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

1998
*
Lecture Notes in Computer Science
*

We show how some

doi:10.1007/3-540-49545-2_18
fatcat:hrlp65zylraxxfgi5yoomcuvki
*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. ...##
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Automated Engineering of Relational and Algebraic Methods in Isabelle/HOL
[chapter]

2011
*
Lecture Notes in Computer Science
*

We present a new integration of

doi:10.1007/978-3-642-21070-9_5
fatcat:oszkpd2pfrbsnovtsp7mvn3sha
*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 ...##
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RALL: Machine-supported proofs for relation algebra
[chapter]

1997
*
Lecture Notes in Computer Science
*

We p r e s e n t a theorem proving

doi:10.1007/3-540-63104-6_36
fatcat:3imsczulfncijg6cdpbnlq24za
*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*. ...##
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A Repository for Tarski-Kleene Algebras

2011
*
Workshop on Automated Theory Engineering
*

Main purposes of the repository are the engineering of

dblp:conf/ate/GuttmannSW11
fatcat:gsxscqdtvfhtfkmx2ikwr6yfci
*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*. ...##
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Sufficient Solvability Conditions for Systems of Partial Fuzzy Relational Equations
[chapter]

2020
*
Communications in Computer and Information Science
*

This paper focuses on searching sufficient conditions for the solvability of

doi:10.1007/978-3-030-50146-4_8
fatcat:ozbmwdahsrgmhiwcn3is4znrvi
*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*...##
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A glimpse of deductive systems in algebra

2010
*
Open Mathematics
*

AbstractThe concept of a deductive

doi:10.2478/s11533-010-0041-4
fatcat:5eb6vjs2lndmdmcyy27ceejg3m
*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. ...##
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Compatible Deductive Systems of Pulexes

2014
*
International Journal of Mathematics and Mathematical Sciences
*

We define the quotient

doi:10.1155/2014/606890
fatcat:zk5qcs3o5ff2tcdowxmg46troe
*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. ...##
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System Description: aRa – An Automatic Theorem Prover for Relation Algebras
[chapter]

2000
*
Lecture Notes in Computer Science
*

Employing results from Tarski/Givant and Maddux we can prove validity

doi:10.1007/10721959_13
fatcat:7qtxqcvyrrhrxdtlcnncllqdom
*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. ...##
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Calculational Relation-Algebraic Proofs in Isabelle/Isar
[chapter]

2004
*
Lecture Notes in Computer Science
*

We propose a collection of theories

doi:10.1007/978-3-540-24771-5_16
fatcat:smzaqpd5uzgw7fg5266chuohqu
*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 ...##
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Two Proof Systems for Peirce Algebras
[chapter]

2004
*
Lecture Notes in Computer Science
*

This paper develops and compares two tableaux-style

doi:10.1007/978-3-540-24771-5_21
fatcat:p5ei2uyfkjhfhjuahtlnhekxuq
*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*. ...##
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On Automated Program Construction and Verification
[chapter]

2010
*
Lecture Notes in Computer Science
*

Based on the standard methods of Dijkstra, Gries and Hoare, it supports a game of

doi:10.1007/978-3-642-13321-3_4
fatcat:xxip7nxymjcfnmslpok5eov27y
*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] . ...##
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Dr.Doodle: A Diagrammatic Theorem Prover
[chapter]

2004
*
Lecture Notes in Computer Science
*

The Dr.Doodle

doi:10.1007/978-3-540-25984-8_24
fatcat:injewouvongungdgmngkrwralm
*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*. ...##
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A note on proximity spaces and connection based mereology

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 ...

##
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On the Congruence Lattice Characterization Theorem

1973
*
Transactions of the American Mathematical Society
*

A simplified

doi:10.2307/1996519
fatcat:lxivnaq27vfb3dz662d2iosuk4
*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. ...##
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On the congruence lattice characterization theorem

1973
*
Transactions of the American Mathematical Society
*

A simplified

doi:10.1090/s0002-9947-1973-0325497-5
fatcat:3ckhzxrhuvegbahnrjdjlzlo24
*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. ...
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