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Minimization and NP multifunctions
 Theoretical Computer Science
, 2000
"... . The implicit characterizations of the polynomialtime computable functions FP given by BellantoniCook and Leivant suggest that this class is the complexitytheoretic analog of the primitive recursive functions. Hence it is natural to add minimization operators to these characterizations and in ..."
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. The implicit characterizations of the polynomialtime computable functions FP given by BellantoniCook and Leivant suggest that this class is the complexitytheoretic analog of the primitive recursive functions. Hence it is natural to add minimization operators to these characterizations and investigate the resulting class of partial functions as a candidate for the analog of the partial recursive functions. We do so in this paper for Cobham's denition of FP by bounded recursion and for BellantoniCook's safe recursion and prove that the resulting classes capture exactly NPMV, the nondeterministic polynomialtime computable partial multifunctions. We also consider the relationship between our schemes and a notion of nondeterministic recursion dened by Leivant and show that the latter characterizes the total functions of NPMV. We view these results as giving evidence that NPMV is the appropriate analog of partial recursive. We reinforce this view by showing that for many ...
POSSIBLE mDIAGRAMS OF MODELS OF ARITHMETIC
"... Abstract. In this paper we investigate the complexity of mdiagrams of models of various completions of firstorder Peano Arithmetic (PA). We obtain characterizations that extend Solovay’s results for open diagrams of models of completions of PA. We first characterize the mdiagrams of models of Tru ..."
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Abstract. In this paper we investigate the complexity of mdiagrams of models of various completions of firstorder Peano Arithmetic (PA). We obtain characterizations that extend Solovay’s results for open diagrams of models of completions of PA. We first characterize the mdiagrams of models of True Arithmetic by showing that the degrees of mdiagrams of nonstandard models A of TA are the same for all m ≥ 0. Next, we obtain a more complicated characterization for arbitrary completions of PA. We then provide examples showing that some of the extra complication is needed. Lastly, we characterize sequences of Turing degrees that occur as (deg(T ∩ Σn))n∈ω, where T is a completion of PA. §1. Introduction. We use P (ω) to denote the class of all subsets of ω. Let LPA be the usual language of PA: relations +, ·, S, and <; and constants 0 and 1. We abbreviate True Arithmetic, the theory of the standard model of PA, by the initials TA. We use S n (0) to denote the numeral for n
JACQUES HERBRAND: LIFE, LOGIC, AND AUTOMATED DEDUCTION
"... The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1 ..."
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The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague1 — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year1925, when he was17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding.
Arithmetic and the Incompleteness Theorems
, 2000
"... this paper please consult me first, via my home page. ..."
UNDECIDABLE PROBLEMS: A SAMPLER
"... Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. 1. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence ..."
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Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. 1. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence from axioms: A single statement is called undecidable if neither it nor its negation can be deduced using the rules of logic from the set of axioms being used. (Example: The continuum hypothesis, that there is no cardinal number strictly between ℵ0 and 2 ℵ0, is undecidable in the ZFC axiom system, assuming that ZFC itself is consistent [Göd40, Coh63, Coh64].) The first examples of statements independent of a “natural ” axiom system were constructed by K. Gödel [Göd31]. 2. Decision problem: A family of problems with YES/NO answers is called undecidable if there is no algorithm that terminates with the correct answer for every problem in the family. (Example: Hilbert’s tenth problem, to decide whether a multivariable polynomial equation with integer coefficients has a solution in integers, is undecidable [Mat70].) Remark 1.1. In modern literature, the word “undecidability ” is used more commonly in sense 2, given that “independence ” adequately describes sense 1. To make 2 precise, one needs a formal notion of algorithm. Such notions were introduced by A. Church [Chu36a] and A. Turing [Tur36] independently in the 1930s. From now on, we interpret algorithm to mean Turing machine, which, loosely speaking, means that it is a computer program that takes as input a finite string of 0s and 1s. The role of the finite string is to specify which problem in the family is to be solved. Remark 1.2. Often in describing a family of problems, it is more convenient to use higherlevel mathematical objects such as polynomials or finite simplicial complexes as input. This is acceptable if these objects can be encoded as finite binary strings. It is not necessary to specify the encoding as long as it is clear that a Turing machine could convert between reasonable encodings imagined by two different readers.
An implication of Gödel’s incompleteness theorem
, 2009
"... A proof of Gödel’s incompleteness theorem is given. With this new proof a transfinite extension of Gödel’s theorem is considered. It is shown that if one assumes the set theory ZFC on the meta level as well as on the object level, a contradiction arises. The cause is shown to be the implicit identif ..."
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A proof of Gödel’s incompleteness theorem is given. With this new proof a transfinite extension of Gödel’s theorem is considered. It is shown that if one assumes the set theory ZFC on the meta level as well as on the object level, a contradiction arises. The cause is shown to be the implicit identification of the meta level and the object level hidden behind the Gödel numbering. An implication of these considerations is stated.
Derivation
, 2001
"... We return to the two firstorder formalizations of arithmetic to make some remarks on how the study of these subjects can be developed further. 1.1 Peano Arithmetic: Definability Since Firstorder Arithmetic is known to be rather intractable (it is undecidable), when looking for an effective approac ..."
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We return to the two firstorder formalizations of arithmetic to make some remarks on how the study of these subjects can be developed further. 1.1 Peano Arithmetic: Definability Since Firstorder Arithmetic is known to be rather intractable (it is undecidable), when looking for an effective approach to number theory, something which one might use for an automated theorem prover, it is natural to consider Peano Arithmetic since, as we said, all standard theorems of number theory which can be formulated as firstorder statements can be derived from PA. To see that this is true one needs to return to the formulations in the previous section and show that they can be used in PA. DEFINITION 1 (a) A relation r ⊆ ω n is defined in PA by the formula ϕ(x1,..., xn) if for each (k1,..., kn) ∈ ω n we have (k1,..., kn) ∈ r = ⇒ PA ⊢ ϕ ( ¯ k1,..., ¯ kn) (k1,..., kn) / ∈ r = ⇒ PA ⊢ ¬ϕ ( ¯ k1,..., ¯ kn) (b) A function f: ω n = ⇒ ω is defined in PA by the formula ϕ(x1,..., xn, y) if for each (k1,..., kn, k) ∈ ω n+1 we have f(k1,..., kn) = k = ⇒ PA ⊢ ϕ ( ¯ k1,..., ¯ kn, ¯ k) f(k1,..., kn) = k = ⇒ PA ⊢ ¬ϕ ( ¯ k1,..., ¯ kn, ¯ k) and PA ⊢ ∃!y ϕ ( ¯ k1,..., ¯ kn, y) In his 1931 paper Gödel showed that the relations and functions discussed in the previous section on firstorder arithmetic are indeed definable in PA 1. Using this expressive power of PA Gödel went on to give explicit firstorder 1 As are all decidable relations and computable functions. 1 sentences which are true in ω but cannot be derived from PA.
Abstract
"... class is an e ectively closed set of reals. We study properties of these classes determined by cardinality, measure and category as well as by the complexity of ..."
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class is an e ectively closed set of reals. We study properties of these classes determined by cardinality, measure and category as well as by the complexity of