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Switch to: References
Citations of:
The Lambda Calculus. Its Syntax and Semantics
Journal of Symbolic Logic 49 (1):301-303 (1984)
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This article presents an historical and conceptual overview on different approaches to logical abstraction. Two main trends concerning abstraction in the history of logic are highlighted, starting from the logical notions of concept and function. This analysis strictly relates to the philosophical discussion on the nature of abstract objects. I develop this issue further with respect to the procedure of abstraction involved by (typed) λ-systems, focusing on the crucial change about meaning and predicability. In particular, the analysis of the nature (...) |
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It is well known that the circumflex notation used by Russell and Whitehead to form complex function names in Principia Mathematica played a role in inspiring Alonzo Church's “lambda calculus” for functional logic developed in the 1920s and 1930s. Interestingly, earlier unpublished manuscripts written by Russell between 1903–1905—surely unknown to Church—contain a more extensive anticipation of the essential details of the lambda calculus. Russell also anticipated Schönfinkel's combinatory logic approach of treating multiargument functions as functions having other functions as value. (...) |
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Pure Type Systems, PTSs, introduced as a generalisation of the type systems of Barendregt's lambda-cube, provide a foundation for actual proof assistants, aiming at the mechanic verification of formal proofs. In this paper we consider simplifications of some of the rules of PTSs. This is of independent interest for PTSs as this produces more flexible PTS-like systems, but it will also help, in a later paper, to bridge the gap between PTSs and systems of Illative Combinatory Logic. First we consider (...) |
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The principal aim of this paper is to present a construction method for nontotal extensional combinatory algebras. This is done in $\S2$ . In $\S0$ we give definitions of some basic notions for partial combinatory algebras from which the corresponding notions for (total) combinatory algebras are obtained as specializations. In $\S1$ we discuss some properties of nontotal extensional combinatory algebras in general. $\S2$ describes a "partial" variant of reflexive complete partial orders yielding nontotal extensional combinatory algebras. Finally, $\S3$ deals with (...) |
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In the first part ("Determination"), I consider different notions of determination, contrast and compare modal with non-modal accounts and then defend two a-modality theses concerning essence and supervenience. I argue, first, that essence is a a-modal notion, i.e. not usefully analysed in terms of metaphysical modality, and then, contra Kit Fine, that essential properties can be exemplified contingently. I argue, second, that supervenience is also an a-modal notion, and that it should be analysed in terms of constitution relations between properties. (...) |
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Mitchell, J.C. and E. Moggi, Kripke-style models for typed lambda calculus, Annals of Pure and Applied Logic 51 99–124. The semantics of typed lambda calculus is usually described using Henkin models, consisting of functions over some collection of sets, or concrete cartesian closed categories, which are essentially equivalent. We describe a more general class of Kripke-style models. In categorical terms, our Kripke lambda models are cartesian closed subcategories of the presheaves over a poset. To those familiar with Kripke models of (...) |
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Traditional combinatory logic uses combinators S and K to represent all Turing-computable functions on natural numbers, but there are Turing-computable functions on the combinators themselves that cannot be so represented, because they access internal structure in ways that S and K cannot. Much of this expressive power is captured by adding a factorisation combinator F. The resulting SF-calculus is structure complete, in that it supports all pattern-matching functions whose patterns are in normal form, including a function that decides structural equality (...) |
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By the theory TT is meant the higher order predicate logic with the following recursively defined types: 1 is the type of individuals and [] is the type of the truth values: [$\tau_l$,..., $\tau_n$] is the type of the predicates with arguments of the types $\tau_l$,..., $\tau_n$. The theory ITT described in this paper is an intensional version of TT. The types of ITT are the same as the types of TT, but the membership of the type 1 of individuals (...) |
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In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead's Principia Mathematica ([71], 1910-1912) and Church's simply typed λ-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege's (...) |
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One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the representation of reasoning and the resulting systems of computer mathematics on the other hand. |
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This essay examines an argument of perennial importance against naive Leibnizian absolute identity theory, originating with Ruth Barcan in 1947 (Barcan, R. 1947. ?The identity of individuals in a strict functional 3 calculus of second order?, Journal of Symbolic Logic, 12, 12?15), and developed by Arthur Prior in 1962 (Prior, A.N. 1962. Formal Logic. Oxford: The Clarendon Press), presented here in the form offered by Nicholas Griffin in his 1977 book, Relative Identity (Griffin, N. 1977. Relative Identity. Oxford: The Clarendon (...) |
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Translations from Lambda calculi into combinatory logics can be used to avoid some implementational problems of the former systems. However, this scheme can only be efficient if the translation produces short output with a small number of combinators, in order to reduce the time and transient storage space spent during reduction of combinatory terms. In this paper we present a combinatory system and an abstraction algorithm, based on the original bracket abstraction operator of Schonfinkel [9]. The algorithm introduces at most (...) |
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In 1990, J. L. Krivine introduced the notion of storage operator to simulate “call by value” in the “call by name” strategy. J. L. Krivine has showed that, using Gödel translation of classical into intuitionistic logic, one can find a simple type for the storage operators in AF2 type system. This paper studies the ∀-positive types and the Gödel transformations of TTR type system. We generalize by using syntactical methods Krivine's theorem about these types and for these transformations. We give (...) |
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Purpose: The paper discusses the concept of a reflexive domain, an arena where the apparent objects as entities of the domain are actually processes and transformations of the domain as a whole. Human actions in the world partake of the patterns of reflexivity, and the productions of human beings, including science and mathematics, can be seen in this light. Methodology: Simple mathematical models are used to make conceptual points. Context: The paper begins with a review of the author's previous work (...) |
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In the proof-theoretic semantics approach to meaning, harmony , requiring a balance between introduction-rules (I-rules) and elimination rules (E-rules) within a meaning conferring natural-deduction proof-system, is a central notion. In this paper, we consider two notions of harmony that were proposed in the literature: 1. GE-harmony , requiring a certain form of the E-rules, given the form of the I-rules. 2. Local intrinsic harmony : imposes the existence of certain transformations of derivations, known as reduction and expansion . We propose (...) |
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Information is a notion of wide use and great intuitive appeal, and hence, not surprisingly, different formal paradigms claim part of it, from Shannon channel theory to Kolmogorov complexity. Information is also a widely used term in logic, but a similar diversity repeats itself: there are several competing logical accounts of this notion, ranging from semantic to syntactic. In this chapter, we will discuss three major logical accounts of information. |
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We present a new proof for the standardization theorem in $\lambda$-calculus, which is largely built upon a structural induction on $\lambda$-terms. We then extract some bounds for the number of $\beta$-reduction steps in the standard $\beta$-reduction sequence obtained from transforming a given $\beta$-reduction sequence, sharpening the standardization theorem. As an application, we establish a super exponential bound for the lengths of $\beta$-reduction sequences from any given simply typed $\lambda$-terms. |
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We show how to encode context-free string grammars, linear context-free tree grammars, and linear context-free rewriting systems as Abstract Categorial Grammars. These three encodings share the same constructs, the only difference being the interpretation of the composition of the production rules. It is interpreted as a first-order operation in the case of context-free string grammars, as a second-order operation in the case of linear context-free tree grammars, and as a third-order operation in the case of linear context-free rewriting systems. This (...) |
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Using the programming-language concept of continuations, we propose a new, multimodal analysis of quantification in Type Logical Grammar. Our approach provides a geometric view of in-situ quantification in terms of graphs, and motivates the limited use of empty antecedents in derivations. Just as continuations are the tool of choice for reasoning about evaluation order and side effects in programming languages, our system provides a principled, type-logical way to model evaluation order and side effects in natural language. We illustrate with an (...) |
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A new logic is presented without predicates—except equality. Yet its expressive power is the same as that of predicate logic, and relations can faithfully be represented in it. In this logic we also develop an alternative for set theory. There is a need for such a new approach, since we do not live in a world of sets and predicates, but rather in a world of things with relations between them. |
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We are used to the idea that computers operate on numbers, yet another kind of data is equally important: the syntax of formal languages, with variables, binding, and alpha-equivalence. The original application of nominal techniques, and the one with greatest prominence in this paper, is to reasoning on formal syntax with variables and binding. Variables can be modelled in many ways: for instance as numbers (since we usually take countably many of them); as links (since they may `point' to a (...) |
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Nominal logic is a variant of first-order logic in which abstract syntax with names and binding is formalized in terms of two basic operations: name-swapping and freshness. It relies on two important principles: equivariance (validity is preserved by name-swapping), and fresh name generation ("new" or fresh names can always be chosen). It is inspired by a particular class of models for abstract syntax trees involving names and binding, drawing on ideas from Fraenkel-Mostowski set theory: finite-support models in which each value (...) |
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The first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-Löf's universe á la Tarski. A set U 0 of codes for small sets is generated inductively at the same time as a function T 0 , which maps a code to the corresponding small set, is defined by recursion on the way the elements of U 0 are generated. In this paper we argue that there is an underlying general notion of simultaneous inductive-recursive definition which is (...) |
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In this paper we re-examine the semantics of classical higher-order logic with the purpose of clarifying the role of extensionality. To reach this goal, we distinguish nine classes of higher-order models with respect to various combinations of Boolean extensionality and three forms of functional extensionality. Furthermore, we develop a methodology of abstract consistency methods needed to analyze completeness of higher-order calculi with respect to these model classes. |
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Because the main difference between combinatory weak equality and λβ-equality is that the rule \begin{equation*}\tag{\xi} X = Y \vdash \lambda x.X = \lambda x.Y\end{equation*} is valid for the latter but not the former, it is easy to assume that another way of defining combinatory β-equality is to add rule (ξ) to the postulates for weak equality. However, to make this true, one must choose the definition of combinatory abstraction in (ξ) very carefully. If one tries to use one of the (...) |
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A λ-theory T is a consistent set of equations between λ-terms closed under derivability. The degree of T is the degree of the set of Godel numbers of its elements. H is the $\lamda$ -theory axiomatized by the set {M = N ∣ M, N unsolvable. A $\lamda$ -theory is sensible $\operatorname{iff} T \supset \mathscr{H}$ , for a motivation see [6] and [4]. In § it is proved that the theory H is ∑ 0 2 -complete. We present Wadsworth's proof (...) |
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Inspired by Pohlers' local predicativity approach to Pure Proof Theory and Howard's ordinal analysis of bar recursion of type zero we present a short, technically smooth and constructive strong normalization proof for Gödel's system T of primitive recursive functionals of finite types by constructing an ε 0 -recursive function [] 0 : T → ω so that a reduces to b implies [a] $_0 > [b]_0$ . The construction of [] 0 is based on a careful analysis of the Howard-Schütte (...) |
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Abramsky, S., Domain theory in logical form, Annals of Pure and Applied Logic 51 1–77. The mathematical framework of Stone duality is used to synthesise a number of hitherto separate developments in theoretical computer science.• Domain theory, the mathematical theory of computation introduced by Scott as a foundation for detonational semantics• The theory of concurrency and systems behaviour developed by Milner, Hennesy based on operational semantics.• Logics of programsStone duality provides a junction between semantics and logics . Moreover, the underlying (...) |
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In this paper the existence or nonexistence of isomorphic mappings between graph models for the untyped lambda calculus is studied. It is shown that Engeler's D A is completely determined, up to isomorphism, by the cardinality of its `atom-set' A. A similar characterization is given for a collection of graph models of the Pω-type; from this some propositions regarding automorphisms are obtained. Also we give an indication of the complexity of the first-order theory of graph models by showing that the (...) |
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The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop's constructive mathematics (BISH). it gives a sketch of both Myhill's axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focusses on the relation between constructive mathematics and programming, with emphasis on Martin-L6f 's theory of types as a formal system for BISH. |
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In this paper we study (self)-applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full self-application and whose provably total functions on W = {0, 1} * are exactly the polynomial time computable functions. Our treatment of PTO is proof-theoretic and very much in the spirit of reductive proof theory. |
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In 1990, J. L. Krivine introduced the notion of storage operator to simulate, for Church integers, the “call by value” in a context of a “call by name” strategy. In the present paper we define for every λ-term S which realizes the successor function on Church integers the notion of S-storage operator. We prove that every storage operator is an S-storage operator. But the converse is not always true. |
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A minimal formula is a formula which is minimal in provable formulas with respect to the substitution relation. This paper shows the following: (1) A β-normal proof of a minimal formula of depth 2 is unique in NJ. (2) There exists a minimal formula of depth 3 whose βη-normal proof is not unique in NJ. (3) There exists a minimal formula of depth 3 whose βη-normal proof is not unique in NK. |
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In 1990, J.L. Krivine introduced the notion of storage operator to simulate 'call by value' in the 'call by name' strategy. J.L. Krivine has shown that, using Gödel translation of classical into intuitionitic logic, we can find a simple type for the storage operators in AF2 type system. This paper studies the $forall$-positive types (the universal second order quantifier appears positively in these types), and the Gödel transformations (a generalization of classical Gödel translation) of TTR type system. We generalize, by (...) |
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A reduction algebra is defined as a set with a collection of partial unary functions (called reduction operators). Motivated by the lambda calculus, the Church-Rosser property is defined for a reduction algebra and a characterization is given for those reduction algebras satisfying CRP and having a measure respecting the reductions. The characterization is used to give (with 20/20 hindsight) a more direct proof of the strong normalization theorem for the impredicative second order intuitionistic propositional calculus. |
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We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and (...) |
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A Partial Combinatory Algebra is completable if it can be extended to a total one. In [1] it is asked (question 11, posed by D. Scott, H. Barendregt, and G. Mitschke) if every PCA can be completed. A negative answer to this question was given by Klop in [12, 11]; moreover he provided a sufficient condition for completability of a PCA (M, ·, K, S) in the form of ten axioms (inequalities) on terms of M. We prove that just one (...) |
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We present a version of system Fω, called image, in which the layer of type constructors is essentially the traditional one of Fω, whereas provability of types is classical. The proof-term calculus accounting for the classical reasoning is a variant of Barbanera and Berardi’s symmetric λ-calculus.We prove that the whole calculus is strongly normalising. For the layer of type constructors, we use Tait and Girard’s reducibility method combined with orthogonality techniques. For the layer of terms, we use Barbanera and Berardi’s (...) |
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Storage operators have been introduced by J. L. Krivine in [5] they are closed λ-terms which, for a data type, allow one to simulate a "call by value" while using the "call by name" strategy. In this paper, we introduce the directed λ-calculus and show that it has the usual properties of the ordinary λ-calculus. With this calculus we get an equivalent--and simple--definition of the storage operators that allows to show some of their properties: $\bullet$ the stability of the set (...) |
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Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators or, in a more direct way, in which derivations are not translated. Both translations are (...) |
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We investigate a theory of Frege structures extended by the Myhill-Flagg hierarchy of implications. We study its relation to a property theory with an approximation operator and we give a proof theoretical analysis of the basic system involved. MSC: 03F35, 03D60. |
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We work in λH, the untyped λ-calculus in which all unsolvables are identified. We resolve a conjecture of Barendregt asserting that the range of a definable map is either infinite or a singleton. This is refuted by constructing a λ-term Ξ such that ΞM = ΞΙ ⟺ ΞM ≠ ΞΩ. The construction generalizes to ranges of any finite size, and to some other sensible lambda theories. |
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