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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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Cited by 6 (3 self)
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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...
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 logic. (We sh ..."
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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...
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 5 (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.
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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Cited by 5 (1 self)
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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
Learnability of TypeLogical Grammars
, 2001
"... A procedure for learning a lexical assignment together with a system of syntactic and semantic categories given a fixed typelogical grammar is briefly described. The logic underlying the grammar can be any cutfree decidable modally enriched extension of the Lambek calculus, but the correspondence ..."
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A procedure for learning a lexical assignment together with a system of syntactic and semantic categories given a fixed typelogical grammar is briefly described. The logic underlying the grammar can be any cutfree decidable modally enriched extension of the Lambek calculus, but the correspondence between syntactic and semantic categories must be constrained so that no infinite set of categories is ultimately used to generate the language. It is shown that under these conditions various linguistically valuable subsets of the range of the algorithm are classes identifiable in the limit from data consisting of sentences labeled by simply typed lambda calculus meaning terms in normal form. The entire range of the algorithm is shown to be not a learnable class, contrary to a mistaken result reported in a preliminary version of this paper. It is informally argued that, given the right type logic, the learnable classes of grammars include members which generate natural languages, and thus that natural languages are learnable in this way.
What does it mean to say that logic is formal?
, 2000
"... Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and ..."
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Cited by 3 (0 self)
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Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and contingent, analytic and synthetic—indeed, it is often invoked to explain these. Nor, it turns out, can it be explained by appeal to schematic inference patterns, syntactic rules, or grammar. What does it mean, then, to say that logic is distinctively formal? Three things: logic is said to be formal (or “topicneutral”) (1) in the sense that it provides constitutive norms for thought as such, (2) in the sense that it is indifferent to the particular identities of objects, and (3) in the sense that it abstracts entirely from the semantic content of thought. Though these three notions of formality are by no means equivalent, they are frequently run together. The reason, I argue, is that modern talk of the formality of logic has its source in Kant, and these three notions come together in the context of Kant’s transcendental philosophy. Outside of this context (e.g., in Frege), they can come apart. Attending to this
On a localstep cutelimination procedure for the intuitionistic sequent calculus
 Proc. of the 13th Int. Conf. on Logic for Programming Artificial Intelligence and Reasoning (LPAR’06), volume 4246 of LNCS
, 2006
"... Abstract. In this paper we investigate, for intuitionistic implicational logic, the relationship between normalization in natural deduction and cutelimination in a standard sequent calculus. First we identify a subset of proofs in the sequent calculus that correspond to proofs in natural deduction. ..."
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Abstract. In this paper we investigate, for intuitionistic implicational logic, the relationship between normalization in natural deduction and cutelimination in a standard sequent calculus. First we identify a subset of proofs in the sequent calculus that correspond to proofs in natural deduction. Then we define a reduction relation on those proofs that exactly corresponds to normalization in natural deduction. The reduction relation is simulated soundly and completely by a cutelimination procedure which consists of local proof transformations. It follows that the sequent calculus with our cutelimination procedure is a proper extension that is conservative over natural deduction with normalization. 1