Book•
Outline of a Mathematical Theory of Computation
01 Nov 1970-
TL;DR: However, Scott does realize that the approach argued for above is simply an argument for an approach that accomodates human understanding of computation and that the operational approach must not be ignored because the machines that the programs of study run on are not capable of dealing with such an abstract level of understanding.
Abstract: However, Scott does realize that the approach argued for above is simply an argument for an approach that accomodates human understanding of computation and that the operational approach must not be ignored because, as he points out, the machines that the programs of study run on are not capable of dealing with such an abstract level of understanding. That is, the computaional approach should not be abandoned because the machines that we build operate on that lower level.
Citations
More filters
IBM1
TL;DR: A new class of computing systems uses the functional programming style both in its programming language and in its state transition rules; these systems have semantics loosely coupled to states—only one state transition occurs per major computation.
Abstract: Conventional programming languages are growing ever more enormous, but not stronger. Inherent defects at the most basic level cause them to be both fat and weak: their primitive word-at-a-time style of programming inherited from their common ancestor—the von Neumann computer, their close coupling of semantics to state transitions, their division of programming into a world of expressions and a world of statements, their inability to effectively use powerful combining forms for building new programs from existing ones, and their lack of useful mathematical properties for reasoning about programs.An alternative functional style of programming is founded on the use of combining forms for creating programs. Functional programs deal with structured data, are often nonrepetitive and nonrecursive, are hierarchically constructed, do not name their arguments, and do not require the complex machinery of procedure declarations to become generally applicable. Combining forms can use high level programs to build still higher level ones in a style not possible in conventional languages.Associated with the functional style of programming is an algebra of programs whose variables range over programs and whose operations are combining forms. This algebra can be used to transform programs and to solve equations whose “unknowns” are programs in much the same way one transforms equations in high school algebra. These transformations are given by algebraic laws and are carried out in the same language in which programs are written. Combining forms are chosen not only for their programming power but also for the power of their associated algebraic laws. General theorems of the algebra give the detailed behavior and termination conditions for large classes of programs.A new class of computing systems uses the functional programming style both in its programming language and in its state transition rules. Unlike von Neumann languages, these systems have semantics loosely coupled to states—only one state transition occurs per major computation.
2,651 citations
Book•
[...]
TL;DR: In this paper, the Curry-Howard isomorphism and the normalisation theorem of a natural deduction system T coherence spaces have been studied in the context of linear logic and linear logic semantics.
Abstract: Sense, denotation and semantics natural deduction the Curry-Howard isomorphism the normalisation theorem Godel's system T coherence spaces denotational semantics of T sums in natural deduction system F coherence semantics of the sum cut elimination (Hauptsatz) strong normalisation for F representation theorem semantics of System F what is linear logic?
1,771 citations
01 Jan 1977
TL;DR: It is argued that a sophisticated question-answering machine that has the capability of making inferences from its data base should employ a certain four-valued logic, the motivating consideration being that minor inconsistencies in its data should not be allowed to lead to irrelevant conclusions.
Abstract: It is argued that a sophisticated question-answering machine that has the capability of making inferences from its data base should employ a certain four-valued logic, the motivating consideration being that minor inconsistencies in its data should not be allowed to lead (as in classical logic) to irrelevant conclusions. The actual form of the four-valued logic is ‘deduced’ from an interplay of this motivating consideration with certain ideas of Dana Scott concerning ‘approximation lattices.’
1,477 citations
Book•
12 Mar 2014TL;DR: This book provides a solid fundament for studying various aspects of computability and complexity in analysis and is written in a style suitable for graduate-level and senior students in computer science and mathematics.
Abstract: Merging fundamental concepts of analysis and recursion theory to a new exciting theory, this book provides a solid fundament for studying various aspects of computability and complexity in analysis. It is the result of an introductory course given for several years and is written in a style suitable for graduate-level and senior students in computer science and mathematics. Many examples illustrate the new concepts while numerous exercises of varying difficulty extend the material and stimulate readers to work actively on the text.
1,330 citations
Proceedings Article•
[...]
15 Jan 1995
TL;DR: In this paper, the authors define abstract bases as the bases of compact elements of algebraic domains and define the notion of ideal completion as the relation with which a basis can be equipped.
Abstract: bases were introduced in [Smy77] where they are called “R-structures”. Examples of abstract bases are concrete bases of continuous domains, of course, where the relation≺ is the restriction of the order of approximation. Axiom (INT) is satisfied because of Lemma 2.2.15 and because we have required bases in domains to have directed sets of approximants for each element. Other examples are partially ordered sets, where (INT) is satisfied because of reflexivity. We will shortly identify posets as being exactly the bases of compact elements of algebraic domains. In what follows we will use the terminology developed at the beginning of this chapter, even though the relation ≺ on an abstract basis need neither be reflexive nor antisymmetric. This is convenient but in some instances looks more innocent than it is. An idealA in a basis, for example, has the property (following from directedness) that for everyx ∈ A there is another element y ∈ A with x ≺ y. In posets this doesn’t mean anything but here it becomes an important feature. Sometimes this is stressed by using the expression ‘ A is a round ideal’. Note that a set of the form↓x is always an ideal because of (INT) but that it need not contain x itself. We will refrain from calling ↓x ‘principal’ in these circumstances. Definition 2.2.21. For a basis〈B,≺〉 let Idl(B) be the set of all ideals ordered by inclusion. It is called theideal completionof B. Furthermore, leti : B → Idl(B) denote the function which maps x ∈ B to ↓x. If we want to stress the relation with whichB is equipped then we write Idl(B,≺) for the ideal completion. Proposition 2.2.22.Let 〈B,≺〉 be an abstract basis.
1,210 citations