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Computability and recursion
 BULL. SYMBOLIC LOGIC
, 1996
"... We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they b ..."
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We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of “computability” and “recursion.” Specifically we recommend that: the term “recursive ” should no longer carry the additional meaning of “computable” or “decidable;” functions defined using Turing machines, register machines, or their variants should be called “computable” rather than “recursive;” we should distinguish the intensional difference between Church’s Thesis and Turing’s Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be “Computability Theory” or simply Computability rather than
The inference problem for template dependencies
, 1982
"... A template dependency is a formalized integrity constraint on a relational database, stating that whenever tuples exist in the database that agree on certain attributes, an additional tuple must also be present that agrees with the others in a specified way. It is shown that the inference problem fo ..."
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Cited by 13 (1 self)
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A template dependency is a formalized integrity constraint on a relational database, stating that whenever tuples exist in the database that agree on certain attributes, an additional tuple must also be present that agrees with the others in a specified way. It is shown that the inference problem for template dependencies is undecidable, that is, there can be no algorithm for determining whether a given dependency is a logical consequence of a given finite set of dependencies. The undecidability result holds whether or not databases are considered to be necessarily finite. INTROD UCTION The goal of dependency theory is to formalize constraints on the data comprising a relational database. In general, a dependency is a statement to the effect that when certain tuples are present in the database, so are certain others. Such statements can be used, for example, to capture the idea that attributes are functionally related or independent in some way. Many varieties of dependencies have been proposed in the literature; see the discussions in Fagin (1980) and Yannakakis and Papadimitriou (1980), for example. The proliferation of varieties is due in part to the desire to balance two opposing forces: on the one hand, dependencies should be of a form general enough to express interesting properties, but on the other hand, the form should not be so general that natural questions about dependencies become undecidable or computationally intractable. A significant question about any class of dependencies is its inference problem: Given a finite set D of dependencies and a single dependency D 0, to determine whether D O is true in every database in which each member of D is true. A solution to the inference problem carries with it the ability to determine whether two sets of
The word problem for cancellation semigroups with zero
 Journal of Symbolic Logic
, 1984
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at. ..."
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Cited by 6 (0 self)
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at.
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
, 2009
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Turing’s Mathematical Work
"... Abstract. We sketch a brief outline of the mathematical, and in particular the logical, achievements of Turing in this, his centenary year. 1. ..."
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Abstract. We sketch a brief outline of the mathematical, and in particular the logical, achievements of Turing in this, his centenary year. 1.