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Semantic Verification of Web Sites Using Natural Semantics
 PROCEEDINGS OF THE 6TH RIAO CONFERENCE  CONTENTBASED MULTIMEDIA INFORMATION ACCESS
, 2000
"... The huge amount of information and knowledge available on the Web leads to the fact that it is more and more difficult to manage this information. Two different ways are commonly explored: giving a syntactical structure to Web sites, and annotating their content to facilitate Web mining. In this pap ..."
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The huge amount of information and knowledge available on the Web leads to the fact that it is more and more difficult to manage this information. Two different ways are commonly explored: giving a syntactical structure to Web sites, and annotating their content to facilitate Web mining. In this paper we explore a different approach inherited from software engineering: specifying the semantics of Web sites, allowing semantic verifications that will help both the conception and the maintenance of Web sites. To achieve this goal, we have experimented with the application of Natural Semantics (traditionally used to specify the semantics of programming languages) to Web sites specification and verification.
On the Logic and Learning of Language
, 2002
"... algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.1 Homomorphisms and free generators . . . . . . . . . . . . 34 3.1.2 Quotient algebras . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.3 Reducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Algebras of la ..."
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algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.1 Homomorphisms and free generators . . . . . . . . . . . . 34 3.1.2 Quotient algebras . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.3 Reducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Algebras of languages . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.1 The algebra of formulae . . . . . . . . . . . . . . . . . . . 38 3.2.2 Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.3 Associated algebras . . . . . . . . . . . . . . . . . . . . . . 40 3.2.4 Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.5 LindenbaumTarski quotient algebras . . . . . . . . . . . . 42 3.3 Algebras of deductive systems . . . . . . . . . . . . . . . . . . . . 44 3.3.1 Determining a class of algebras . . . . . . . . . . . . . . . 45 3.3.2 Algebra of a sequent calculus . . . . . . . . . . . . . . . . . 46 3.3.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4 Subsuming special cases: an example . . . . . . . . . . . . . . . . 49 3.4.1 The sequent system GL . . . . . . . . . . . . . . . . . . . . 49 3.4.2 The equivalent system t(GL) . . . . . . . . . . . . . . . . . 51 3.4.3 Algebraic models for GL . . . . . . . . . . . . . . . . . . . 52 3.5 Kripke semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4 Categorial type logics 61 4.1 The typed lambda calculus . . . . . . . . . . . . . . . . . . . . . . 62 4.2 Categorial grammar . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3 Forms of Lambek's calculus . . . . . . . . . . . . . . . . . . . . . . 69 4.3.1 Classical CG revisited . . . . . . . . . . . . . . . . . . . . . 70 4.3.2 The nonassociative productfree system . . . . . . . . . . . 70 4.3.3 Addin...
Combining Derivations and Refutations for Cutfree Completeness in BiIntuitionistic Logic
, 2008
"... Biintuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cutfree ” sequent calculus has recently been shown to fail cutelimination. We present a new cutfree se ..."
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Biintuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cutfree ” sequent calculus has recently been shown to fail cutelimination. We present a new cutfree sequent calculus for biintuitionistic logic, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between intuitionistic implication and dual intuitionistic exclusion, similarly to future and past modalities in tense logic. Our calculus handles this interaction using derivations and refutations as first class citizens. We employ extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of refutations, and rules which compose certain refutations and derivations to form derivations. Automated deduction using terminating backward search is also possible, although this is not our main purpose. 1
Labelled Natural Deduction for Substructural Logics
 Logic Journal of the IGPL
, 1997
"... In this paper a uniform methodology to perform Natural Deduction over the family of linear, relevance and intuitionistic logics is proposed. The methodology follows the Labelled Deductive Systems (LDS) discipline, where the deductive process manipulates declarative units  formulas label led acc ..."
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In this paper a uniform methodology to perform Natural Deduction over the family of linear, relevance and intuitionistic logics is proposed. The methodology follows the Labelled Deductive Systems (LDS) discipline, where the deductive process manipulates declarative units  formulas label led according to a labelling algebra. In the system described here, labels are either ground terms or variables of a given labelling language and inference rules manipulate formulas and labels simultaneously, generating (whenever necessary) constraints on the labels used in the rules. A set of natural deduction style inference rules is given, and the notion of a derivation is defined which associates a labelled natural deduction style "structural derivation" with a set of generated constraints. Algorithmic procedures, based on a technique called resource abduction, are defined to solve the constraints generated within a derivation, and their termination conditions discussed. A natural deduc...
A cutfree sequent calculus for biintuitionistic logic: extended version
, 2007
"... Abstract. Biintuitionistic logic is the extension of intuitionistic logic with a connective dual to implication. Biintuitionistic logic was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cutfree ” sequent calculus for BiInt has recently been s ..."
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Abstract. Biintuitionistic logic is the extension of intuitionistic logic with a connective dual to implication. Biintuitionistic logic was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cutfree ” sequent calculus for BiInt has recently been shown by Uustalu to fail cutelimination. We present a new cutfree sequent calculus for BiInt, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between implication and its dual, similarly to future and past modalities in tense logic. Our calculus handles this interaction using extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of failed derivation trees. Our simple termination argument allows our calculus to be used for automated deduction, although this is not its main purpose. 1
Constructing a Tractable Reasoning Framework upon a FineGrained Structural Operational Semantics
, 2008
"... The primary focus of this thesis is the semantic gap between a finegrained structural operational semantics and a set of rely/guaranteestyle development rules. The semantic gap is bridged by considering the development rules to be a part of the same logical framework as the operational semantics, ..."
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Cited by 6 (4 self)
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The primary focus of this thesis is the semantic gap between a finegrained structural operational semantics and a set of rely/guaranteestyle development rules. The semantic gap is bridged by considering the development rules to be a part of the same logical framework as the operational semantics, and a set of soundness proofs show that the development rules, though making development easier for a developer, do not add any extra power to the logical framework as a whole. The soundness proofs given are constructed to take advantage of the structural nature of the language and its semantics; this allows for the addition of new development rules in a modular fashion. The particular language semantics allows for very finegrained concurrency. The language itself includes a construct for nested parallel execution of statements, and the semantics is written so that statements can interfere with each other between individual variable reads. The language also includes an atomic block construct for which the semantics is an embodiment of a form of software transactional memory. The inclusion of the atomic construct helps illustrate the inherent expressive weakness present in the rely/guarantee rules with respect to termination properties. As such, two development rules are proposed for the atomic construct, one of which has serious restrictions in its application, and another for which the termination property does not hold.
Program Extraction, Simplified ProofTerms and Realizability
, 1996
"... This paper forms part of a programme for extracting programs from proofs. Many people have done such work. What distinguishes our work is that our aim is to start with "real proofs", that is to say, proofs in a mathematics book  as opposed to (e.g. computer generated) proofs in formal log ..."
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Cited by 5 (5 self)
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This paper forms part of a programme for extracting programs from proofs. Many people have done such work. What distinguishes our work is that our aim is to start with "real proofs", that is to say, proofs in a mathematics book  as opposed to (e.g. computer generated) proofs in formal logic. (We shall assume that all these proofs are correct.) Then we proceed to produce programs from them in a variant of the lambda calculus. The additions to the lambda calculus are essentially projections, definition by cases and recursion together with (names for) the functions, relations and elements from the mathematical system considered. In order to carry out this programme we do, initially, consider proofs in formal logic. However, as we progress we shall speed up in the same way as a young mathematician develops by treating larger and larger proofs as single steps in order to prove a big theorem. In this paper we show how to extract the programs from proofs in formal logic. Our programs have the advantage of being reusable. That is to say, when we use a theorem A again in the proof of a later theorem B we only need to reuse the old program for A in order to get the new program for B. We do not need to write out the whole proof of A within the proof of B. Full details of this process will be presented in [1]. In the present paper we first sketch our method for extracting the programs. (We shall give a full account in [1].) Then we consider a variant of lambda calculus using the constructs we have mentioned. Our main result here is that the formal terms thus constructed are realizers i.e. we give a semantics for the system of mathematics we consider. For clarity we restrict our attention to arithmetic, including induction. The technique, however, applies immediately to any first o...
Craig’s interpolation theorem formalised and mechanised
 in Isabelle/HOL. Logic in Computer Science
, 2006
"... We formalise and mechanise a construtive, proof theoretic proof of Craig’s Interpolation Theorem in Isabelle/HOL. We give all the definitions and lemma statements both formally and informally. We also transcribe informally the formal proofs. We detail the main features of our mechanisation, such as ..."
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We formalise and mechanise a construtive, proof theoretic proof of Craig’s Interpolation Theorem in Isabelle/HOL. We give all the definitions and lemma statements both formally and informally. We also transcribe informally the formal proofs. We detail the main features of our mechanisation, such as the formalisation of binding for first order formulae. We also give some applications of Craig’s Interpolation Theorem. 1.
External and internal syntax of the λcalculus
 In: Buchberger, Ida, Kutsia (Eds.), Proc. of the AustrianJapanese Workshop on Symbolic Computation in Software Science, SCSS 2008. No. 08–08 in RISCLinz Report Series
"... There is growing interest in the study of the syntactic structure of expressions equipped with a variable binding mechanism. The importance of this study can be justified for various reasons, e.g. educational, scientific and engineering reasons. This study is educationally important since in logic a ..."
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There is growing interest in the study of the syntactic structure of expressions equipped with a variable binding mechanism. The importance of this study can be justified for various reasons, e.g. educational, scientific and engineering reasons. This study is educationally important since in logic and computer science, we cannot avoid teaching the