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davis r truth deduction computation 1989

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Textbook in PDF format The study of programming languages is central to the study of computer science. When studying a programming language, one must learn both the syntax and the semantics of the language in order to understand it, rather than simply be able to use it. Everyone knows and agrees upon what is meant by the syntax of a language, but what do we mean by “semantics” ? What one requires of a description of the semantics of a language is dependent upon one’s point of view. A compiler writer is happy with a description of the operational semantics of a language; someone interested in proving properties of a given program might prefer to use the axiomatic semantics of the language. Because a solid foundation in mathematical logic is also critical to the study of computer science, I have combined an introduction to formal logic with the study of three approaches to semantics of languages. In particular, I chose to investigate the concepts of truth (denotational semantics), deduction (axiomatic semantics), and computation (operational semantics) as they apply to mathematical logic and computer science formalisms (i.e., languages) and to observe how the relationships among these ideas vary as the expressive power of the formalism increases. The investigation of each formalism begins with a description of the syntax of the language. This is simply a description of the symbols allowed and the rules by which the symbols may be combined to form legal sentences of the language. Truth is an intuitive notion. Since the symbols of a formalism (language) have no intrinsic meaning, we speak of truth in an interpretation of a formalism. An interpretation gives meaning to the symbols and sentences of a language by mapping them onto objects in the domain of interpretation. For example, the French map the sequence of symbols “pomme ” onto the object we call “apple.” Deduction involves the formal manipulation of symbols according to prescribed rules. A deductive system, or formal theory, specifies the rules by which derivations can be made. These derivations are strictly syntactic, so they have no intrinsic meaning. In particular, proof is a syntactic notion; its relationship with truth depends upon the formal theory being investigated. Computation is also syntactic in nature and closely related to deduction. Computation suggests an algorithm; we are interested not only in what the legal moves are, but in how we might execute them. Thus a computational scheme must provide some information to control the use of the rules for rewriting symbols

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Davis R. Truth, deduction, and computation...1989.pdf 11.9 MB

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davis r truth deduction computation 1989

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