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Rewrite Rule Inference Using Equality Saturation [article]

Chandrakana Nandi, Max Willsey, Amy Zhu, Yisu Remy Wang, Brett Saiki, Adam Anderson, Adriana Schulz, Dan Grossman, Zachary Tatlock
2021 arXiv   pre-print
This paper explores how equality saturation, a promising technique that uses e-graphs to apply rewrite rules, can also be used to infer rewrite rules.  ...  We show that equality saturation efficiently shrinks both sets, leading to faster synthesis of smaller, more general rulesets. We prototyped these strategies in a tool dubbed ruler.  ...  We are grateful to Andrew Reynolds for answering our questions about CVC4's rule synthesis component, to Pavel Panchekha and Oliver Flatt for helping us run Herbie with Ruler's rules, and to Sorawee Porncharoenwase  ... 
arXiv:2108.10436v1 fatcat:254trb66wjbcpku7ifyzsz3mpq

Cut Admissibility by Saturation [chapter]

Guillaume Burel
2014 Lecture Notes in Computer Science  
This also leads us to introduce an extension of deduction modulo with conditional term rewriting rules.  ...  In this paper, we show that, in the same way that it is possible to recover confluence using Knuth-Bendix completion, one can regain cut admissibility in the general case using standard saturation techniques  ...  From Γ, A one can therefore deduce B ∨ L 1 ⊥ ∨ L n ⊥ , Let us show that all new clauses in the derivation of ⊓ ⊔ do not involve equality: Since Γ is saturated up to compositeness, all inferences using  ... 
doi:10.1007/978-3-319-08918-8_9 fatcat:iac7kwgrqzecdojd2rdjeumjqy

On word problems in Horn theories [chapter]

Emmanuel Kounalis, Michael Rusinowitch
1988 Lecture Notes in Computer Science  
We interpret Horn clauses as conditional rewrite rules. Then we give sufficient conditions so that the word problem can be decided by conditional normalization in some Horn theories.  ...  For instance, the clauses 10 and 11 in Example 2 will never be used as rewrite rules.  ...  A Complete Set of Inference Rules for Horn Logic with Equality A long-standing problem in theorem proving is to efficiently handle the equality relation.  ... 
doi:10.1007/3-540-19242-5_12 fatcat:fd7yjxw25zakdl7wrc43nflbea

On word problems in horn theories

Emmanuel Kounalis, Michael Rusinowitch
1991 Journal of symbolic computation  
We interpret Horn clauses as conditional rewrite rules. Then we give sufficient conditions so that the word problem can be decided by conditional normalization in some Horn theories.  ...  For instance, the clauses 10 and 11 in Example 2 will never be used as rewrite rules.  ...  A Complete Set of Inference Rules for Horn Logic with Equality A long-standing problem in theorem proving is to efficiently handle the equality relation.  ... 
doi:10.1016/s0747-7171(08)80134-4 fatcat:fmjkysa7yvbmtlkujlpdbn4vgm

On narrowing, refutation proofs and constraints [chapter]

Robert Nieuwenhuis
1995 Lecture Notes in Computer Science  
Re nements of narrowing (basic, LSE, etc.) are instances of the technique, but are also de ned here for arbitrary (possibly ordering and/or equality constrained or not yet convergent or saturated) Horn  ...  We develop a proof technique for dealing with narrowing and refutational theorem proving in a uniform way, clarifying the exact relationship between the existing results in both elds and allowing us to  ...  for saturated (under an inference system similar to the one used here) sets S coincides with the unique perfect model for S under the given term ordering.  ... 
doi:10.1007/3-540-59200-8_47 fatcat:xx5kk5ztfffmxmmfbjxjrucfpm

Associative-commutative superposition [chapter]

Leo Bachmair, Harald Ganzinger
1995 Lecture Notes in Computer Science  
It is compatible with an abstract notion of redundancy that covers such simpli cation techniques as tautology deletion, subsumption, and simpli cation by (associative-commutative) rewriting.  ...  Using saturation up to redundancy, we may infer that C 00 is true in I C .  ...  Saturation of clause sets can be achieved by fair application of inference rules.  ... 
doi:10.1007/3-540-60381-6_1 fatcat:5e5ihgzlhfhvvbo266oruhs4ky

From Search to Computation: Redundancy Criteria and Simplification at Work [chapter]

Thomas Hillenbrand, Ruzica Piskac, Uwe Waldmann, Christoph Weidenbach
2013 Lecture Notes in Computer Science  
When executed by a theorem prover, the inference rules of these calculi usually generate a tremendously growing search space.  ...  This includes a variant of contextual rewriting to decide a subclass of Euclidean geometry, ordered chaining techniques for Church-Rosser and priority queue proofs, contextual rewriting and history-dependent  ...  The paramodulation rule embodies the ideas of the resolution calculus and the operation of "replacing equals by equals" that is fundamental for term rewriting.  ... 
doi:10.1007/978-3-642-37651-1_7 fatcat:fr4uqtwvubbtvgjepamaamojeu

Combining algebra and universal algebra in first-order theorem proving: The case of commutative rings [chapter]

Leo Bachmair, Harald Ganzinger, Jürgen Stuber
1995 Lecture Notes in Computer Science  
Associative-commutative superposition forms the deductive core of our method, while a convergent rewrite system for commutative rings provides a starting point for more specialized inferences tailored  ...  This result is obtained by the judicious application of the simpli cation techniques a orded by convergent rewriting and by a process called symmetrization that embeds inferences between single clauses  ...  Secondly, the rewrite rules in R e may be used to simplify clauses in N. The latter aspect will be discussed in the next section.  ... 
doi:10.1007/bfb0014420 fatcat:h7fr556bpra3tjrlf4jymclqq4

Rewrite-based Equational Theorem Proving with Selection and Simplification

LEO BACHMAIR, HARALD GANZINGER
1994 Journal of Logic and Computation  
It is often possible to compute the closure of nontrivial sets of clauses under application of non-redundant inferences.  ...  We present various refutationally complete calculi for first-order clauses with equality that allow for arbitrary selection of negative atoms in clauses.  ...  One of the more successful approaches to equational theorem proving is based on the use of equations as (one-way) rewrite rules.  ... 
doi:10.1093/logcom/4.3.217 fatcat:unag3wwpmzdgzcqdjx7dklezhq

Subsumption Demodulation in First-Order Theorem Proving [article]

Bernhard Gleiss, Laura Kovacs, Jakob Rath
2020 arXiv   pre-print
Motivated by applications of first-order theorem proving to software analysis, we introduce a new inference rule, called subsumption demodulation, to improve support for reasoning with conditional equalities  ...  Our experiments, using the TPTP and SMT-LIB repositories, show that subsumption demodulation in Vampire can solve many new problems that could so far not be solved by state-of-the-art reasoners.  ...  An inference rule is a set of inferences and an inference system is a set of inference rules. An inference system is sound if all its inference rules are sound.  ... 
arXiv:2001.10213v1 fatcat:mlqi3hptvnflfchhkhp55w25py

Subsumption Demodulation in First-Order Theorem Proving [chapter]

Bernhard Gleiss, Laura Kovács, Jakob Rath
2020 Lecture Notes in Computer Science  
Motivated by applications of first-order theorem proving to software analysis, we introduce a new inference rule, called subsumption demodulation, to improve support for reasoning with conditional equalities  ...  Our experiments, using the TPTP and SMT-LIB repositories, show that subsumption demodulation in Vampire can solve many new problems that could so far not be solved by state-of-the-art reasoners.  ...  An inference rule is a set of (concrete) inferences and an inference system is a set of inference rules. An inference system is sound if all its inference rules are sound.  ... 
doi:10.1007/978-3-030-51074-9_17 fatcat:l3bzolwqqvbb5n2w7vkg2ay7vm

Editorial

Leo Bachmair, Jieh Hsiang
1991 Journal of symbolic computation  
Kounalis & Rusinowitch suggest a notion of saturated set for Horn clauses and give sufficient conditions under which the use of saturated sets as rewrite rules provides a decision procedure for the word  ...  Both inference systems refine the resolution/paramodulation calculus and incorporate inference rules for simplification by rewriting.  ...  For researchers interested in other aspects of rewriting and its applications, we recommend the survey, Rewrite Systems by N. Dershowitz and J.-P.  ... 
doi:10.1016/s0747-7171(08)80129-0 fatcat:524cajbdfze73njq3r7nknovmy

Automatic 'Descente Infinie' Induction Reasoning [chapter]

Sorin Stratulat
2005 Lecture Notes in Computer Science  
proof, and ii) reasoning modules that perform the computation and allow for modular design of the concrete inference rules.  ...  We present a framework and a methodology to build and analyse automatic provers using the 'Descente Infinie' induction principle.  ...  We thank the anonymous referees and Claus-Peter Wirth for useful comments on previous versions of this paper.  ... 
doi:10.1007/11554554_20 fatcat:sen7aduonbcerludou3fsda57m

Theorem proving in cancellative abelian monoids (extended abstract) [chapter]

Harald Ganzinger, Uwe Waldmann
1996 Lecture Notes in Computer Science  
Strong ordering constraints allow us to restrict to inferences that involve the maximal term of the maximal sum in the maximal literal.  ...  We describe a re ned superposition calculus for cancellative abelian monoids which requires neither explicit inferences with the theory clauses nor extended equations or clauses.  ...  As soon as the clauses contain non-constant free function symbols, we also have to use the inference rules of the traditional superposition calculus, i. e., equality resolution, standard superposition,  ... 
doi:10.1007/3-540-61511-3_102 fatcat:ft66vqfawbhy3pwnakgyhh6vvq

Ordered chaining for total orderings [chapter]

Leo Bachmair, Harald Ganzinger
1994 Lecture Notes in Computer Science  
We design new inference systems for total orderings by applying rewrite techniques to chaining calculi.  ...  We demonstrate that our inference systems are compatible with a concept of (global) redundancy for clauses and inferences that covers such widely used simplification techniques as tautology deletion, subsumption  ...  Rewrite systems Let I be a set of ground atomic formulas and ≻ be a simplification ordering. We use the equalities and inequalities in I as rewrite rules.  ... 
doi:10.1007/3-540-58156-1_32 fatcat:losit2c3prhvtdxl7mf5qylwoy
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