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

Harald Ganzinger
1996 Lecture Notes in Computer Science  
Proof-search in intuitionistic logic with equality, or back to simultaneous rigid E-Unification p. 32 Extensions to a generalization critic for inductive proof p. 47 Learning domain knowledge to improve  ...  semantics for production rule systems: theory and applications p. 508 Experiments in the heuristic use of past proof experience p. 523 Lemma discovery in automating induction p. 538 Advanced indexing  ... 
doi:10.1007/3-540-61511-3_64 fatcat:iizfu32trnfadmj4z7xcvwds5y

Method of marks for propositional linear temporal logic

Regimantas Pliuškevičius
2014 Lietuvos matematikos rinkinys  
In this paper, some loop-check free saturation-like decision procedure based on some technique of marks is proposed.  ...  It is known that traditional techniques used to ensure termination of a decision procedure in non-classical logics are based on loop-checking, in general.  ...  It is known that combination of these temporal operators requires to use induction-like tools. To determine such tools, some simple saturation procedure, based on the technique of marks, is proposed.  ... 
doi:10.15388/lmr.a.2014.09 fatcat:spthugufxfce5i2apszu6nlsiu

Deduction versus Computation: The Case of Induction [chapter]

Eric Deplagne, Claude Kirchner
2002 Lecture Notes in Computer Science  
When mechanizing proof construction, explicit induction is used in proof assistants and implicit induction is used in rewrite based automated theorem provers.  ...  In this work, we propose such an approach based on the general notion of deduction modulo.  ...  When mechanizing proof construction, explicit induction is used in proof assistants and implicit induction is used in rewrite based automated theorem provers.  ... 
doi:10.1007/3-540-45470-5_3 fatcat:4bjblblahbhkrhbmrrmxx2ivsu

Proofs by Deduction [chapter]

2017 Logic and Language Models for Computer Science  
Models and proofs even play an integral role within deductive tools as major methods for efficient proof search rely on refining a simultaneous search for a model or a proof.  ...  The topic is in a sense evergreen: models and proofs will always be an integral part of deduction.  ...  Using models to guide the search for proofs and vice versa. An upcoming next generation of reasoning procedures employ (partial) models/proofs for proof search.  ... 
doi:10.1142/9789813229211_0003 fatcat:cqpgqtobz5gx3db5bq54gxrd4m

Superposition with Structural Induction [chapter]

Simon Cruanes
2017 Lecture Notes in Computer Science  
Superposition-based provers have been successfully used to discharge proof obligations stemming from proof assistants. However, many such obligations require induction to be proved.  ...  Lemmas obtained by generalization or theory exploration can be introduced during search, used, and proved, all in the same search space.  ...  out a lot of related works and limitations in several occasions).  ... 
doi:10.1007/978-3-319-66167-4_10 fatcat:jzgkbftdsvdsthbtkxg4ipmeqm

Induction and Skolemization in saturation theorem proving [article]

Stefan Hetzl, Jannik Vierling
2021 arXiv   pre-print
In a practically relevant setting we establish a Skolem-free characterization of refutation in saturation-based proof systems with induction.  ...  We consider a typical integration of induction in saturation-based theorem provers and investigate the effects of Skolem symbols occurring in the induction formulas.  ...  Typically saturation-based theorem provers operate in a clausal setting because clauses have less structure and are therefore better suited for automated proof search.  ... 
arXiv:2105.07734v1 fatcat:clt5v76wpvgcjinsvdhej4vf5q

Towards a foundation of completion procedures as semidecision procedures

Maria Paola Bonacina, Jieh Hsiang
1995 Theoretical Computer Science  
The application of completion to disprove inductive conjectures, i.e. the so-called inductionless induction method, is also covered as a semidecision process.  ...  Finally, we present according to our framework, some equational completion procedures based on unfailing Knuth-Bendix completion.  ...  ., Proc. 2nd CTRS Workshop, held in Montreal, in June 1990, and available in Springer, Lecture Notes in Computer Science, Vol. 516, pp. 206-232, and "On fairness of completion-based theorem proving strategies  ... 
doi:10.1016/0304-3975(94)00187-n fatcat:fjtaagpgbvclxekxustvzjs7pe

Which simple types have a unique inhabitant?

Gabriel Scherer, Didier Rémy
2015 Proceedings of the 20th ACM SIGPLAN International Conference on Functional Programming - ICFP 2015  
Backward search in this logic gives an effective algorithm that returns either zero, one or two distinct inhabitants for any given type.  ...  We present a saturating focused logic that introduces irreducible cuts on positive types "as soon as possible".  ...  In the literature on proof search for propositional logic, the usual termination argument is based on the subformula property: in a closed, fully cut-eliminated proof, the formulas that appear in subderivations  ... 
doi:10.1145/2784731.2784757 dblp:conf/icfp/SchererR15 fatcat:adazx34yafdctcjyrgftd5cekm

Which simple types have a unique inhabitant?

Gabriel Scherer, Didier Rémy
2015 SIGPLAN notices  
Backward search in this logic gives an effective algorithm that returns either zero, one or two distinct inhabitants for any given type.  ...  We present a saturating focused logic that introduces irreducible cuts on positive types "as soon as possible".  ...  In the literature on proof search for propositional logic, the usual termination argument is based on the subformula property: in a closed, fully cut-eliminated proof, the formulas that appear in subderivations  ... 
doi:10.1145/2858949.2784757 fatcat:g5ik2xr4dbfz5k5d2ewc6fdu44

Cut Admissibility by Saturation [chapter]

Guillaume Burel
2014 Lecture Notes in Computer Science  
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  ...  Deduction modulo is a framework in which theories are integrated into proof systems such as natural deduction or sequent calculus by presenting them using rewriting rules.  ...  ; it is equivalent to the completeness of the proof search procedures based on deduction modulo R; etc.  ... 
doi:10.1007/978-3-319-08918-8_9 fatcat:iac7kwgrqzecdojd2rdjeumjqy

Using Stålmarck's Algorithm to Prove Inequalities [chapter]

Byron Cook, Georges Gonthier
2005 Lecture Notes in Computer Science  
In this paper we characterize a set of formulae with bitvector-inequalities for which 1-saturation is sufficient to prove unsatisfiability.  ...  Stålmarck's 1-saturation algorithm is an incomplete but fast method for computing partial equivalence relations over propositional formulae.  ...  R ≤ Saturate(n, R, F ) Proof. By induction on n with Lemma 7 in the base case and an argument based on Lemma 2 and Lemma 1 in the inductive case. Lemma 9 . 9 Saturate computes a fixpoint.  ... 
doi:10.1007/11576280_23 fatcat:famael6wdbbermltjvmafkyh6e

Automatic 'Descente Infinie' Induction Reasoning [chapter]

Sorin Stratulat
2005 Lecture Notes in Computer Science  
A stronger connection between different proof techniques like those based on implicit induction and saturation is established by uniformly and explicitly representing them as applications of this principle  ...  proof, and ii) reasoning modules that perform the computation and allow for modular design of the concrete inference rules.  ...  To witness that the above proof techniques share the same logic, the methodology is applied in Section 3 to build a concrete implicit induction prover and analyse an existing saturation-based inference  ... 
doi:10.1007/11554554_20 fatcat:sen7aduonbcerludou3fsda57m

Strategic Issues, Problems and Challenges in Inductive Theorem Proving

Bernhard Gramlich
2005 Electronical Notes in Theoretical Computer Science  
Both in general TP and in ITP, strategies for guiding the proof search process are of fundamental importance, in automated as well as in interactive or mixed settings.  ...  In the paper we will analyze and discuss the most important strategic and proof search issues in ITP, compare ITP with TP, and argue why ITP is in a sense much more challenging.  ...  in the proof search process is very much essential and highly important in practice. · The proof search control in (first-order) ITP is extremely challenging, for almost every non-trivial inductive conjecture  ... 
doi:10.1016/j.entcs.2005.01.006 fatcat:reml2iecwfexhlnmr4d22a7sj4

AUTO2, A Saturation-Based Heuristic Prover for Higher-Order Logic [chapter]

Bohua Zhan
2016 Lecture Notes in Computer Science  
The prover is designed to make use of human-specified heuristics when searching for proofs.  ...  The core algorithm is a best-first search through the space of propositions derivable from the initial assumptions, where new propositions are added by user-defined functions called proof steps.  ...  On the other hand it has a robost, saturation-based search mechanism. We discuss these features and their motivations in Section 2.  ... 
doi:10.1007/978-3-319-43144-4_27 fatcat:zwlitqglmnawnp2laenjfhwnla

Substitutional Definition of Satisfiability in Classical Propositional Logic [chapter]

Anton Belov, Zbigniew Stachniak
2005 Lecture Notes in Computer Science  
The syntactic framework of the so-called saturated substitutions is defined and used to obtain new characterizations of SAT as well as the classes of minimal and maximal models of formulas of classical  ...  formulas (in areas such as model-based diagnosis, model-based reasoning, or planning, cf  ...  This includes the complete and incomplete methods as well as algorithms for SAT based on such techniques as the search in Hamming Balls [3] and Satisfiability Coding Lemma [9] .  ... 
doi:10.1007/11499107_3 fatcat:6kmfrlx6o5ebdnijc5vqassy4i
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