsumption on the “set at a time” basis, we combine sequences of instructions incode trees.We show that this technique yields a new way of indexing clauses. Some experimental results are given.The code trees technique may be used in various procedures, including binary resolution, hyper-resolution, UR-resolution, the inverse method, paramodulation and rewriting, OLDT-resolution, SLD-AL-resolution, bottom-up evaluation of logic programs, and disjunctive logic programs.

Sine Qua non for large theory reasoning

One possible way to deal with large theories is to have a good selection method for relevant axioms. This is confirmed by the fact that the largest available first-order knowledge base (the Open CYC) contains over 3 million axioms, while answering queries to it usually requires not more than a few dozen axioms. A method for axiom selection has been proposed by the first author in the Sumo INference Engine (SInE) system. SInE has won the large theory division of CASC in 2008. The method turned out to be so successful that the next two years it was used by the winner as well as by several other competing systems. This paper contains the presentation of the method and describes experiments with it in the theorem prover Vampire.

The Inverse Method

Part I. History Chapter 1. The Early History of Automated Deduction (Martin Davis). 1. Presburgers Procedure. 2. Newell, Shaw & Simon, and H. Gelernter. 3. First-Order Logic. Bibliography. Index. Part II. Classical Logic. Chapter 2. Resolution Theorem Proving (Leo Bachmair, Harald Ganzinger). 1. Introduction. 2. Preliminaries. 3. Standard Resolution. 4. A Framework for Saturation-Based Theorem Proving. 5. General Resolution. 6. Basic Resolution Strategies. 7. Refined Techniques for Defining Orderings and Selection Functions. 8. Global Theorem Proving Methods. 9. First-Order Resolution Methods. 10. Effective Saturation of First-Order Theories. 11. Concluding Remarks. Bibliography. Index. Chapter 3. Tableaux and Related Methods (Reiner Hahnle). 1. Introduction. 2. Preliminaries. 3. The Tableau Method. 4. Clause Tableaux. 5. Tableaux as a Framework. 6. Comparing Calculi. 7. Historical Remarks & Resources Bibliography. Notation. Index. Chapter 4. The Inverse Method (Anatoli Degtyarev, Andrei Voronkov). 1. Introduction. 2. Preliminaries. 3. Cooking classical logic. 4. Applying the recipe to nonclassical logics. 5. Naming and connections with resolution. 6. Season your meal: strategies and redundancies 7. Path calculi 8. Logics without the contraction rules 9. Conclusion Bibliography. Index Chapter 5. Normal Form Transformations (Matthias Baaz, Uwe Egly, Alexander Leitsch). 1. Introduction. 2. Notation and Definitions. 3. On the Concept of Normal Form. 4. Equivalence-Preserving Normal Forms. 5. Skolem Normal Form. 6. Conjunctive Normal Form. 7. Normal Forms in Nonclassical Logics. 8. Conclusion. Bibliography. Index. Chapter 6. Computing Small Clause Normal Forms (Andreas Nonnengart, Christoph Weidenbach). 1. Introduction. 2. Preliminaries. 3. Standard CNF-Translation. 4. Formula Renaming. 5. Skolemization. 6. Simplification. 7. Bibliographic Notes. 8. Implementation Notes. Bibliography. Index. Part III. Equality and other theories. Chapter 7. Paramodulation-Based Theorem Proving (Robert Nieuwenhuis, Albert Rubio). 1. About this chapter. 2. Preliminaries. 3. Paramodulation calculi. 4. Saturation procedures. 5. Paramodulation with constrained clauses. 6. Paramodulation with built-in equational theories. 7. Symbolic constraint solving. 8. Extensions. 9. Perspectives. Bibliography. Index. Chapter 8. Unification Theory (Franz Baader, Wayne Snyder). 1. Introduction. 2. Syntactic unification. 3. Equational unification. 4. Syntactic methods for E-unification. 5. Semantic approaches to E-unification. 6. Combination of unification algorithms. 7. Further topics. Bibliography. Index. Chapter 9. Rewriting (Nachum Dershowitz, David A. Plaisted). 1. Introduction. 2. Terminology. 3. Normal Forms and Validity. 4. Termination Properties. 5. Church-Rosser Properties. 6. Completion. 7. Relativized Rewriting. 8. Equational Theorem Proving. 9. Conditional Rewriting. 10. Programming. Bibliography. Index. Chapter 10. Equality Reasoning in Sequent-Based Calculi (Anatoli Degtyarev, Andrei Voronkov). 1. Introduction. 2. Translation of logic with equality into logic without equality. 3. Free variable systems. 4. Early history. 5. Simultaneous rigid E-unification. 6. Incomplete procedures for rigid E-unification. 7. Sequent-based calculi and paramodulation. 8. Equality elimination. 9. Equality reasoning in nonclassical logics. 10. Conclusion and open problems. Bibliography. Calculi and inference rules. Index. Chapter 11. Automated Reasoning in Geometry (Shang-Ching Chou, Xiao-Shan Gao). 1. A history review of automated reasoning in geometry. 2. Algebraic approaches to automated reasoning in geometry. 3. Coordinate-free approaches to automated reasoning in geometry. 4. AI approaches to automated reasoning in geometry. 5. Final remarks. Bibliography. Index. Chapter 12. Solving Numerical Constraints (Alexander Bockmayr, Volker Weispfenning). 1. Introduction. 2. Linear constraints over fields. 3. Linear diophantine constraints. 4. Non-linear constraints over continuous domains. 5. Non-linear diophantine constraints. Bibliography. Index. Part IV. Induction. Chapter 13. The Automation of Proof by Mathematical Induction (Alan Bundy). 1. Introduction. 2. Induction Rules. 3. Recursive Definitions and Datatypes. 4. Inductive Proof Techniques. 5. Theoretical Limitations of Inductive Inference. 6. Special Search Control Problems. 7. Rippling. 8. The Productive Use of Failure. 9. Existential Theorems. 10. Interactive Theorem Proving. 11. Inductive Theorem Provers. 12. Conclusion. Bibliography. Main Index. Name Index. Chapter 14. Inductionless Induction (Hubert Comon). 1. Introduction. 2. Formal background. 3. General Setting of the Inductionless Induction Method. 4. Inductive completion methods. 5. Examples of Axiomatizations A from the Literature. 6. Ground Reducibility. 7. A comparison between inductive proofs and proofs by consistency. Bibliography. Index.

Integrating linear arithmetic into superposition calculus

We present a method of integrating linear rational arithmetic into superposition calculus for first-order logic. One of our main results is completeness of the resulting calculus under some finiteness assumptions.

Limited resource strategy in resolution theorem proving

For most applications of first-order theorem provers a proof should be found within a fixed time limit. When the time limit is set, systems can perform much better by using algorithms other than the ordinary complete ones. In this paper we describe the limited resource strategy (LRS) intended to improve performance of the OTTER saturation algorithm when a fixed limit is imposed on the time of a run. The strategy is adaptive in the following sense: it adjusts the limit on the weight of clauses according to some statistics collected on the earlier stages of proof search. We give experimental evidence that the LRS gives a considerable improvement over the OTTER saturation algorithm. We also show that it is superior to the DISCOUNT algorithm, which does not use passive clauses for simplification, and to the non-adaptive weight-based algorithms.

Interpolation and Symbol Elimination

We prove several results related to local proofs, interpolation and superposition calculus and discuss their use in predicate abstraction and invariant generation. Our proofs and results suggest that symbol-eliminating inferences may be an interesting alternative to interpolation.