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Soundness and Completeness Proofs by Coinductive Methods

Jasmin Christian Blanchette, Andrei Popescu, Dmitriy Traytel
2016 Journal of automated reasoning  
We show how codatatypes can be employed to produce compact, high-level proofs of key results in logic: the soundness and completeness of proof systems for variations of first-order logic.  ...  Soundness becomes interesting as soon as one allows infinite proofs of first-order formulas.  ...  Popescu was partially supported by the EPSRC project Verification of Web-based Systems (VOWS, grant EP/N019547/1) and by the DFG project Security Type Systems and Deduction (grant NI 491/ 13-3).  ... 
doi:10.1007/s10817-016-9391-3 fatcat:vnleinry7jckjfp3qvgginqs34

Coinductive Soundness of Corecursive Type Class Resolution [chapter]

František Farka, Ekaterina Komendantskaya, Kevin Hammond
2017 Lecture Notes in Computer Science  
This paper shows, for the first time, that corecursive type class resolution and its recent extensions are coinductively sound with respect to the greatest Herbrand models of logic programs and that they  ...  Horn clauses and first-order resolution are commonly used for the implementation of type classes in Haskell.  ...  ), by COST Action IC1202 (TACLe), supported by COST (European Cooperation in Science and Technology), and by EPSRC grant EP/K031864/1-2 "'Coalgebraic Logic Programming for Type Inference".  ... 
doi:10.1007/978-3-319-63139-4_18 fatcat:jmfbgvz36jgsfiaboiiox5bniy

Coinductive Soundness of Corecursive Type Class Resolution [article]

František Farka, Ekaterina Komendantskaya, Kevin Hammond
2016 arXiv   pre-print
This paper shows, for the first time, that corecursive type class resolution and its extensions are coinductively sound with respect to the greatest Herbrand models of logic programs and that they are  ...  We establish incompleteness results for various fragments of the proof system.  ...  Applications -a Software Engineering Approach" (ICT-644235), and by COST Action IC1202 (TACLe), supported by COST (European Cooperation in Science and Technology).  ... 
arXiv:1608.05233v2 fatcat:w5h4djfwizgsroan7gbmac2rie

Coinduction in Uniform: Foundations for Corecursive Proof Search with Horn Clauses [chapter]

Henning Basold, Ekaterina Komendantskaya, Yue Li
2019 Lecture Notes in Computer Science  
with recursion controlled by the later modality; and soundness of both proof systems is proven relative to a novel coalgebraic description of complete Herbrand models.  ...  We establish proof-theoretic, constructive and coalgebraic foundations for proof search in coinductive Horn clause theories.  ...  We would like to thank Damien Pous and the anonymous reviewers for their valuable feedback.  ... 
doi:10.1007/978-3-030-17184-1_28 fatcat:7azord5jkjhlvdxyir54hjsdqq

Structural Resolution: a Framework for Coinductive Proof Search and Proof Construction in Horn Clause Logic [article]

Ekaterina Komendantskaya, Patricia Johann, Martin Schmidt
2017 arXiv   pre-print
Finite SLD-computations are inductively sound and complete with respect to least Herbrand models of logic programs.  ...  This puts coinductive methods in LP on par with productivity-based observational approaches to coinduction in ITP and TRS.  ...  and Tom Schrijvers.  ... 
arXiv:1511.07865v2 fatcat:6lqbu7vyqzegnchxtiws7ty3fu

Towards Coinductive Theory Exploration in Horn Clause Logic: Position Paper

Ekaterina Komendantskaya, Yue Li
2018 Electronic Proceedings in Theoretical Computer Science  
To this aim, we propose a coinductive extension of Miller et al's framework of uniform proofs and prove its soundness relative to coinductive models of Horn clause logic.  ...  Coinduction occurs in two guises in Horn clause logic: in proofs of self-referencing properties and relations, and in proofs involving construction of (possibly irregular) infinite data.  ...  Intuitively, the proof proceeds by defining a scheme by which we can reconstruct a corresponding non-terminating derivation, and then showing that the proofs are sound w.r.t greatest complete Herbrand  ... 
doi:10.4204/eptcs.278.5 fatcat:e5z3z3pn4rbuvbx3dral2fxqny

A Coalgebraic Foundation for Coinductive Union Types [chapter]

Marcello Bonsangue, Jurriaan Rot, Davide Ancona, Frank de Boer, Jan Rutten
2014 Lecture Notes in Computer Science  
We obtain inclusion of tree languages as a sound and complete method to show semantic subtyping of recursive types with basic types, product and union, interpreted coinductively.  ...  We give a sound and complete characterization of semantic subtyping in terms of inclusion of maximal traces.  ...  Existing coinductive proof methods such as [3] [4] [5] are incomplete and involve complex soundness proofs.  ... 
doi:10.1007/978-3-662-43951-7_6 fatcat:dxksvgigpne67kbrfqjtjiyiim

Tower Induction and Up-to Techniques for CCS with Fixed Points [chapter]

Steven Schäfer, Gert Smolka
2017 Lecture Notes in Computer Science  
We present a refinement of Pous' companion-based coinductive proof technique and apply it to CCS with general fixed points.  ...  Our induction principle implies a new sufficient condition for soundness of up-to techniques subsuming respectfulness and compatibility.  ...  [5] have introduced another enhancement of the coinductive proof method in the form of parameterized coinduction.  ... 
doi:10.1007/978-3-319-57418-9_17 fatcat:5oehvrfjgzbnria3scuejuli4e

Coinductive Uniform Proofs [article]

Ekaterina Komendantskaya, Yue Li
2019 arXiv   pre-print
To this aim, we propose a coinductive extension of Miller et al framework of uniform proofs and prove its soundness relative to coinductive models of Horn clause logic.  ...  Coinduction occurs in two guises in Horn clause logic: in proofs of circular properties and relations, and in proofs involving construction of infinite data.  ...  By transitivity of ⊆, I x ⊆ T (I), for all I x . So I ⊆ T (I), by the construction of I . D.3 Proof of Theorem 1 (Soundness of Coinductive Uniform Proofs) Theorem 1 (Soundness of co-hohh proofs).  ... 
arXiv:1903.07371v1 fatcat:wbq75w5wd5dfxefhevozlz7frm

Coinductive big-step operational semantics for type soundness of Java-like languages

Davide Ancona
2011 Proceedings of the 13th Workshop on Formal Techniues for Java-Like Programs - FTfJP '11  
We prove that such a semantics is sound w.r.t. the usual small-step operational semantics, and then prove soundness of a conventional nominal type system w.r.t. the coinductive semantics.  ...  We define a coinductive semantics for a simple Java-like language by simply interpreting coinductively the rules of a standard big-step operational semantics.  ...  The complete proof with all necessary lemmas can be found in a companion paper [2] . Then we show that soundness of the type system w.r.t.  ... 
doi:10.1145/2076674.2076679 dblp:conf/ecoop/Ancona11 fatcat:2v5n5wmxlfdhvptssuj4vto45u

Proving partial-correctness and invariance properties of transition-system models

Vlad Rusu, Gilles Grimaud, Michaël Hauspie
2019 Science of Computer Programming  
We embed the proof system in the Coq proof assistant and formally prove its soundness and completeness.  ...  The complete proof strategy reduces partial correctness to invariance.  ...  We thank the anonymous reviewers of earlier versions of this paper for their useful hints and suggestions, especially the suggestion to use coinduction, and David Nowak for his lessons on coinduction in  ... 
doi:10.1016/j.scico.2019.102342 fatcat:aqregmbhdfau7i6ncnibu37n7m

Circular Coinduction with Special Contexts [chapter]

Dorel Lucanu, Grigore Roşu
2009 Lecture Notes in Computer Science  
, intuitively Behavioral Specifications, Intuitively 2 Circular Coinduction Circular Coinduction, intuitively Proof System 3 Special Contexts Intuition Special Hypothesis Defined by Special Contexts  ...  Cuza Univ. of Iasi (UAIC, RO) and Univ. of Illinois at Urbana-Champaign (UIUC, US) CIRC implements circular coinduction and circular induction completely automated 1 Introduction Behavioral Equivalence  ...  Achievements: circular coinduction is a simple an powerful proof method by coinduction special hypotheses defined by the special contexts significantly improves the circular coinduction proof systems the  ... 
doi:10.1007/978-3-642-10373-5_33 fatcat:q4y7vcdz2nektjfypxisvofqoy

Coinductive Proof Techniques for Language Equivalence [chapter]

Jurriaan Rot, Marcello Bonsangue, Jan Rutten
2013 Lecture Notes in Computer Science  
Language equivalence can be checked coinductively by establishing a bisimulation on suitable deterministic automata.  ...  Then we extend our enhanced proof method to incorporate language complement and intersection.  ...  Conclusions We presented a proof method for language equivalence, based on coinduction and bisimulation-up-to.  ... 
doi:10.1007/978-3-642-37064-9_42 fatcat:fy6mexeq5bawpovgqdlgae6sqy

The New Normal: We Cannot Eliminate Cuts in Coinductive Calculi, But We Can Explore Them [article]

Ekaterina Komendantskaya and Dmitry Rozplokhas and Henning Basold
2020 arXiv   pre-print
We show that CoLP by Gupta et al. gives rise to cut-free proofs in CLJ with fixpoint terms, and we formulate and implement a novel method of coinductive theory exploration that provides several heuristics  ...  For example, Gentzen's classical and intuitionistic calculi LK and LJ enjoy cut elimination. The property is less studied in coinductive extensions of sequent calculi.  ...  Simon et al. (2006) have shown that the method of loop detection is sound relative to the complete Herbrand models of logic programs.  ... 
arXiv:2008.03714v1 fatcat:a2vx23xm6za2tbuuz5jxl54tsm

The New Normal: We Cannot Eliminate Cuts in Coinductive Calculi, But We Can Explore Them

Ekaterina Komendantskaya, Dmitry Rozplokhas, Henning Basold
2020 Theory and Practice of Logic Programming  
We show that CoLP by Gupta et al. gives rise to cut-free proofs in CLJ with fixpoint terms, and we formulate and implement a novel method of coinductive theory exploration that provides several heuristics  ...  For example, Gentzen's classical and intuitionistic calculi LK and LJ enjoy cut elimination. The property is less studied in coinductive extensions of sequent calculi.  ...  Simon et al. (2006) have shown that the method of loop detection is sound relative to the complete Herbrand models of logic programs.  ... 
doi:10.1017/s1471068420000423 fatcat:nsdrc3y2kfb5jab5taalswsxfe
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