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Automated Reasoning: Past Story and New Trends*
"... We overview the development of firstorder automated reasoning systems starting from their early years. Based on the analysis of current and potential applications of such systems, we also try to predict new trends in firstorder automated reasoning. Our presentation will be centered around two main ..."
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We overview the development of firstorder automated reasoning systems starting from their early years. Based on the analysis of current and potential applications of such systems, we also try to predict new trends in firstorder automated reasoning. Our presentation will be centered around two main motives: efficiency and usefulness for existing and future potential applications. This paper expresses the views of the author on past, present, and future of theorem proving in firstorder logic gained during ten years of working on the development, implementation, and applications of the theorem prover Vampire, see [Riazanov and Voronkov, 2002a]. It reflects our recent experience with applications of Vampire in verification, proof assistants, theorem proving, and semantic Web, as well as the analysis of future potential applications. 1 Theorem Proving in FirstOrder Logic The idea of automatic theorem proving has a long history both in mathematics and computer science. For a long time, it was believed by many that hard theorems in mathematics can be proved in a completely automatic way, using the ability of computers to perform fast combinatorial calculations. The very first experiments in automated theorem proving have shown that the purely combinatorial methods of proving firstorder theorems are too week even for proving theorems regarded as relatively easy by mathematicians. Provability in firstorder logic is a very hard combinatorial problem. Firstorder logic is undecidable, which means that there is no terminating procedure checking provability of formulas. There are decidable classes of firstorder formulas but formulas of these classes do not often arise in applications. Due to undecidability, very short formulas may turn out to be extremely complex, while very long ones rather easy. Sometimes firstorder provers find proofs consisting of several thousand steps in a few seconds, but sometimes it takes hours to find a tenstep proof. The theory of firstorder reasoning is centered around the completeness theorems while in practice completeness is often not an issue due to the intrinsic * Partially supported by a grant from EPSRC.
The Barcelona Prover
"... . Here we describe the equational theorem prover Barcelona, in its version that participated in the CADE'96 theorem proving competition. The system was built on top of our toolkit of data structures and algorithms for automated deduction in firstorder logic with equality, and was devised mainl ..."
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. Here we describe the equational theorem prover Barcelona, in its version that participated in the CADE'96 theorem proving competition. The system was built on top of our toolkit of data structures and algorithms for automated deduction in firstorder logic with equality, and was devised mainly to test the performance of this toolkit. Key words: Automated theorem proving, competition, Barcelona, data structures and algorithms, implementation 1. Introduction During the last decade, research on automated deduction in our group has mainly focussed on theoretical results for firstorder logic with equality. New techniques for e.g., clausal rewriting and deduction with constrained clauses have been developed and completeness results established. Many necessary underlying results on term orderings, constraint solving and answer computation have been given, with their decidability and complexity characteristics (cf. http://wwwlsi.upc.es/dept/sectp.html). In order to better understand thes...
Practical Algorithms for Deciding Path Ordering Constraint Satisfaction
 Yale University/ Glasgow University
, 2001
"... this paper we introduce some new notions of solved form, where, in addition to the closure under the classical RPO decomposition rules, a restricted form of transitivity through variables is applied. It is proved that if C is a solved form in this sense, then it is satisfiable under extended signatu ..."
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this paper we introduce some new notions of solved form, where, in addition to the closure under the classical RPO decomposition rules, a restricted form of transitivity through variables is applied. It is proved that if C is a solved form in this sense, then it is satisfiable under extended signatures if, and only if, it has no cycle (Section 5)
SPASS Input Syntax Version 1.5
"... This document introduces the SPASS input syntax. It came out of the DFG syntax format that was thought to be a format that can easily be parsed such that it forms a compromise between the needs of the different groups. 1 ..."
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This document introduces the SPASS input syntax. It came out of the DFG syntax format that was thought to be a format that can easily be parsed such that it forms a compromise between the needs of the different groups. 1
An Implementation Kernel for Theorem Proving . . .
 IN PROC. OF THE 1996 JOINT CONFERENCE ON DECLARATIVE PROGRAMMING APPIAGULPPRODE'96
, 1996
"... We provide a standard abstract architecture around which highperformance theorem provers for full clausal logic with equality can be built. A WAMlike heap structure for storing terms (as DAG's, with structure sharing) and several substitution trees [Gra95b] are central in the architecture ..."
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We provide a standard abstract architecture around which highperformance theorem provers for full clausal logic with equality can be built. A WAMlike heap structure for storing terms (as DAG's, with structure sharing) and several substitution trees [Gra95b] are central in the architecture. These two data structures turn out to be surprisingly well combinable due to conceptual similarities. Indexing techniques based on substitution trees outperform previous methods, and are integrated in such a way that e.g. no writing on the heap is needed during (manytoone) term unification. Static clause (sub)sets can be compiled in this framework into efficient abstract machine code for inference computation and redundancy proving. Finally, as an example, a toy equational completion system based on the framework is described.
From Search to Computation: Redundancy Criteria and Simplification at Work
"... The concept of redundancy and simplification has been an ongoing theme in Harald Ganzinger’s work from his first contributions to equational completion to the various variants of the superposition calculus. When executed by a theorem prover, the inference rules of these calculi usually generate a t ..."
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The concept of redundancy and simplification has been an ongoing theme in Harald Ganzinger’s work from his first contributions to equational completion to the various variants of the superposition calculus. When executed by a theorem prover, the inference rules of these calculi usually generate a tremendously growing search space. The redundancy and simplification concept is indispensable for cutting down this search space to a manageable size. For a number of subclasses of firstorder logic appropriate redundancy and simplification concepts even turn the superposition calculus into a decision procedure. Hence, the key to successfully applying firstorder theorem proving to a problem domain is to find those simplifications and redundancy criteria that fit this domain and can be effectively implemented. We present Harald Ganzinger’s work in the light of the simplification and redundancy techniques that have been developed for concrete problem areas. This includes a variant of contextual rewriting to decide a subclass of Euclidean geometry, ordered chaining techniques for ChurchRosser and priority queue proofs, contextual rewriting and historydependent complexities for the completion of conditional rewrite systems, rewriting with equivalences for theorem proving in set theory, soft typing for the exploration of sort information in the context of equations, and constraint inheritance for automated complexity analysis.
Modular Redundancy for Theorem Proving
, 2000
"... . We introduce a notion of modular redundancy for theorem proving. It can be used to exploit redundancy elimination techniques (like tautology elimination, subsumption, demodulation or other more refined methods) in combination with arbitrary existing theorem provers, in a refutation complete wa ..."
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. We introduce a notion of modular redundancy for theorem proving. It can be used to exploit redundancy elimination techniques (like tautology elimination, subsumption, demodulation or other more refined methods) in combination with arbitrary existing theorem provers, in a refutation complete way, even if these provers are not (or not known to be) complete in combination with the redundancy techniques when applied in the usual sense. 1 Introduction The concept of saturation in theorem proving is nowadays a wellknown, widely recognized useful concept. The main idea of saturation is that a theorem proving procedure does not need to compute the closure of a set of formulae w.r.t. a given inference system, but only the closure up to redundancy. Examples of early notions of redundancy (in the context of resolution) are the elimination of tautologies and subsumption. Bachmair and Ganzinger gave more general abstract notions of redundancy for inferences and formulae (see, e.g., [BG94...
SPAIN: Search spaces and Proofs Analyzed INdependently
"... . SPAIN is a system that can easily be applied to existing clausal e.g. resolutionbased theorem provers for verifying the correctness of inference and redundancy steps and for analysing how the search space is explored. An advantage of its approach for proof verification is that provers do no ..."
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. SPAIN is a system that can easily be applied to existing clausal e.g. resolutionbased theorem provers for verifying the correctness of inference and redundancy steps and for analysing how the search space is explored. An advantage of its approach for proof verification is that provers do not need to mantain any proof information. Regarding search space analysis, the system is able to provide a large amount of useful information globally and for each level of the search tree, like the number of clauses (generated, kept, removed,...), different size measures like the number (or depth) of variables, symbols, literals or terms, and showing for each measure maximum, minimum, average, etc. 1 Introduction Normally a user of an automated theorem prover is not satisfied with a yes/no answer. It is necessary to reconstruct the proof in order to check its correctness, to analyze its properties and, probably, to translate it into a more easily understandable form. For obvious reaso...
Coalescing: Syntactic Abstraction for Reasoning in FirstOrder Modal Logics ∗
, 2014
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
From Search to Computation: Redundancy Criteria and Simplification at Work
"... Abstract. The concept of redundancy and simplification has been an ongoing theme in Harald Ganzinger’s work from his first contributions to equational completion to the various variants of the superposition calculus. When executed by a theorem prover, the inference rules of logic calculi usually gen ..."
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Abstract. The concept of redundancy and simplification has been an ongoing theme in Harald Ganzinger’s work from his first contributions to equational completion to the various variants of the superposition calculus. When executed by a theorem prover, the inference rules of logic calculi usually generate a tremendously huge search space. The redundancy and simplification concept is indispensable for cutting down this search space to a manageable size. For a number of subclasses of firstorder logic appropriate redundancy and simplification concepts even turn the superposition calculus into a decision procedure. Hence, the key to successfully applying firstorder theorem proving to a problem domain is to find those simplifications and redundancy criteria that fit this domain and can be effectively implemented. We present Harald Ganzinger’s work in the light of the simplification and redundancy techniques that have been developed for concrete problem areas. This includes a variant of contextual rewriting to decide a subclass of Euclidean geometry, ordered chaining techniques for ChurchRosser and priority queue proofs, contextual rewriting and historydependent complexities for the completion of conditional rewrite systems, rewriting with equivalences for theorem proving in set theory, soft typing for the exploration of sort information in the context of equations, and constraint inheritance for automated complexity analysis. 1