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The Analysis of Programming Structure
 ACM SIGACT News
, 1997
"... This paper has explored three examples of good semantical analyses of programming structures. The three examples share two characteristics: the semantic models are abstract enough to be applicable in many situations, and the models lead to proofs of noncomputability. Other examples of programming s ..."
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This paper has explored three examples of good semantical analyses of programming structures. The three examples share two characteristics: the semantic models are abstract enough to be applicable in many situations, and the models lead to proofs of noncomputability. Other examples of programming structures have been omitted from this short essay: foundations for objectoriented languages, descriptions of languages with local variables, and the theory of database query languages. Each of these examples have corresponding semantical theories that enjoy the two characteristics above. The richness of programming structure suggests a corollary: it is folly to look for one universal model to explain all programming structures. Of course, as a theoretical subject, semantics benefits from the reduction of many concepts to a primitive, common level. Nevertheless, reduction must often be resisted. We have seen how computability theory loses all kinds of relevant distinctions. Another example is the naive semantics of PCF based on dcpos: the model is not abstract enough,
1 What Is an Algorithm?
"... When algorithms are defined rigorously in Computer Science literature machines, mathematical models of computers, sometimes idealized by allowing access to “unbounded memory”. 1 My aims here are to argue ..."
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When algorithms are defined rigorously in Computer Science literature machines, mathematical models of computers, sometimes idealized by allowing access to “unbounded memory”. 1 My aims here are to argue
DETAILED PROOF OF THEOREM 4.1 IN SENSE AND DENOTATION AS ALGORITHM AND VALUE
"... §4. Sense identity and indirect reference. Van Heijenoort [30] quotes an extensive passage from a 1906 letter from Frege to Husserl which begins with the following sentence: “It seems to me that we must have an objective criterion for recognizing a thought as the same thought, since without such a c ..."
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§4. Sense identity and indirect reference. Van Heijenoort [30] quotes an extensive passage from a 1906 letter from Frege to Husserl which begins with the following sentence: “It seems to me that we must have an objective criterion for recognizing a thought as the same thought, since without such a criterion a logical analysis is not possible.” This could be read as asserting the existence of a decision procedure for sense identity, but unfortunately, the letter goes on to suggest that logically equivalent sentences have the same sense, a position which is contrary to the whole spirit of [12]. It is apparently not clear what Frege thought of this question or if he seriously considered it at all. Kreisel and Takeuti [17] raise explicitly the question of synonymity of sentences which may be the same as that of identity of sense. If we identify sense with referential intension, the matter is happily settled by a theorem. Theorem 4.1. For each recursor structure A = (U1,..., Uk, f1,..., fn) of finite signature, the relation ∼A of intensional identity on the terms of FLR interpreted on A is decidable. For each structure A and arbitrary integers n, m, let SA(n, m) ⇐ ⇒ n, m are Gödel numbers of sentences or terms θn, θm of FLR (58) and θn ∼A θm. The rigorous meaning of 4.1 is that this relation SA is decidable, i.e., computable by a Turing machine. By the usual coding methods then, we get immediately: Corollary 4.2. The relation SA of intensional identity on Gödel numbers of expressions of FLR is elementary (definable in LPC), over each acceptable structure A. The Corollary is useful because it makes it possible to talk indirectly about FLR intensions within FLR. In general, we cannot do this directly because the intensions of a structure A are higher type objects over A which are not ordinarily 15 members of any basic set of the universe of A. One reason we might want to discuss FLR intensions within FLR is to express indirect reference, where Frege’s treatment deviates from his general doctrine of separate compositionality This manuscript is posted on
Spring School on Infinite Games and its Applications 2005 Games and Semantics
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Ordinal Machines and Admissible Recursion Theory
"... We generalize standard Turing machines working in time ω on a tape of length ω to abstract machines with time α and tape length α, for α some limit ordinal. This model of computation determines an associated computability theory: αcomputability theory. We compare the new theory to αrecursion theor ..."
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We generalize standard Turing machines working in time ω on a tape of length ω to abstract machines with time α and tape length α, for α some limit ordinal. This model of computation determines an associated computability theory: αcomputability theory. We compare the new theory to αrecursion theory, which was developed by G. Sacks and his school. For α an admissible ordinal, the basic notions of αcomputable and αrecursive as well as αcomputably enumerable and αrecursively enumerable completely agree. Moreover there is an isomorphism of parts of the degree structure induced by αcomputability and of a degree structure in αrecursion theory, which allows us to transfer, e.g., the SacksSimpson theorem or Shore’s density theorem to αcomputability theory. We emphasize the algorithmic approach by giving a proof of the SacksSimpson theorem, which is solely based on αmachines and does not rely on constructibility theory.
Exact Representations of and Computability on Real
"... This master's thesis explores computability of real functions via representations of real numbers. ..."
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This master's thesis explores computability of real functions via representations of real numbers.
Transfinite Machine Models
, 2011
"... In recent years there has emerged the study of discrete computational models which are allowed to act transfinitely. By ‘discrete ’ we mean that the machine models considered are not analogue machines, but compute by means of distinct stages or in units of time. The paradigm of such models is, of co ..."
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In recent years there has emerged the study of discrete computational models which are allowed to act transfinitely. By ‘discrete ’ we mean that the machine models considered are not analogue machines, but compute by means of distinct stages or in units of time. The paradigm of such models is, of course, Turing’s original
INFINITARY LANGUAGES by
"... We begin with the following quotation from Karp [1964]: My interest in infinitary logic dates back to a February day in 1956 when I remarked to my thesis supervisor, Professor Leon Henkin, that a particularly vexing problem would be so simple if only I could write a formula which would say x = 0 or ..."
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We begin with the following quotation from Karp [1964]: My interest in infinitary logic dates back to a February day in 1956 when I remarked to my thesis supervisor, Professor Leon Henkin, that a particularly vexing problem would be so simple if only I could write a formula which would say x = 0 or x = 1 or x = 2 etc. To my surprise, he replied, &quot;Well, go ahead.&quot;