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Knowledge Archives in Theorema: A LogicInternal Approach
"... Archives are implemented as an extension of Theorema for representing mathematical repositories in a natural way. An archive can be conceived as one large formula in a language consisting of higherorder predicate logic together with a few constructs for structuring knowledge: attaching labels to ..."
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Archives are implemented as an extension of Theorema for representing mathematical repositories in a natural way. An archive can be conceived as one large formula in a language consisting of higherorder predicate logic together with a few constructs for structuring knowledge: attaching labels to subhierarchies, disambiguating symbols by the use of namespaces, importing symbols from other namespaces and specifying the domains of categories and functors as namespaces with variable operations. All these constructs are logicinternal in the sense that they have a natural translation to higherorder logic so that certain aspects of Mathematical Knowledge Management can be realized in the object logic itself. There are a variety of operations on archives, though in this paper we can only sketch a few of them: knowledge retrieval and theory exploration, merging and splitting, insertion and translation to predicate logic.
Publication/citation: A prooftheoretic approach to mathematical knowledge management
, 2005
"... There are many reallife examples of formal systems that support constructions or proofs, but that do not provide direct support for remembering them so that they can be recalled and reused in the future. In this paper we examine the operations of publication (remembering a proof) and citation (reca ..."
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There are many reallife examples of formal systems that support constructions or proofs, but that do not provide direct support for remembering them so that they can be recalled and reused in the future. In this paper we examine the operations of publication (remembering a proof) and citation (recalling a proof for reuse), regarding them as forms of common subexpression elimination on proof terms. We then develop this idea from a proof theoretic perspective, describing a simple complete proof system for universal Horn equational logic using three new proof rules, publish, cite and forget. These rules can provide a prooftheoretic infrastructure for proof reuse in any system. 1
Mathematical Knowledge Archives in Theorema
"... Archives are implemented as an extension of Theorema for representing large bodies of mathematics. They provide various constructs for organizing knowledge bases in a natural way: breaking formulae across cells, grouping them in a hierarchical structure, attaching labels to subhierarchies, disambigu ..."
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Archives are implemented as an extension of Theorema for representing large bodies of mathematics. They provide various constructs for organizing knowledge bases in a natural way: breaking formulae across cells, grouping them in a hierarchical structure, attaching labels to subhierarchies, disambiguating symbols by the use of namespaces, importing symbols from other namespaces, addressing the domains of categories and functors as namespaces with variable operations. All constructs are logic–internal in the sense that they have a natural translation to higher–order logic so that "mathematical knowledge management" can be treated by the object logic itself.
KAT and PHL in Coq
"... In this article we describe an implementation of Kleene algebra with tests (KAT) in the Coq theorem prover. KAT is an equational system that has been successfully applied in program verification and, in particular, it subsumes the propositional Hoare logic (PHL). We also present an PHL encoding in K ..."
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In this article we describe an implementation of Kleene algebra with tests (KAT) in the Coq theorem prover. KAT is an equational system that has been successfully applied in program verification and, in particular, it subsumes the propositional Hoare logic (PHL). We also present an PHL encoding in KAT, by deriving its deduction rules as theorems of KAT. Some examples of simple program's formal correctness are given. This work is part of a study of the feasibility of using KAT in the automatic production of certificates in the context of (sourcelevel) ProofCarryingCode (PCC).
Using Probabilistic Kleene Algebra pKA for Protocol Verification
"... We propose a method for verification of probabilistic distributed systems in which a variation of Kozen’s Kleene Algebra with Tests [11] is used to take account of the wellknown interaction of probability and “adversarial ” scheduling [17]. We describe pKA, a probabilistic Kleenestyle algebra, bas ..."
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We propose a method for verification of probabilistic distributed systems in which a variation of Kozen’s Kleene Algebra with Tests [11] is used to take account of the wellknown interaction of probability and “adversarial ” scheduling [17]. We describe pKA, a probabilistic Kleenestyle algebra, based on a widely accepted model of probabilistic/demonic computation [7,25,17]. Our technical aim is to express probabilistic versions of Cohen’s separation theorems[4]. Separation theorems simplify reasoning about distributed systems, where with purely algebraic reasoning they can reduce complicated interleaving behaviour to “separated ” behaviours each of which can be analysed on its own. Until now that has not been possible for probabilistic distributed systems. We present two case studies. The first treats a simple voting mechanism in the algebraic style, and the second — based on Rabin’s Mutual exclusion with bounded waiting [12] — is one where verification problems have already occurred: the original presentation was later shown to have subtle flaws [24]. It motivates our interest in algebras, where assumptions relating probability and secrecy are clearly exposed and, in some cases, can be given simple characterisations in spite of their intricacy. Finally we show how the algebraic proofs for these theorems can be automated using a modification of Kozen and AboulHosn’s KATML [3].
UDC 004.421 KAT and PHL in Coq
"... Abstract. In this article we describe an implementation of Kleene algebra with tests (KAT) in the Coq theorem prover. KAT is an equational system that has been successfully applied in program verification and, in particular, it subsumes the propositional Hoare logic (PHL). We also present an PHL enc ..."
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Abstract. In this article we describe an implementation of Kleene algebra with tests (KAT) in the Coq theorem prover. KAT is an equational system that has been successfully applied in program verification and, in particular, it subsumes the propositional Hoare logic (PHL). We also present an PHL encoding in KAT, by deriving its deduction rules as theorems of KAT. Some examples of simple program's formal correctness are given. This work is part of a study of the feasibility of using KAT in the automatic production of certificates in the context of (sourcelevel) ProofCarryingCode (PCC).