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Heterogeneous fibring of deductive systems via abstract proof systems
, 2005
"... Fibring is a metalogical constructor that applied to two logics produces a new logic whose formulas allow the mixing of symbols. Homogeneous fibring assumes that the original logics are presented in the same way (e.g via Hilbert calculi). Heterogeneous fibring, allowing the original logics to have ..."
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Fibring is a metalogical constructor that applied to two logics produces a new logic whose formulas allow the mixing of symbols. Homogeneous fibring assumes that the original logics are presented in the same way (e.g via Hilbert calculi). Heterogeneous fibring, allowing the original logics to have different presentations (e.g. one presented by a Hilbert calculus and the other by a sequent calculus), has been an open problem. Herein, consequence systems are shown to be a good solution for heterogeneous fibring when one of the logics is presented in a semantic way and the other by a calculus and also a solution for the heterogeneous fibring of calculi. The new notion of abstract proof system is shown to provide a better solution to heterogeneous fibring of calculi namely because derivations in the fibring keep the constructive nature of derivations in the original logics. Preservation of compactness and semidecidability is investigated.
A History of Satisfiability
, 2009
"... 1.1. Preface: the concept of satisfiability Interest in Satisfiability is expanding for a variety of reasons, not in the least because nowadays more problems are being solved faster by SAT solvers than other means. This is probably because Satisfiability stands at the crossroads of logic, graph theo ..."
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1.1. Preface: the concept of satisfiability Interest in Satisfiability is expanding for a variety of reasons, not in the least because nowadays more problems are being solved faster by SAT solvers than other means. This is probably because Satisfiability stands at the crossroads of logic, graph theory, computer science, computer engineering, and operations research. Thus, many problems originating in one of these fields typically have multiple translations to Satisfiability and there exist many mathematical tools available to the SAT solver to assist in solving them with improved performance. Because of the strong links to so many fields, especially logic, the history of Satisfiability can best be understood as it unfolds with respect to its logic roots. Thus, in addition to timelining events specific to Satisfiability, the chapter follows the presence of Satisfiability in logic as it was developed to model human thought and scientific reasoning through its use in computer design and now as modeling tool for solving a variety of practical problems. In order to succeed in this, we must introduce many ideas that have arisen during numerous attempts to reason
The Role of True Finiteness in the Admissible Recursively Enumerable Degrees
 Memoirs of the American Mathematical Society
"... Abstract. When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In ..."
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Abstract. When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of αfiniteness. As examples we discuss both codings of models of arithmetic into the recursively enumerable degrees, and nondistributive lattice embeddings into these degrees. We show that if an admissible ordinal α is effectively close to ω (where this closeness can be measured by size or by cofinality) then such constructions may be performed in the αr.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the firstorder language of partially ordered sets, and so these results also show that there are natural elementary differences between the structures of αr.e. degrees for various classes of admissible ordinals α. Together with coding work which shows that for some α, the theory of the αr.e. degrees is complicated, we get that for every admissible ordinal
LambdaCalculus and Functional Programming
"... This paper deals with the problem of a program that is essentially the same over any of several types but which, in the older imperative languages must be rewritten for each separate type. For example, a sort routine may be written with essentially the same code except for the types for integers, bo ..."
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This paper deals with the problem of a program that is essentially the same over any of several types but which, in the older imperative languages must be rewritten for each separate type. For example, a sort routine may be written with essentially the same code except for the types for integers, booleans, and strings. It is clearly desirable to have a method of writing a piece of code that can accept the specific type as an argument. Milner developed his ideas in terms of type assignment to lambdaterms. It is based on a result due originally to Curry (Curry 1969) and Hindley (Hindley 1969) known as the principal typescheme theorem, which says that (assuming that the typing assumptions are sufficiently wellbehaved) every term has a principal typescheme, which is a typescheme such that every other typescheme which can be proved for the given term is obtained by a substitution of types for type variables. This use of type schemes allows a kind of generality over all types, which is known as polymorphism.
Arithmetic and the Incompleteness Theorems
, 2000
"... this paper please consult me first, via my home page. ..."