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

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 ...

##
###
Method of marks for propositional linear temporal logic

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. ...

##
###
Deduction versus Computation: The Case of Induction
[chapter]

2002
*
Lecture Notes in Computer Science
*

When mechanizing

doi:10.1007/3-540-45470-5_3
fatcat:4bjblblahbhkrhbmrrmxx2ivsu
*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. ...##
###
Proofs by Deduction
[chapter]

2017
*
Logic and Language Models for Computer Science
*

Models and

doi:10.1142/9789813229211_0003
fatcat:cqpgqtobz5gx3db5bq54gxrd4m
*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*. ...##
###
Superposition with Structural Induction
[chapter]

2017
*
Lecture Notes in Computer Science
*

Superposition-

doi:10.1007/978-3-319-66167-4_10
fatcat:jzgkbftdsvdsthbtkxg4ipmeqm
*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). ...##
###
Induction and Skolemization in saturation theorem proving
[article]

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*. ...

##
###
Towards a foundation of completion procedures as semidecision procedures

1995
*
Theoretical Computer Science
*

The application of completion to disprove

doi:10.1016/0304-3975(94)00187-n
fatcat:fjtaagpgbvclxekxustvzjs7pe
*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 ...##
###
Which simple types have a unique inhabitant?

2015
*
Proceedings of the 20th ACM SIGPLAN International Conference on Functional Programming - ICFP 2015
*

Backward

doi:10.1145/2784731.2784757
dblp:conf/icfp/SchererR15
fatcat:adazx34yafdctcjyrgftd5cekm
*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 ...##
###
Which simple types have a unique inhabitant?

2015
*
SIGPLAN notices
*

Backward

doi:10.1145/2858949.2784757
fatcat:g5ik2xr4dbfz5k5d2ewc6fdu44
*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 ...##
###
Cut Admissibility by Saturation
[chapter]

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. ...

##
###
Using Stålmarck's Algorithm to Prove Inequalities
[chapter]

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. ...

##
###
Automatic 'Descente Infinie' Induction Reasoning
[chapter]

2005
*
Lecture Notes in Computer Science
*

A stronger connection between different

doi:10.1007/11554554_20
fatcat:sen7aduonbcerludou3fsda57m
*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 ...##
###
Strategic Issues, Problems and Challenges in Inductive Theorem Proving

2005
*
Electronical Notes in Theoretical Computer Science
*

Both

doi:10.1016/j.entcs.2005.01.006
fatcat:reml2iecwfexhlnmr4d22a7sj4
*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 ...##
###
AUTO2, A Saturation-Based Heuristic Prover for Higher-Order Logic
[chapter]

2016
*
Lecture Notes in Computer Science
*

The prover is designed to make use of human-specified heuristics when

doi:10.1007/978-3-319-43144-4_27
fatcat:zwlitqglmnawnp2laenjfhwnla
*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. ...##
###
Substitutional Definition of Satisfiability in Classical Propositional Logic
[chapter]

2005
*
Lecture Notes in Computer Science
*

The syntactic framework of the so-called

doi:10.1007/11499107_3
fatcat:6kmfrlx6o5ebdnijc5vqassy4i
*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] . ...
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