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Higherorder narrowing with definitional trees
 Neural Computation
, 1996
"... Functional logic languages with a sound and complete operational semantics are mainly based on narrowing. Due to the huge search space of simple narrowing, steadily improved narrowing strategies have been developed in the past. Needed narrowing is currently the best narrowing strategy for firstord ..."
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Cited by 78 (23 self)
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Functional logic languages with a sound and complete operational semantics are mainly based on narrowing. Due to the huge search space of simple narrowing, steadily improved narrowing strategies have been developed in the past. Needed narrowing is currently the best narrowing strategy for firstorder functional logic programs due to its optimality properties w.r.t. the length of derivations and the number of computed solutions. In this paper, we extend the needed narrowing strategy to higherorder functions and λterms as data structures. By the use of definitional trees, our strategy computes only incomparable solutions. Thus, it is the first calculus for higherorder functional logic programming which provides for such an optimality result. Since we allow higherorder logical variables denoting λterms, applications go beyond current functional and logic programming languages.
Incremental Dynamics
, 1998
"... An incremental semantics for a logic with dynamic binding is developed on the basis of a variable free notation for dynamic logic. The variable free indexing mechanism guarantees that active registers are never overwritten by new quantifier actions. The resulting system has the same expressive power ..."
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Cited by 17 (4 self)
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An incremental semantics for a logic with dynamic binding is developed on the basis of a variable free notation for dynamic logic. The variable free indexing mechanism guarantees that active registers are never overwritten by new quantifier actions. The resulting system has the same expressive power as Dynamic Predicate Logic or Discourse Representation Theory, but comes with a more well behaved consequence relation. A calculus for dynamic reasoning with anaphora is presented and its soundness and completeness are established. Incremental dynamic logic provides an explicit account of anaphoric context and yields new insight into the dynamics of anaphoric linking in reasoning. 1991 Mathematics Subject Classification: 03B65, 68Q55 1991 Computing Reviews Classification System: F.3.1, F.3.2, I.2.4, I.2.7 Keywords and Phrases: dynamic semantics of natural language, complete calculus for dynamic reasoning with anaphora, incremental interpretation, monotonic semantics, anaphora and context ...
Composition and Compilation in Functional Programming Languages
, 1994
"... Functional programming languages, such as Backus' FP, and high level expression oriented languages, such as APL, are examples of programming languages in which the primary method of program construction is the process of composition. In this paper we describe an approach to generating code for langu ..."
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Cited by 6 (2 self)
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Functional programming languages, such as Backus' FP, and high level expression oriented languages, such as APL, are examples of programming languages in which the primary method of program construction is the process of composition. In this paper we describe an approach to generating code for languages based on compositions. The approach involves finding an intermediate representation which grows in size very slowly as additional terms are composed. In particular, the size of the intermediate representation of a composed object should be considerably smaller, and easier to interpret, than the sum of the sizes of the internal representations of the individual elements. We illustrate this technique by showing how to generate conventional code for Backus' language FP. The general technique, however, is applicable to other languages, as well as other architectures. 1 Introduction The purposes of this paper are twofold. First, we want to describe an approach to compilation and code gener...
Execution time of lambdaterms via non uniform semantics and intersection types. Research report
, 2006
"... Abstract. The relational semantics for Linear Logic induces a semantics for the type free Lambda Calculus. This one is built on nonidempotent intersection types. We give a principal typing property for this type system. We then prove that the size of the derivations is closely related to the execut ..."
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Cited by 4 (2 self)
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Abstract. The relational semantics for Linear Logic induces a semantics for the type free Lambda Calculus. This one is built on nonidempotent intersection types. We give a principal typing property for this type system. We then prove that the size of the derivations is closely related to the execution time of lambdaterms in a particular environment machine, Krivine’s machine.
NOTES on LAMBDA CALCULUS
"... INTRODUCTION Lambda calculus, invented by Alonzo Church in the 1930s, is a general but syntactically simple model of computation. It was conceived as part of a system of higherorder logic and function theory. The first undecidability results were for lambda calculus; similar results for Turing mac ..."
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INTRODUCTION Lambda calculus, invented by Alonzo Church in the 1930s, is a general but syntactically simple model of computation. It was conceived as part of a system of higherorder logic and function theory. The first undecidability results were for lambda calculus; similar results for Turing machines came later. In addition to its purely mathematical applications, lambda calculus is important in the study of computer programming languages. It has served as a basic linguistic prototype from which LISP, ALGOLlike languages, and functional programming languages have been derived. It also serves as a basic metalanguage for expressing the denotational semantics of programming languages. These notes are a brief introduction to the typefree lambda calculus. Two versions of the typefree lambda calculus are presented: the callbyname (CBN) and the callbyvalue (CBV) calculi. The CBN calculus was the original version of the lambda
1 HigherOrder Narrowing with Definitional Trees 1
"... Functional logic languages with a sound and complete operational semantics are mainly based on an inference rule called narrowing. Narrowing extends functional evaluation by goal solving capabilities as in logic programming. Due to the huge search space of simple narrowing, steadily improved narrowi ..."
Abstract
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Functional logic languages with a sound and complete operational semantics are mainly based on an inference rule called narrowing. Narrowing extends functional evaluation by goal solving capabilities as in logic programming. Due to the huge search space of simple narrowing, steadily improved narrowing strategies have been developed in the past. Needed narrowing is currently the best narrowing strategy for firstorder functional logic programs due to its optimality properties w.r.t. the length of derivations and the number of computed solutions. In this paper, we extend the needed narrowing strategy to higherorder functions and λterms as data structures. By the use of definitional trees, our strategy computes only independent solutions. Thus, it is the first calculus for higherorder functional logic programming which provides for such an optimality result. Since we allow higherorder logical variables denoting λterms, applications go beyond current functional and logic programming languages. We show soundness and completeness of our strategy with respect to LNT reductions, a particular form of higherorder reductions defined via definitional trees. A general completeness result is only provided for terminating rewrite systems due to the lack of an overall theory of higherorder reduction which is outside the scope of this paper. 1
c Departamento de Matemática, FCEyN,
"... In classical logics, the meaning of a formula is invariant with respect to the renaming of bound variables. This property, normally taken for granted, has been shown not to hold in the case of Information Friendly (IF) logics. In this paper we argue that this is not an inherent characteristic of the ..."
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In classical logics, the meaning of a formula is invariant with respect to the renaming of bound variables. This property, normally taken for granted, has been shown not to hold in the case of Information Friendly (IF) logics. In this paper we argue that this is not an inherent characteristic of these logics but a defect in the way in which the compositional semantics given by Hodges for the regular fragment was generalized to arbitrary formulas. We fix this by proposing an alternative formalization, based on a variation of the classical notion of valuation. Basic metatheoretical results are proven. We present these results for Hodges ’ slash logic (from which these can be easily transferred to other IFlike logics) and we also consider the flattening operator, for which we give novel gametheoretical semantics. Key words: Independence friendly logic, regular formulas, signaling, valuation, compositional semantics, full abstraction, flattening operator. 1.
DOI 10.3233/FI2010306 IOS Press Church–Rosser Made Easy
"... Abstract. The Church–Rosser theorem states that the λcalculus is confluent under βreductions. The standard proof of this result is due to Tait and MartinLöf. In this note, we present an alternative proof based on the notion of acceptable orderings. The technique is easily modified to give conflue ..."
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Abstract. The Church–Rosser theorem states that the λcalculus is confluent under βreductions. The standard proof of this result is due to Tait and MartinLöf. In this note, we present an alternative proof based on the notion of acceptable orderings. The technique is easily modified to give confluence of the βηcalculus. Keywords: lambdacalculus, confluence, Church–Rosser theorem