Showing papers on "Computational logic published in 1972"

A mathematical introduction to logic

Abstract: USEFUL FACTS ABOUT SETS. SENTENTIAL LOGIC. FIRST-ORDER LOGIC. UNDECIDABILITY. SECOND-ORDER LOGIC.

Implementation and applications of Scott's logic for computable functions

Abstract: The basis for this paper is a logic designed by Dana Scott [1] in 1969 for formalizing arguments about computable functions of higher type. This logic uses typed combinators, and we give a more or less direct translation into typed l-calculus, which is an easier formalism to use, though not so easy for the metatheory because of the presence of bound variables. We then describe, by example only, a proof-checker program which has been implemented for this logic; the program is fully described in [2]. We relate the induction rule which is central to the logic to two more familiar rules - Recursion Induction and Structural Induction - showing that the former is a theorem of the logic, and that for recursively defined structures the latter is a derived rule of the logic. Finally we show how the syntax and semantics of a simple programming language may be described completely in the logic, and we give an example of a theorem which relates syntactic and semantic properties of programs and which can be stated and proved within the logic.

Introduction to elementary mathematical logic

Abstract: This introduction covers the calculus of propositions as well as quantification theory. Presupposing no more than a familiarity with the most elementary principles of logic and mathematics, the book is accessible to the high-school student or the layman desiring a clear and straightforward presentation of the subject that will prepare him to take on the standard, more advanced texts. The book is carefully designed to be self-sufficient for purposes of self-study, and the exercises following the end of each section can be used by the student to gauge the level of his understanding: they serve as "feedback signals" that tell him if he should proceed or review the material just covered.Of particular interest is the application of the logic of propositions to the analysis and synthesis of digital systems, which is presented at the end of the first of the three sections.The first section develops the fundamentals of the logic of propositions. Taken up in turn are objects and operations; formulas, equivalent formulas, and identically true formulas; applications; normal and minimum forms of functions; and applications to digital systems.The second section covers the calculus of propositions. Material is presented on the axiomatic approach, including the consistency, independence, and completeness of a system of axioms in the calculus of propositions.The logic of predicates is the subject of the last section. It treats in some detail operations on sets, defects in the logic of propositions, operations on predicates, quantifiers, equivalent and generally valid formulas, aspects of traditional logic, equality relations, and the axiomatic derivation of the mathematical theory.References to more advanced studies available in English are provided.

Language and logic

Abstract: "What we cannot speak about we must pass over in silence." (Wittgenstein) A puzzle Anybody with just one pound is a poor man. Anybody with just one pound more than a poor man is poor themselves. Is it impossible not to be poor? If not why not? This week's lecture continues on from using Natural Deduction to prove formulae in propositional logic and covers a number of topics which should be dealt with before we look at predicate calculas. General stratagies for proving in Natural Deduction. By now you should be reasonably good at proving simple theorems in Natural Deduction and probably have a set of "rules of thumb"for when to try to solve a problem in a particular way. Tomassi (Logic pp 96-117, 1999) offers a "Golden rule" which is works most of the time: 1. Is the main connective in the conclusion a conditional? If so, assume the antecedent of the conditional and then try and prove the consequent using the premises given. 2. Is the main connective of any of the premises a disjunction? If so try to use the vE rule i.e. assume the first disjunct and try to prove the conclusion and then assume the second disjunct and try to prove the conclusion. Finally draw the conclusion from the original disjunctive premise using vE. 3. Try RAA (reducto ad absurdium) i.e. assume the opposite of what you're trying to prove and then prove that this leads to a contradiction. The double negation rule will then allow you to finish the proof. Let's try these out.

Universal Function Theory and Galois Logic Studies

Abstract: : The report advances work in two areas relevant to logic design in an MSI/LSI technology. First, the theory of universal functions is advanced of the full generality of finite mathematical structures. Second, the theory of Galois logic design is enhanced in various ways. Various Galois lattice-like operations are defined and compared. Next, methods of converting Galois multiplication gates to binary addition gates and to Galois linear gates are described. Finally, techniques are offered using the Galois linear module to reduce hardware at the cost of switching speed.