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Unification with Abstraction and Theory Instantiation in Saturation-Based Reasoning [chapter]

Giles Reger, Martin Suda, Andrei Voronkov
2018 Lecture Notes in Computer Science  
We make a new contribution to the field by providing a new method of using SMT solvers in saturation-based reasoning.  ...  The second rule is unification with abstraction where the notion of unification is extended to introduce constraints where theory terms may not otherwise unify.  ...  This rule will be used in Sect. 3 to justify ignoring certain literals when performing instantiation. Saturation-Based Proof Search (and Theory Reasoning).  ... 
doi:10.1007/978-3-319-89960-2_1 fatcat:qmvu5nhrdzdevgqlyq7eckcplu

EasyChair Preprint Unification with Abstraction and Theory Instantiation in Saturation-based Reasoning Unification with Abstraction and Theory Instantiation in Saturation-based Reasoning *

Giles Reger, Martin Suda, Andrei Voronkov, Giles Reger, Martin Suda, Andrei Voronkov
2017 unpublished
This paper explores two new inference rules for reasoning with quantifiers and theories in a saturation-based first-order theorem prover.  ...  The second new rule is unification with abstraction where the notion of unification is extended to introduce constraints where theory terms may not otherwise unify e.g. p(2) may unify with ¬p(x + 1) ∨  ...  This rule will be used in Section 3 to justify ignoring certain literals when performing instantiation. Saturation-Based Proof Search (and Theory Reasoning).  ... 
fatcat:756jur7hhvgb7nxw6fyh63odji

Saturation-based theorem proving: Past successes and future potential [chapter]

Harald Ganzinger
1996 Lecture Notes in Computer Science  
dynamically stratified programs p. 328 Unification in pseudo-linear sort theories is decidable p. 343 Theorem proving with group presentations: examples and questions p. 358 Transforming termination  ...  Proof search with set variable instantiation in the calculus of constructions p. 658Search strategies for resolution in temporal logics p. 673Optimal axiomatizations for multiple-valued operators and quantifiers  ... 
doi:10.1007/3-540-61511-3_64 fatcat:iizfu32trnfadmj4z7xcvwds5y

Comparing Approaches To Resolution Based Higher-Order Theorem Proving

Christoph Benzmüller
2002 Synthese  
We investigate several approaches to resolution based automated theorem proving in classical higher-order logic (based on Church's simply typed λ-calculus) and discuss their requirements with respect to  ...  In particular we focus on Andrews' higher-order resolution (Andrews 1971), Huet's constrained resolution (Huet 1972), higher-order E-resolution, and extensional higher-order resolution (Benzmüller and  ...  ACKNOWLEDGEMENTS I want to thank Volker Sorge and the anonymous referee of this paper for their useful comments and contributions.  ... 
doi:10.1023/a:1020840027781 fatcat:sghuiqlg6vd3dhgr2f6clo4ssq

iProver-Eq: An Instantiation-Based Theorem Prover with Equality [chapter]

Konstantin Korovin, Christoph Sticksel
2010 Lecture Notes in Computer Science  
Acknowledgements Our work is based on the iProver system, as in described in Korovin (2008) and Korovin and Sticksel (2010) . We thank Renate Schmidt for helpful discussions.  ...  Figure 1: Saturation process in the iProver-Eq system The saturation process is based on a given clause algorithm, which partitions the set of clauses into two disjoint sets, namely the Inst-Active and  ...  The common idea is to instantiate clauses and to employ efficient propositional or more general ground reasoning methods in order to prove unsatisfiability or to find a model.  ... 
doi:10.1007/978-3-642-14203-1_17 fatcat:kibzdvfmrfcpnegq5rv6jcp4se

Extensional Paramodulation for Higher-Order Logic and Its Effective Implementation Leo-III

Alexander Steen
2019 Künstliche Intelligenz  
A primary focus of this thesis is the provision of evidence that a paramodulation-based proof calculus can effectively be employed for performant equational reasoning in Extensional Type Theory (higher-order  ...  for reasoning with choice.  ...  In resolution-based theorem proving and, more generally, modern saturation-based theorem proving, the initial conjecture is negated, transformed into a set of clauses, and successively saturated.  ... 
doi:10.1007/s13218-019-00628-8 fatcat:b6yfenxgxndb3myskqushsm6uy

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

Konstantin Korovin
2013 Lecture Notes in Computer Science  
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.  ...  Acknowledgements I am very grateful to Harald Ganzinger who introduced me to the area of instantiationbased reasoning and with whom I had the great pleasure of investigating this area.  ... 
doi:10.1007/978-3-642-37651-1_10 fatcat:q4tfqya64zcahjg273c6c3dyh4

An application of abstract interpretation in source level program transformation [chapter]

Daniel Schreye, Maurice Bruynooghe
1989 Lecture Notes in Computer Science  
to those of their imperative equivalents' Also, there is the ease with which abstract interpretation can be described and implemented within the setting of logic programming.  ...  The increasing attention that abstract interpretation has obtained from researchers active in the field of logic programming can beexplained by severat reasons.  ...  However, in this easy example, the abstractions x and y still constitute Prolog variables and the unification mechanism of Prolog can deal with them precisely in the w a y we expect it to.  ... 
doi:10.1007/3-540-50820-1_39 fatcat:v3bwjopefzgmnggf6mtax3owuu

Natural deduction as higher-order resolution

Lawrence C. Paulson
1986 The Journal of Logic Programming  
Resolution gives both forwards and backwards proof, supporting a large class of logics. Isabelle has been used to prove theorems in Martin-Lofs constructive type theory.  ...  Sets and functions are semantic; formal type theories and Zermelo-Fraenkel set theory are syntactic. The ultimate semanticnotion is truth, faintly approximated by theorems of formal logic.  ...  An LCF theory is a data base of constants, types, axioms, and theorems. At present, Isabelle allows only simple abbreviations. We need methods for combining theories and working in different logics.  ... 
doi:10.1016/0743-1066(86)90015-4 fatcat:d2jkwnbuyzeo7ko2ymi6qvxvpm

Extensional higher-order resolution [chapter]

Christoph Benzmüller, Michael Kohlhase
1998 Lecture Notes in Computer Science  
In this paper we present an extensional higher-order resolution calculus that is complete relative to Henkin model semantics.  ...  The treatment of the extensionality principles -necessary for the completeness result -by specialized (goal-directed) inference rules is of practical applicability, as an implentation of the calculus in  ...  The authors are grateful to Peter Andrews and Frank Pfenning for stimulating discussions.  ... 
doi:10.1007/bfb0054248 fatcat:l7x7x6a7r5exjb5pbmr35pm4rm

Making Theory Reasoning Simpler [chapter]

Giles Reger, Johannes Schoisswohl, Andrei Voronkov
2021 Lecture Notes in Computer Science  
AbstractReasoning with quantifiers and theories is at the core of many applications in program analysis and verification.  ...  Our previous work proposed an instantiation rule for theory reasoning that produced pragmatically useful instances.  ...  The previously mentioned unification with abstraction and theory instantiation [20] rules support lazy unification modulo theories and pragmatic instantiation.  ... 
doi:10.1007/978-3-030-72013-1_9 fatcat:scvyq5encvhmbib3e437pij4g4

Combining Decision Algorithms for Matching in the Union of Disjoint Equational Theories

Christophe Ringeissen
1996 Information and Computation  
The concept of matching is ubiquitous in declarative programming and in automated reasoning.  ...  For instance, it is a key mechanism to run rule-based programs and to simplify clauses generated by theorem provers.  ...  In these reasoning tools, the capabilities of an intruder are modeled using equational theories [1] , and the reasoning is supported by decision procedures and solvers modulo equational theories, including  ... 
doi:10.1006/inco.1996.0042 fatcat:l4ohlj7wq5arvkyshxioglxv7m

Building and Combining Matching Algorithms [chapter]

Christophe Ringeissen
2019 Lecture Notes in Computer Science  
The concept of matching is ubiquitous in declarative programming and in automated reasoning.  ...  For instance, it is a key mechanism to run rule-based programs and to simplify clauses generated by theorem provers.  ...  In these reasoning tools, the capabilities of an intruder are modeled using equational theories [1] , and the reasoning is supported by decision procedures and solvers modulo equational theories, including  ... 
doi:10.1007/978-3-030-22102-7_24 fatcat:6nx7ollvnzchrcuacrpa522f6q

Can a Higher-Order and a First-Order Theorem Prover Cooperate? [chapter]

Christoph Benzmüller, Volker Sorge, Mateja Jamnik, Manfred Kerber
2005 Lecture Notes in Computer Science  
While it has been shown in the past that higher-order reasoning systems can solve problems of this kind automatically, the complexity inherent in their calculi and their inefficiency in dealing with large  ...  We present a solution to this challenge by combining a higher-order and a first-order automated theorem prover, both based on the resolution principle, in a flexible and distributed environment.  ...  Some problems are much harder in higher-order, for instance, unification is undecidable, strong constraining term-and literal-orderings are not available, extensionality reasoning and set variable instantiation  ... 
doi:10.1007/978-3-540-32275-7_27 fatcat:vllvzeicvbdixmzafcjhtee7ca

Parsing Head-driven Phrase Structure Grammar

Derek Proudian, Carl Pollard
1985 Proceedings of the 23rd annual meeting on Association for Computational Linguistics -  
We believe that this makes the HPSG system ,nique in its combination of linguistic theory and practical application.  ...  But, unlike other implementations of linguistic theories, this system is not a toy, as it deals with a variety of practical problems not covered in the theoretical literature.  ...  Unification Another important difference between HPSG and other unification based syntactic theories concerns the form of the expressions which are actually unified.  ... 
doi:10.3115/981210.981231 dblp:conf/acl/ProudianP85 fatcat:6fklsmhopjekpapyd6dr7uulwa
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