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A simple proof of Parsons' theorem
"... Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1formulas. More than three decades ago, Charles Parsons showed that the provably total functions of I# 1 are exactly the primitive recursive functions. In this paper, we observe that Parsons' resul ..."
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Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1formulas. More than three decades ago, Charles Parsons showed that the provably total functions of I# 1 are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is a consequence of Herbrand's theorem concerning the of universal theories. We give a selfcontained proof requiring only basic knowledge of mathematical logic.
P is not equal to NP
, 2009
"... SAT ̸ ∈ P is true, and provable in a simply consistent extension B ′ of a first order theory B of computing, with a single finite axiom B characterizing a universal Turing machine. Therefore P ̸ = N P is true, and provable in a simply consistent extension B ′ ′ of B. ..."
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SAT ̸ ∈ P is true, and provable in a simply consistent extension B ′ of a first order theory B of computing, with a single finite axiom B characterizing a universal Turing machine. Therefore P ̸ = N P is true, and provable in a simply consistent extension B ′ ′ of B.
Double Exponential Inseparability Of Robinson Subsystem Q+  From The Unsatisfiable Sentences In The Language Of Addition
, 1992
"... this report is organized in the following way. As shown by a careful analysis in the first sections, a small set of properties of addition and successor are sufficient to define an exp(3; k)representation ..."
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this report is organized in the following way. As shown by a careful analysis in the first sections, a small set of properties of addition and successor are sufficient to define an exp(3; k)representation
unknown title
, 2000
"... Neil Leslie asserts his moral right to be identified as the author of this work. cfl Neil Leslie Abstract We explain how to program with continuations in MartinL "of's theory of types (MLTT). MLTT is a theory designed to formalize constructive mathematics. By appealing to the Curry ..."
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Neil Leslie asserts his moral right to be identified as the author of this work. cfl Neil Leslie Abstract We explain how to program with continuations in MartinL &quot;of's theory of types (MLTT). MLTT is a theory designed to formalize constructive mathematics. By appealing to the CurryHoward `propositions as types ' analogy, and to the BrouwerHeytingKolmogorov interpretation of intuitionistic logic we can treat MLTT as a framework for the specification and derivation of correct functional programs. However, programming in MLTT has two weaknesses: ffl we are limited in the functions that we can naturally express; ffl the functions that we do write naturally are often inefficient. Programming with continuations allows us partially to address these problems. The continuationpassing programming style is known to offer a number of advantages to the functional programmer. We can also observe a relationship between continuation passing and type lifting in categorial grammar. We present computation rules which allow us to use continuations with inductivelydefined types, and with types not presented inductively. We justify the new elimination rules using the usual prooftheoretic semantics. We show that the new rules preserve the consistency of the theory. We show how to use wellorderings to encode continuationpassing operators for inductively defined types. Acknowledgements An earlier version of some of the material in Chapter 6 appeared as [70]. I would like to thank: ffl Peter Kay of Massey University's Albany campus, and Steve Reeves of Waikato University, for providing invaluable support and guidance; ffl Ross Renner, and the School of Mathematical and Computing Sciences at Victoria University of Wellington, for indulging me with time and money to visit Waikato University to talk with Steve; ffl Mhairi for caring (again) that I should finish a thesis; ffl Keir and Ailidh for not caring about theses at all.