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A Mechanised Proof System for Relation Algebra Using Display Logic [chapter]

Jeremy E. Dawson, Rajeev Goré
1998 Lecture Notes in Computer Science  
We show how some relation algebraic theorems proved using our system can be put in the form of structural rules of Display Logic, facilitating later re-use.  ...  Our implementation is the rst mechanisation of any display calculus, but also provides a useful interactive proof assistant for relation algebra.  ...  The rst author wishes to thank the Australian Defence Science and Technology Organization for support during the period when the work described in this paper was done.  ... 
doi:10.1007/3-540-49545-2_18 fatcat:hrlp65zylraxxfgi5yoomcuvki

Automated Engineering of Relational and Algebraic Methods in Isabelle/HOL [chapter]

Simon Foster, Georg Struth, Tjark Weber
2011 Lecture Notes in Computer Science  
We present a new integration of relational and algebraic methods in the Isabelle/HOL theorem proving environment.  ...  It consists of a fine grained hierarchy of algebraic structures based on Isabelle's type classes and locales, and a repository of more than 800 facts obtained by automated theorem proving.  ...  Having established, for instance, that binary relations form a Kleene algebra, Isabelle immediately makes all abstract theorems of Kleene algebra proved in the system available for binary relations as  ... 
doi:10.1007/978-3-642-21070-9_5 fatcat:oszkpd2pfrbsnovtsp7mvn3sha

RALL: Machine-supported proofs for relation algebra [chapter]

David Oheimb, Thomas F. Gritzner
1997 Lecture Notes in Computer Science  
We p r e s e n t a theorem proving system for abstract relation algebra called RALL (= Relation-Algebraic Language and Logic), based on the generic theorem prover Isabelle.  ...  On the one hand, the system is an advanced case study for Isabelle/HOL, and on the other hand, a quite mature proof assistant for research on the relational calculus.  ...  Furthermore, we h a ve bene ted from discussions with Lamia Labed-Jilani, one of Ali Mili's students, who is undertaking a project on relation-algebraic proofs using the OTTER system.  ... 
doi:10.1007/3-540-63104-6_36 fatcat:3imsczulfncijg6cdpbnlq24za

A Repository for Tarski-Kleene Algebras

Walter Guttmann, Georg Struth, Tjark Weber
2011 Workshop on Automated Theory Engineering  
Main purposes of the repository are the engineering of algebraic theories for computing systems and their application in formal program development.  ...  It covers variants of Kleene algebras and relation algebras with many of their models. Most theorems have been obtained by automated theorem proving within Isabelle.  ...  It calls 5 external ATP systems, all of which find a proof within a few seconds. They return the set of lemmas from our Kleene algebra repository used in the proof.  ... 
dblp:conf/ate/GuttmannSW11 fatcat:gsxscqdtvfhtfkmx2ikwr6yfci

Sufficient Solvability Conditions for Systems of Partial Fuzzy Relational Equations [chapter]

Nhung Cao, Martin Štěpnička
2020 Communications in Computer and Information Science  
This paper focuses on searching sufficient conditions for the solvability of systems of partial fuzzy relational equations. In the case of solvable systems, we provide solutions of the systems.  ...  In this investigation, we consider seven most-known algebras of undefined values in partial fuzzy set theory such as the Bochvar, Bochvar external, Sobociński, McCarthy, Nelson, Kleene, and the Lukasiewicz  ...  Bochvar External Algebra and Sobociński Algebra In this section, we present the investigation of the solvability of systems of partial fuzzy relational equations in the case of the Bochvar external algebra  ... 
doi:10.1007/978-3-030-50146-4_8 fatcat:ozbmwdahsrgmhiwcn3is4znrvi

A glimpse of deductive systems in algebra

Dumitru Buşneag, Sergiu Rudeanu
2010 Open Mathematics  
AbstractThe concept of a deductive system has been intensively studied in algebraic logic, per se and in connection with various types of filters.  ...  In this paper we introduce an axiomatization which shows how several resembling theorems that had been separately proved for various algebras of logic can be given unique proofs within this axiomatic framework  ...  In particular a referee called our attention to the coincidence of Hertz algebras with relatively pseudocomplemented semilattices and to an error in a previous version.  ... 
doi:10.2478/s11533-010-0041-4 fatcat:5eb6vjs2lndmdmcyy27ceejg3m

Compatible Deductive Systems of Pulexes

Shokoofeh Ghorbani
2014 International Journal of Mathematics and Mathematical Sciences  
We define the quotient algebra induced by a compatible deductive system and study its properties.  ...  We also define a congruence relation on a pulex and show that there is a bijective correspondence between the compatible deductive systems and the congruence relations.  ...  Conclusion We introduced the notions of compatible deductive systems and congruence relations on a pulex and obtained some related results. Then we studied the relation between them.  ... 
doi:10.1155/2014/606890 fatcat:zk5qcs3o5ff2tcdowxmg46troe

System Description: aRa – An Automatic Theorem Prover for Relation Algebras [chapter]

Carsten Sinz
2000 Lecture Notes in Computer Science  
Employing results from Tarski/Givant and Maddux we can prove validity in the theories of simple semi-associative relation algebras, relation algebras and representable relation algebras using the calculi  ...  aRa is an automatic theorem prover for various kinds of relation algebras.  ...  The RALF system [HBS94] offers no automatic proof search, but has particular strengths in proof presentation.  ... 
doi:10.1007/10721959_13 fatcat:7qtxqcvyrrhrxdtlcnncllqdom

Calculational Relation-Algebraic Proofs in Isabelle/Isar [chapter]

Wolfram Kahl
2004 Lecture Notes in Computer Science  
We propose a collection of theories in the proof assistant Isabelle/Isar that support calculational reasoning in and about heterogeneous relational algebras and Kleene algebras.  ...  Recent years have seen a rapid growth of interest in computer science applications of Kleene algebras and related structures, studied from a relational perspective, see e.g. [DM01] .  ...  Acknowledgements I am grateful to Millie Rhoss de Guzman and Hitoshi Furusawa for their comments on previous versions of this paper and many related discussions, and also to the anonymous reviewers for  ... 
doi:10.1007/978-3-540-24771-5_16 fatcat:smzaqpd5uzgw7fg5266chuohqu

Two Proof Systems for Peirce Algebras [chapter]

Renate A. Schmidt, Ewa Orłowska, Ullrich Hustadt
2004 Lecture Notes in Computer Science  
This paper develops and compares two tableaux-style proof systems for Peirce algebras. One is a tableau refutation proof system, the other is a proof system in the style of Rasiowa-Sikorski.  ...  Since Peirce algebras are, in a sense, a join of a Boolean algebra and a relation algebra, the proof systems presented in the paper inherit many features, on the one hand, from the proof systems for first-order  ...  First, we develop two proof systems for the class of Peirce algebras, namely a Rasiowa-Sikorski-style system and tableau system.  ... 
doi:10.1007/978-3-540-24771-5_21 fatcat:p5ei2uyfkjhfhjuahtlnhekxuq

On Automated Program Construction and Verification [chapter]

Rudolf Berghammer, Georg Struth
2010 Lecture Notes in Computer Science  
Based on the standard methods of Dijkstra, Gries and Hoare, it supports a game of proof and refutation with automated theorem provers, model search tools and computer algebra systems combined with "hidden  ...  " domain-specific algebraic theories that have been designed and optimised for automation.  ...  To execute our relational programs and develop them in a relational setting, we use the RelView computer algebra system [5] .  ... 
doi:10.1007/978-3-642-13321-3_4 fatcat:xxip7nxymjcfnmslpok5eov27y

Dr.Doodle: A Diagrammatic Theorem Prover [chapter]

Daniel Winterstein, Alan Bundy, Corin Gurr
2004 Lecture Notes in Computer Science  
The Dr.Doodle system was developed for the domain of metric-space analysis (a geometric domain, but traditionally taught using a dry algebraic formalism).  ...  This paper presents the Dr.Doodle system, an interactive theorem prover that uses diagrammatic representations.  ...  Insight is often more clearly perceived in these diagrammatic proofs than in the corresponding algebraic proofs.  ... 
doi:10.1007/978-3-540-25984-8_24 fatcat:injewouvongungdgmngkrwralm

A note on proximity spaces and connection based mereology

Dimiter Vakarelov, Ivo Düntsch, Brandon Bennett
2001 Proceedings of the international conference on Formal Ontology in Information Systems - FOIS '01  
In this note, we prove a topological representation theorem for a connection based class of systems, using methods and tools from the theory of proximity spaces.  ...  The key novelty is a new proximity semantics for connection relations.  ...  Ivo Düntsch acknowledges a brief discussion with John Stell in 1997 about proximity spaces; however, the topic was not pursued further. References  ... 
doi:10.1145/505168.505182 dblp:conf/fois/VakarelovDB01 fatcat:n7jvnk7flrgwrfij2idd6trgzi

On the Congruence Lattice Characterization Theorem

William A. Lampe
1973 Transactions of the American Mathematical Society  
A simplified proof is given for the theorem characterizing the congruence lattice of a universal algebra.  ...  While in later proofs, [6], [9] or this one, certain systems of congruence relations have been more emphasized.  ...  Incidentally, unary- algebraic systems accomplish in this proof that which íí-closed systems accom- plish in 19], but the concepts are not in general equivalent.  ... 
doi:10.2307/1996519 fatcat:lxivnaq27vfb3dz662d2iosuk4

On the congruence lattice characterization theorem

William A. Lampe
1973 Transactions of the American Mathematical Society  
A simplified proof is given for the theorem characterizing the congruence lattice of a universal algebra.  ...  While in later proofs, [6], [9] or this one, certain systems of congruence relations have been more emphasized.  ...  Incidentally, unary- algebraic systems accomplish in this proof that which íí-closed systems accom- plish in 19], but the concepts are not in general equivalent.  ... 
doi:10.1090/s0002-9947-1973-0325497-5 fatcat:3ckhzxrhuvegbahnrjdjlzlo24
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