Let F be their rings of integers. If there exists an elliptic curve E over F such that rk E(F ) = rk E(K) = 1, then there exists a diophantine definition of .

There has been a great effort in giving machine independent, algebraic characterizations of complexity classes, especially of functions. Astonishingly, no satisfactory characterization of the prominent class #P has been known up to now. Here, we characterize #P as the closure of a set of simple arithmetical functions under summation and weak product. Building on that result, the hierarchy of counting functions, which is the closure of #P under substitution, is characterized; remarkably without using the operator of substitution, since we can show that in the context of this hierarchy the operation of modified subtraction is as powerful as substitution. This leads us to a number of consequences concerning closure of #P under certain arithmetical operations. Analogous results are achieved for the class Gap-P which is the closure of #P under subtraction.

this paper is to show that every recursively enumerable set is represented by formulas of each of the two following types: 9b9c " & =1 9d[P (a; b; c) < D (a; b; c)d < Q (a; b; c)]; (1) 9b9c8f [f F (a; b; c) )W (a; b; c; f) > 0]: (2) 3. Let R be a recursively enumerable set. Let us nd formula of the type 9h 1 : : : 9h [R(a; h 1 ; : : : ; h ) = 0]; (3) which represents the set R (existence of such formula is shown, for instance, in [6-9,14]). Let us denote the degree of the polynomial R by . Without loss of generality we may assume that 1. The original article appeared in: Teoriya Algorifmov i Matematicheskaya Logika (volume dedicated to A. A. Markov), Vychislitel'nyi Tsentr Akademii Nauk SSSR, Moscow, 1974, p.112-123 (in Russian). Translated into English by M.A.Vsemirnov. 1 2 YURI MATIYASEVICH, JULIA ROBINSON If formula (2) is equivalent to formula (3), then the pair hb; ci, whose existence is stated in (2), must carry the whole information about the -tuple hh 1 ; : : : h i, whose existence is stated in (3). Many ways to code tuples of nonnegative integers by means of one or two nonnegative integers are known. We choose one nonstandard method, which allows us to check the truth of the relation R(a; h 1 ; : : : ; h ) = 0; (4) by the code directly, without nding the numbers h 1 ; : : : ; h . Let us denote by B(h 1 ; : : : ; h ; k) the polynomial

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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.

The axiomatization of arithmetical properties in theorem proving creates many straightforward inference steps. In analyzing mathematical proofs with the CERES (Cut-Elimination by Resolution) system, it is convenient to hide these inferences. The central topic of the thesis is the extension of the CERES method to allow reasoning modulo equational theories. For this, the inference systems of Sequent Calculus modulo and Extended Narrowing and Resolution replace their non-equational counterparts in CERES. The method is illustrated by examples comparing inference modulo the theory of associativity and commutativity with unit element to inference in the empty theory.