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A proof of the Kepler conjecture
 Math. Intelligencer
, 1994
"... This section describes the structure of the proof of ..."
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Cited by 107 (11 self)
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This section describes the structure of the proof of
Knowledge Representation and Classical Logic
"... Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspe ..."
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Cited by 10 (4 self)
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Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspective of knowledge representation, their formal languages were not sufficiently expressive. On the other hand, most logicians were not concerned about the possibility of automated reasoning; from the perspective of knowledge representation, they were often too generous in the choice of syntactic constructs. In spite of these differences, classical mathematical logic has exerted significant influence on knowledge representation research, and it is appropriate to begin this handbook with a discussion of the relationship between these fields. The language of classical logic that is most widely used in the theory of knowledge representation is the language of firstorder (predicate) formulas. These are the formulas that John McCarthy proposed to use for representing declarative knowledge in his advice taker paper [176], and Alan Robinson proposed to prove automatically using resolution [236]. Propositional logic is, of course, the most important subset of firstorder logic; recent
A proof of the dodecahedral conjecture
, 1998
"... This article gives a proof of Fejes Tóth’s Dodecahedral conjecture: the volume of a Voronoi polyhedron in a threedimensional packing of balls of unit radius is at least the volume of a regular dodecahedron of unit inradius. 1 ..."
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Cited by 5 (2 self)
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This article gives a proof of Fejes Tóth’s Dodecahedral conjecture: the volume of a Voronoi polyhedron in a threedimensional packing of balls of unit radius is at least the volume of a regular dodecahedron of unit inradius. 1
Certified exact real arithmetic using coinduction in arbitrary integer base
 in "Functional and Logic Programming Symposium (FLOPS)", Lecture Notes in Computer Science
"... arbitrary integer base ..."
Formalising the πcalculus using Nominal Logic
"... Abstract. We formalise the picalculus using the nominal datatype package, a package based on ideas from the nominal logic by Pitts et al., and demonstrate an implementation in Isabelle/HOL. The purpose is to derive powerful induction rules for the semantics in order to conduct machine checkable pro ..."
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Abstract. We formalise the picalculus using the nominal datatype package, a package based on ideas from the nominal logic by Pitts et al., and demonstrate an implementation in Isabelle/HOL. The purpose is to derive powerful induction rules for the semantics in order to conduct machine checkable proofs, closely following the intuitive arguments found in manual proofs. In this way we have covered many of the standard theorems of bisimulation equivalence and congruence, both late and early, and both strong and weak in a unison manner. We thus provide one of the most extensive formalisations of a process calculus ever done inside a theorem prover. A significant gain in our formulation is that agents are identified up to alphaequivalence, thereby greatly reducing the arguments about bound names. This is a normal strategy for manual proofs about the picalculus, but that kind of hand waving has previously been difficult to incorporate smoothly in an interactive theorem prover. We show how the nominal logic formalism and its support in Isabelle accomplishes this and thus significantly reduces the tedium of conducting completely formal proofs. This improves on previous work using weak higher order abstract syntax since we do not need extra assumptions to filter out exotic terms and can keep all arguments within a familiar firstorder logic.
Flyspeck in a Semantic Wiki Collaborating on a Large Scale Formalization of the Kepler Conjecture
"... Abstract. Semantic wikis have been successfully applied to many problems in knowledge management and collaborative authoring. They are particularly appropriate for scientific and mathematical collaboration. In previous work we described an ontology for mathematical knowledge based on the semantic ma ..."
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Abstract. Semantic wikis have been successfully applied to many problems in knowledge management and collaborative authoring. They are particularly appropriate for scientific and mathematical collaboration. In previous work we described an ontology for mathematical knowledge based on the semantic markup language OMDoc and a semantic wiki using both. We are now evaluating these technologies in concrete application scenarios. In this paper we evaluate the applicability of our infrastructure to mathematical knowledge management by focusing on the Flyspeck project, a formalization of Thomas Hales ’ proof of the Kepler Conjecture. After describing the Flyspeck project and its requirements in detail, we evaluate the applicability of two wiki prototypes to Flyspeck, one based on Semantic MediaWiki and another on our mathematicsspecific semantic wiki SWiM. 1 Scientific Communication and the Flyspeck Project Scientific communication consists mainly of exchanging documents, and a great deal of scientific work consists of collaboratively authoring them. Common instances are writing down first hypotheses, commenting on results of experiments or project steps, and structuring, annotating, or reorganizing existing items of knowledge, as depicted in Buchberger’s figure on the right.
An interpretation of isabelle/hol in hol light
 In Furbach and Shankar [20
"... Abstract. We define an interpretation of the Isabelle/HOL logic in HOL Light and its metalanguage, OCaml. Some aspects of the Isabelle logic are not representable directly in the HOL Light object logic. The interpretation thus takes the form of a set of elaboration rules, where features of the Isabe ..."
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Abstract. We define an interpretation of the Isabelle/HOL logic in HOL Light and its metalanguage, OCaml. Some aspects of the Isabelle logic are not representable directly in the HOL Light object logic. The interpretation thus takes the form of a set of elaboration rules, where features of the Isabelle logic that cannot be represented directly are elaborated to functors in OCaml. We demonstrate the effectiveness of the interpretation via an implementation, translating a significant part of the Isabelle standard library into HOL Light. 1
Animating the Formalised Semantics of a Javalike Language
"... Abstract. Considerable effort has gone into the techniques of extracting executable code from formal specifications and animating them. We show how to apply these techniques to the large JinjaThreads formalisation. It models a substantial subset of multithreaded Java source and bytecode in Isabelle/ ..."
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Abstract. Considerable effort has gone into the techniques of extracting executable code from formal specifications and animating them. We show how to apply these techniques to the large JinjaThreads formalisation. It models a substantial subset of multithreaded Java source and bytecode in Isabelle/HOL and focuses on proofs and modularity whereas code generation was of little concern in its design. Employing Isabelle’s code generation facilities, we obtain a verified Java interpreter that is sufficiently efficient for running small Java programs. To this end, we present refined implementations for common notions such as the reflexive transitive closure and Russell’s definite description operator. From our experience, we distill simple guidelines on how to develop future formalisations with executability in mind. 1
Mechanized quantifier elimination for linear realarithmetic in Isabelle/HOL
"... Abstract. We integrate Ferrante and Rackoff’s quantifier elimination procedure for linear real arithmetic in Isabelle/HOL in two manners: (a) tacticstyle, i.e. for every problem instance a proof is generated by invoking a series of inference rules, and (b) reflection, where the whole algorithm is i ..."
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Abstract. We integrate Ferrante and Rackoff’s quantifier elimination procedure for linear real arithmetic in Isabelle/HOL in two manners: (a) tacticstyle, i.e. for every problem instance a proof is generated by invoking a series of inference rules, and (b) reflection, where the whole algorithm is implemented and verified within Isabelle/HOL. We discuss the performance obtained for both integrations. 1
Proof Pearl: A Probabilistic Proof for the GirthChromatic Number Theorem
"... Abstract. The GirthChromatic number theorem is a theorem from graph theory, stating that graphs with arbitrarily large girth and chromatic number exist. We formalize a probabilistic proof of this theorem in the Isabelle/HOL theorem prover, closely following a standard textbook proof and use this to ..."
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Abstract. The GirthChromatic number theorem is a theorem from graph theory, stating that graphs with arbitrarily large girth and chromatic number exist. We formalize a probabilistic proof of this theorem in the Isabelle/HOL theorem prover, closely following a standard textbook proof and use this to explore the use of the probabilistic method in a theorem prover. 1