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Clarifying Utterances
 ENSCHEDE. UNIVERSITEIT TWENTE, FACULTEIT INFORMATICA
, 1998
"... The paper argues for the importance of utterances (as opposed to "sentences in context") as basic units in dialogue. The main case study is clarification: a characterization is provided within KOS, a recently developed synthesis of situation semantics and dialogue games, of the clarifications th ..."
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Cited by 20 (1 self)
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The paper argues for the importance of utterances (as opposed to "sentences in context") as basic units in dialogue. The main case study is clarification: a characterization is provided within KOS, a recently developed synthesis of situation semantics and dialogue games, of the clarifications that can follow up on an utterance. This involves outlining: (a) a theory of understanding for dialogue, (b) a semantics for clarification acts, and (c) an account of how utterances update contexts. I will show that a relational view of meaning, whose original motivation was purely logical, can provide an important component for a theory of utterance understanding and utterance updates.
Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a pla ..."
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Cited by 6 (0 self)
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Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
On Lachlan's major subdegree problem, to
 in: Set Theory and the Continuum, Proceedings of Workshop on Set Theory and the Continuum
, 1989
"... The Major Subdegree Problem of A. H. Lachlan (first posed in 1967) has become a longstanding open question concerning the structure of the computably enumerable (c.e.) degrees. Its solution has important implications for Turing definability and for the ongoing programme of fully characterising the ..."
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Cited by 2 (2 self)
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The Major Subdegree Problem of A. H. Lachlan (first posed in 1967) has become a longstanding open question concerning the structure of the computably enumerable (c.e.) degrees. Its solution has important implications for Turing definability and for the ongoing programme of fully characterising the theory of the c.e. Turing degrees. A c.e. degree a is a major subdegree of a c.e. degree b> a if for any c.e. degree x, 0 ′ = b ∨ x if and only if 0 ′ = a ∨ x. In this paper, we show that every c.e. degree b ̸ = 0 or 0 ′ has a major subdegree, answering Lachlan’s question affirmatively. 1
Strong minimal covers and a question of Yates: the story so far
 the proceedings of the ASL meeting
, 2006
"... Abstract. An old question of Yates as to whether all minimal degrees have a strong minimal cover remains one of the longstanding problems of degree theory, apparently largely impervious to present techniques. We survey existing results in this area, focussing especially on some recent progress. 1. ..."
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Abstract. An old question of Yates as to whether all minimal degrees have a strong minimal cover remains one of the longstanding problems of degree theory, apparently largely impervious to present techniques. We survey existing results in this area, focussing especially on some recent progress. 1.
THE STRENGTH OF THE RAINBOW RAMSEY THEOREM
, 2009
"... The Rainbow Ramsey Theorem is essentially an “antiRamsey” theorem which states that certain types of colorings must be injective on a large subset (rather than constant on a large subset). Surprisingly, this version follows easily from Ramsey’s Theorem, even in the weak system RCA0 of reverse mathe ..."
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The Rainbow Ramsey Theorem is essentially an “antiRamsey” theorem which states that certain types of colorings must be injective on a large subset (rather than constant on a large subset). Surprisingly, this version follows easily from Ramsey’s Theorem, even in the weak system RCA0 of reverse mathematics. We answer the question of the converse implication for pairs, showing that the Rainbow Ramsey Theorem for pairs is in fact strictly weaker than Ramsey’s Theorem for pairs over RCA0. The separation involves techniques from the theory of randomness by showing that every 2random bounds an ωmodel of the Rainbow Ramsey Theorem for pairs. These results also provide as a corollary a new proof of Martin’s theorem that the hyperimmune degrees have measure one.
The Incomputable Alan Turing
"... The last century saw dramatic challenges to the Laplacian predictability which had underpinned scientific research for around 300 years. Basic to this was Alan Turing’s 1936 discovery (along with Alonzo Church) of the existence of unsolvable problems. This paper focuses on incomputability as a power ..."
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The last century saw dramatic challenges to the Laplacian predictability which had underpinned scientific research for around 300 years. Basic to this was Alan Turing’s 1936 discovery (along with Alonzo Church) of the existence of unsolvable problems. This paper focuses on incomputability as a powerful theme in Turing’s work and personal life, and examines its role in his evolving concept of machine intelligence. It also traces some of the ways in which important new developments are anticipated by Turing’s ideas in logic.
Computational Processes, Observers and Turing Incompleteness
"... We propose a formal definition of Wolfram’s notion of computational process based on iterated transducers together with a weak observer, a model of computation that captures some aspects of physicslike computation. These processes admit a natural classification into decidable, intermediate and comp ..."
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We propose a formal definition of Wolfram’s notion of computational process based on iterated transducers together with a weak observer, a model of computation that captures some aspects of physicslike computation. These processes admit a natural classification into decidable, intermediate and complete, where intermediate processes correspond to recursively enumerable sets of intermediate degree in the classical setting. It is shown that a standard finite injury priority argument will not suffice to establish the existence of an intermediate computational process.
LIMITS ON JUMP INVERSION FOR STRONG REDUCIBILITIES
"... Abstract. We show that Sacks ’ and Shoenfield’s analogs of jump inversion fail for both tt and wttreducibilities in a strong way. In particular we show that there is a ∆0 2 set B>tt ∅ ′ such that there is no c.e. set A with A ′ ≡wtt B. We also show that there is a Σ0 2 set C>tt ∅ ′ such that there ..."
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Abstract. We show that Sacks ’ and Shoenfield’s analogs of jump inversion fail for both tt and wttreducibilities in a strong way. In particular we show that there is a ∆0 2 set B>tt ∅ ′ such that there is no c.e. set A with A ′ ≡wtt B. We also show that there is a Σ0 2 set C>tt ∅ ′ such that there is no ∆0 2 set D with D ′ ≡wtt C. 1.
Formal Proof: Reconciling Correctness and Understanding
"... A good proof is a proof that makes us wiser. Manin [41, p. 209]. Abstract. Hilbert’s concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of pr ..."
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A good proof is a proof that makes us wiser. Manin [41, p. 209]. Abstract. Hilbert’s concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of proof assistants was followed by an impressive record of deep mathematical theorems formally proved. Formal proof is practically achievable. With formal proof, correctness reaches a standard that no penandpaper proof can match, but an essential component of mathematics — the insight and understanding — seems to be in short supply. So, what makes a proof understandable? To answer this question we first suggest a list of symptoms of understanding. We then propose a vision of an environment in which users can write and check formal proofs as well as query them with reference to the symptoms of understanding. In this way, the environment reconciles the main features of proof: correctness and understanding. 1