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Fresh Logic
 Journal of Applied Logic
, 2007
"... Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with resp ..."
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Cited by 183 (21 self)
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Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with respect to metavariables. We present oneandahalfthorder logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of oneandahalfthorder logic derivability, show them equivalent, show that the derivations satisfy cutelimination, and prove correctness of an interpretation of firstorder logic within it. We discuss the technicalities in a wider context as a casestudy for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation
A Shortest 2Basis for Boolean Algebra in Terms of the Sheffer Stroke
 J. Automated Reasoning
, 2003
"... In this article, we present a short 2basis for Boolean algebra in terms of the Sheffer stroke and prove that no such 2basis can be shorter. We also prove that the new 2basis is unique (for its length) up to applications of commutativity. Our proof of the 2basis was found by using the method of p ..."
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Cited by 9 (6 self)
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In this article, we present a short 2basis for Boolean algebra in terms of the Sheffer stroke and prove that no such 2basis can be shorter. We also prove that the new 2basis is unique (for its length) up to applications of commutativity. Our proof of the 2basis was found by using the method of proof sketches and relied on the use of an automated reasoning program.
On Automating the Calculus of Relations
 In: Proc. IJCAR. Vol. 5195. LNCS
, 2008
"... Abstract. Relation algebras provide abstract equational axioms for the calculus of binary relations. They name an established area of mathematics with various applications in computer science. We prove more than hundred theorems of relation algebras with offtheshelf automated theorem provers. This ..."
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Cited by 7 (2 self)
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Abstract. Relation algebras provide abstract equational axioms for the calculus of binary relations. They name an established area of mathematics with various applications in computer science. We prove more than hundred theorems of relation algebras with offtheshelf automated theorem provers. This yields a basic calculus from which more advance applications can be explored. Here, we present two examples from the formal methods literature. Our experiments not only further underline the feasibility of automated deduction in complex algebraic structures and provide theorem proving benchmarks, they also pave the way for lifting established formal methods such as B or Z to a new level of automation. 1
NonBoolean Descriptions for MindMatter Problems
"... A framework for the mindmatter problem in a holistic universe which has no parts is outlined. The conceptual structure of modern quantum theory suggests to use complementary Boolean descriptions as elements for a more comprehensive nonBoolean description of a world without an apriorigiven mindmat ..."
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Cited by 6 (0 self)
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A framework for the mindmatter problem in a holistic universe which has no parts is outlined. The conceptual structure of modern quantum theory suggests to use complementary Boolean descriptions as elements for a more comprehensive nonBoolean description of a world without an apriorigiven mindmatter distinction. Such a description in terms of a locally Boolean but globally nonBoolean structure makes allowance for the fact that Boolean descriptions play a privileged role in science. If we accept the insight that there are no ultimate building blocks, the existence of holistic correlations between contextually chosen parts is a natural consequence. The main problem of a genuinely nonBoolean description is to find an appropriate partition of the universe of discourse. If we adopt the idea that all fundamental laws of physics are invariant under time translations, then we can consider a partition of the world into a tenseless and a tensed domain. In the sense of a regulative principle, the material domain is defined as the tenseless domain with its homogeneous time. The tensed domain contains the mental domain with a tensed time characterized by a privileged position, the Now. Since this partition refers to two complementary descriptions which are not given apriori,wehavetoexpectcorrelations between these two domains. In physics it corresponds to Newton’s separation of universal laws of nature and contingent initial conditions. Both descriptions have a nonBoolean structure and can be encompassed into a single nonBoolean description. Tensed and tenseless time can be synchronized by holistic correlations. 1.
Automated discovery of single axioms for ortholattices
 Algebra Universalis
, 2005
"... Abstract. We present short single axioms for ortholattices, orthomodular lattices, and modular ortholattices, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. We also give multiequation bases in terms of the Sheffer stroke and in terms of join, meet, and complemen ..."
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Cited by 4 (1 self)
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Abstract. We present short single axioms for ortholattices, orthomodular lattices, and modular ortholattices, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. We also give multiequation bases in terms of the Sheffer stroke and in terms of join, meet, and complementation. Proofs are omitted but are available in an associated technical report and on the Web. We used computers extensively to find candidates, reject candidates, and search for proofs that candidates are single axioms. 1.
A Grand Challenge of Theorem Discovery
 Proceedings of the Workshop on Challenges and Novel Applications for Automated Reasoning, 19th International Conference on Automated Reasoning
, 2003
"... Abstract. A primary mode of operation of ATP systems is to supply the system with axioms and a conjecture, and to then ask the system to produce a proof (or at least an assurance that there is a proof) that the conjecture is a theorem of the axioms. This paper challenges ATP to a new mode of operati ..."
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Cited by 4 (1 self)
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Abstract. A primary mode of operation of ATP systems is to supply the system with axioms and a conjecture, and to then ask the system to produce a proof (or at least an assurance that there is a proof) that the conjecture is a theorem of the axioms. This paper challenges ATP to a new mode of operation, by which interesting theorems are generated from a set of axioms. The challenge requires solutions to both theoretical and computational issues. 1
Automated theorem proving in quasigroup and loop theory
 NORTHERN MICHIGAN UNIVERSITY, MARQUETTE, MI 49855 USA
"... We survey all known results in the area of quasigroup and loop theory to have been obtained with the assistance of automated theorem provers. We provide both informal and formal descriptions of selected problems, and compare the performance of selected stateofthe art first order theorem provers on ..."
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Cited by 3 (2 self)
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We survey all known results in the area of quasigroup and loop theory to have been obtained with the assistance of automated theorem provers. We provide both informal and formal descriptions of selected problems, and compare the performance of selected stateofthe art first order theorem provers on them. Our analysis yields some surprising results, e.g., the theorem prover most often used by loop theorists does not necessarily yield the best performance.
Short equational bases for ortholattices
 Preprint ANL/MCSP10870903, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL
, 2004
"... Short single axioms for ortholattices, orthomodular lattices, and modular ortholattices are presented, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. Other equational bases in terms of the Sheffer stroke and in terms of join, meet, and complement are presented. ..."
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Cited by 3 (3 self)
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Short single axioms for ortholattices, orthomodular lattices, and modular ortholattices are presented, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. Other equational bases in terms of the Sheffer stroke and in terms of join, meet, and complement are presented. Proofs are omitted but are available in an associated technical report. Computers were used extensively to find candidates, reject candidates, and search for proofs that candidates are single axioms. The notion of computer proof is addressed. 1
Discovering Boundary Algebra: A Simple Notation for Boolean Algebra and the Truth Functors
 International Journal of General Systems
, 2003
"... Boundary algebra is a new and simple notation for the Boolean algebra 2 and the truth functors. The primary arithmetic [PA] is built up from the atoms, ‘() ’ and the blank page, by enclosure between ‘( ‘ and ‘)’, denoting the primitive notion of distinction, and concatenation. Inserting letters deno ..."
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Cited by 1 (0 self)
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Boundary algebra is a new and simple notation for the Boolean algebra 2 and the truth functors. The primary arithmetic [PA] is built up from the atoms, ‘() ’ and the blank page, by enclosure between ‘( ‘ and ‘)’, denoting the primitive notion of distinction, and concatenation. Inserting letters denoting the presence or absence of () into a PA formula yields boundary algebra [BA], a simpler notation for SpencerBrown’s (1969) primary algebra [pa]. The BA axioms are “()()=()”, and “(()) [=⊥] may be written or erased at will.” Repeated application of these axioms to a PA formula yields a member of B={(),⊥}, its simplification. If (a)b [dually (a(b))] ⇔ a≤b, then ⊥≤() [()≤⊥] follows trivially, so that B is a poset. BA is a selfdual notation for the Boolean algebra 2: (a) ⇔ a′, () ⇔ 1 [0] so that B is the carrier for 2, and ab ⇔ a∪b [a∩b]. The basis abc=bca (Dilworth 1938), a(ab) = a(b) (Bricken 2002), and a(a)=() facilitates clausal reasoning and proof by calculation. BA also simplifies the usual normal forms and Quine’s (1982) truth value analysis. () ⇔ true [false] yields boundary logic.
Using Automated Theorem Provers in Nonassociative Algebra
"... We present a case study on how mathematicians use automated theorem provers to solve open problems in (nonassociative) algebra. 1 ..."
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We present a case study on how mathematicians use automated theorem provers to solve open problems in (nonassociative) algebra. 1