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42
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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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.
Discrete Event Calculus Deduction using FirstOrder Automated Theorem Proving
"... Abstract. The event calculus is a powerful and highly usable formalism for reasoning about action and change. The discrete event calculus limits time to integers. This paper shows how discrete event calculus problems can be encoded in firstorder logic, and solved using a firstorder logic automated ..."
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Abstract. The event calculus is a powerful and highly usable formalism for reasoning about action and change. The discrete event calculus limits time to integers. This paper shows how discrete event calculus problems can be encoded in firstorder logic, and solved using a firstorder logic automated theorem proving system. The following techniques are discussed: reification is used to convert event and fluent atoms into firstorder terms, uniquenessofnames axioms are generated to ensure uniqueness of event and fluent terms, predicate completion is used to convert secondorder circumscriptions into firstorder formulae, and a limited firstorder axiomatization of integer arithmetic is developed. The performance of firstorder automated theorem proving is compared to that of satisfiability solving. 1
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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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
CLASSIFICATION RESULTS IN QUASIGROUP AND LOOP THEORY VIA A COMBINATION OF AUTOMATED REASONING TOOLS
"... Abstract. We present some novel classification results in quasigroup and loop theory. For quasigroups up to size 5 and loops up to size 7, we describe a unique property which determines the isomorphism (and in the case of loops, the isotopism) class for any example. These invariant properties were g ..."
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Abstract. We present some novel classification results in quasigroup and loop theory. For quasigroups up to size 5 and loops up to size 7, we describe a unique property which determines the isomorphism (and in the case of loops, the isotopism) class for any example. These invariant properties were generated using a variety of automated techniques – including machine learning and computer algebra – which we present here. Moreover, each result has been automatically verified, again using a variety of techniques – including automated theorem proving, computer algebra and satisfiability solving – and we describe our bootstrapping approach to the generation and verification of these classification results. 1.
RESTRUCTURING FORMAL MATHEMATICS FOR NATURAL TEXTS
, 2004
"... In the presence of growing collections of formal mathematics, and renewed interest in formal mathematics and automated theorem proving for new domains such as hardware or code verification, it is vital to be able to present formal content accessibly to broad audiences. We propose a novel approach to ..."
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In the presence of growing collections of formal mathematics, and renewed interest in formal mathematics and automated theorem proving for new domains such as hardware or code verification, it is vital to be able to present formal content accessibly to broad audiences. We propose a novel approach to constructing a content planner for formal mathematics produced by a tacticstyle prover which capitalizes on the inherent structure of the formal proofs. Though it had been posited that highlevel formal structure is unsuitable as a source of information for text generation, due to its heuristic nature and necessary lack of details, we are able to show that this is not the case. Tacticstyle proofs share significant structural commonality with the discourse structure of corresponding texts. These commonalities allow a content planner to be constructed which need only use lowlevel logical content as a supplementary information source to the generation process. To show that this is the case, we collected two corpora of texts generated to communicate the proof content of a series of formal proofs produced by the Nuprl
Proof Generation for Saturating FirstOrder Theorem Provers
"... Firstorder Automated Theorem Proving (ATP) is one of the oldest and most developed areas of automated reasoning. Today, the most widely used firstorder provers are fully automatic and process firstorder logic with equality. Many stateoftheart ATP systems consist of a clausifier, translating a ..."
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Firstorder Automated Theorem Proving (ATP) is one of the oldest and most developed areas of automated reasoning. Today, the most widely used firstorder provers are fully automatic and process firstorder logic with equality. Many stateoftheart ATP systems consist of a clausifier, translating a full firstorder
Automating Algebraic Specifications of Nonfreely Generated Data Types
"... Abstract. Nonfreely generated data types are widely used in case studies carried out in the theorem prover KIV. The most common examples are stores, sets and arrays. We present an automatic method that generates finite counterexamples for wrong conjectures and therewith offers a valuable support fo ..."
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Abstract. Nonfreely generated data types are widely used in case studies carried out in the theorem prover KIV. The most common examples are stores, sets and arrays. We present an automatic method that generates finite counterexamples for wrong conjectures and therewith offers a valuable support for proof engineers saving their time otherwise spent on unsuccessful proof attempts. The approach is based on the finite model finding and uses Alloy Analyzer [1] to generate finite instances of theories in KIV [6]. Most definitions of functions or predicates on infinite structures do not preserve the semantics if a transition to arbitrary finite substructures is made. We propose the constraints which should be satisfied by the finite substructures, identify a class of amenable definitions and present a practical realization using Alloy. The technique is evaluated on the library of basic data types as well as on some examples from case studies in KIV.
On deciding satisfiability by DPLL(Γ + T) and unsound theorem proving
"... Abstract. Applications in software verification often require determining the satisfiability of firstorder formulæ with respect to some background theories. During development, conjectures are usually false. Therefore, it is desirable to have a theorem prover that terminates on satisfiable instance ..."
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Abstract. Applications in software verification often require determining the satisfiability of firstorder formulæ with respect to some background theories. During development, conjectures are usually false. Therefore, it is desirable to have a theorem prover that terminates on satisfiable instances. Satisfiability Modulo Theories (SMT) solvers have proven highly scalable, efficient and suitable for integrated theory reasoning. Superpositionbased inference systems are strong at reasoning with equalities, universally quantified variables, and Horn clauses. We describe a calculus that tightly integrates Superposition and SMT solvers. The combination is refutationally complete if background theory symbols only occur in ground formulæ, and nonground clauses are variable inactive. Termination is enforced by introducing additional axioms as hypotheses. The calculus detects any unsoundness introduced by these axioms and recovers from it. 1