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21
A Formal Definition of Intelligence Based on an Intensional Variant of Algorithmic Complexity
 In Proceedings of the International Symposium of Engineering of Intelligent Systems (EIS'98
, 1998
"... Machine Due to the current technology of the computers we can use, we have chosen an extremely abridged emulation of the machine that will effectively run the programs, instead of more proper languages, like lcalculus (or LISP). We have adapted the "toy RISC" machine of [Hernndez & H ..."
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Cited by 38 (19 self)
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Machine Due to the current technology of the computers we can use, we have chosen an extremely abridged emulation of the machine that will effectively run the programs, instead of more proper languages, like lcalculus (or LISP). We have adapted the "toy RISC" machine of [Hernndez & Hernndez 1993] with two remarkable features inherited from its objectoriented coding in C++: it is easily tunable for our needs, and it is efficient. We have made it even more reduced, removing any operand in the instruction set, even for the loop operations. We have only three registers which are AX (the accumulator), BX and CX. The operations Q b we have used for our experiment are in Table 1: LOOPTOP Decrements CX. If it is not equal to the first element jump to the program top.
Information and Computation: Classical and Quantum Aspects
 REVIEWS OF MODERN PHYSICS
, 2001
"... Quantum theory has found a new field of applications in the realm of information and computation during the recent years. This paper reviews how quantum physics allows information coding in classically unexpected and subtle nonlocal ways, as well as information processing with an efficiency largely ..."
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Cited by 36 (3 self)
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Quantum theory has found a new field of applications in the realm of information and computation during the recent years. This paper reviews how quantum physics allows information coding in classically unexpected and subtle nonlocal ways, as well as information processing with an efficiency largely surpassing that of the present and foreseeable classical computers. Some outstanding aspects of classical and quantum information theory will be addressed here. Quantum teleportation, dense coding, and quantum cryptography are discussed as a few samples of the impact of quanta in the transmission of information. Quantum logic gates and quantum algorithms are also discussed as instances of the improvement in information processing by a quantum computer. We provide finally some examples of current experimental
Is Independence an Exception?
, 1994
"... Gödel's Incompleteness Theorem asserts that any sufficiently rich, sound, and recursively axiomatizable theory is incomplete. We show that, in a quite general topological sense, incompleteness is a rather common phenomenon: With respect to any reasonable topology the set of true and unprovable ..."
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Cited by 19 (12 self)
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Gödel's Incompleteness Theorem asserts that any sufficiently rich, sound, and recursively axiomatizable theory is incomplete. We show that, in a quite general topological sense, incompleteness is a rather common phenomenon: With respect to any reasonable topology the set of true and unprovable statements of such a theory is dense and in many cases even corare.
Positivity problems for loworder linear recurrence sequences
 In Proc. Symp. on Discrete Algorithms (SODA). ACMSIAM
, 2014
"... We consider two decision problems for linear recurrence sequences (LRS) over the integers, namely the Positivity Problem (are all terms of a given LRS positive?) and the Ultimate Positivity Problem (are all but finitely many terms of a given LRS positive?). We show decidability of both problems for ..."
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Cited by 14 (7 self)
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We consider two decision problems for linear recurrence sequences (LRS) over the integers, namely the Positivity Problem (are all terms of a given LRS positive?) and the Ultimate Positivity Problem (are all but finitely many terms of a given LRS positive?). We show decidability of both problems for LRS of order 5 or less, with complexity in the Counting Hierarchy for Positivity, and in polynomial time for Ultimate Positivity. Moreover, we show by way of hardness that extending the decidability of either problem to LRS of order 6 would entail major breakthroughs in analytic number theory, more precisely in the field of Diophantine approximation of transcendental numbers. 1
Is Complexity a Source of Incompleteness?
 IS COMPLEXITY A SOURCE OF INCOMPLETENESS
, 2004
"... ..."
Total Termination of Term Rewriting is Undecidable
 Journal of Symbolic Computation
, 1995
"... Usually termination of term rewriting systems (TRS's) is proved by means of a monotonic wellfounded order. If this order is total on ground terms, the TRS is called totally terminating. In this paper we prove that total termination is an undecidable property of finite term rewriting systems. T ..."
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Cited by 9 (3 self)
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Usually termination of term rewriting systems (TRS's) is proved by means of a monotonic wellfounded order. If this order is total on ground terms, the TRS is called totally terminating. In this paper we prove that total termination is an undecidable property of finite term rewriting systems. The proof is given by means of Post's Correspondence Problem. 1 Introduction Termination of term rewriting systems (TRS's) is an important property. Often termination proofs are given by defining an order that is wellfounded, and proving that for every rewrite step the value of the term decreases according this order. In many cases the order is monotonic, and it suffices to prove that l oe ? r oe for all rewrite rules l ! r and all ground substitutions oe. Standard techniques following this approach include recursive path order and KnuthBendix order, see for example [17]. It is an interesting question whether these orders are total or can be extended to a total monotonic order, or are essen...
Commutation Problems on Sets of Words and Formal Power Series
, 2002
"... We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series  the semilinear and the bounded series  both natural generaliza ..."
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Cited by 5 (3 self)
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We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series  the semilinear and the bounded series  both natural generalizations of the semilinear languages, and we study their behaviour under rational operations, morphisms, Hadamard product, and difference. Turning to commutation on sets of words, we then study the notions of centralizer of a language  the largest set commuting with a language , of root and of primitive root of a set of words. We answer a question raised by Conway more than thirty years ago  asking whether or not the centralizer of any rational language is rational  in the case of periodic, binary, and ternary sets of words, as well as for rational ccodes, the most general results on this problem. We also prove that any code has a unique primitive root and that two codes commute if and only if they have the same primitive root, thus solving two conjectures of Ratoandromanana, 1989. Moreover, we prove that the commutation with an ccode X can be characterized similarly as in free monoids: a language commutes with X if and only if it is a union of powers of the primitive root of X.
Ultimate Positivity is Decidable for Simple Linear Recurrence Sequences?
"... Abstract. We consider the decidability and complexity of the Ultimate Positivity Problem, which asks whether all but finitely many terms of a given rational linear recurrence sequence (LRS) are positive. Using lower bounds in Diophantine approximation concerning sums of Sunits, we show that for sim ..."
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Cited by 4 (3 self)
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Abstract. We consider the decidability and complexity of the Ultimate Positivity Problem, which asks whether all but finitely many terms of a given rational linear recurrence sequence (LRS) are positive. Using lower bounds in Diophantine approximation concerning sums of Sunits, we show that for simple LRS (those whose characteristic polynomial has no repeated roots) the Ultimate Positivity Problem is decidable in polynomial space. If we restrict to simple LRS of a fixed order then we obtain a polynomialtime decision procedure. As a complexity lower bound we show that Ultimate Positivity for simple LRS is at least as hard as the decision problem for the universal theory of the reals: a problem that is known to lie between coNP and PSPACE. 1
Primitive Solutions Of The Post Correspondence Problem
, 1996
"... s in both places for creating a very pleasant working environment. For financial support I would like to thank the Academy of Finland and Suomen Kulttuurirahasto (Elina, Sofia and Yrjo Arffman foundation). Finally, I would like to acknowledge my debt to Keijo, not only for his support, but also for ..."
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Cited by 2 (2 self)
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s in both places for creating a very pleasant working environment. For financial support I would like to thank the Academy of Finland and Suomen Kulttuurirahasto (Elina, Sofia and Yrjo Arffman foundation). Finally, I would like to acknowledge my debt to Keijo, not only for his support, but also for making some of my algorithms into programs. Real patience was needed when one of the computer runs took three weeks. Turku, February 21st, 1996 Marjo Lipponen Contents 1 Introduction 7 1.1 Basics of language theory : : : : : : : : : : : : : : : : : : : : : 9 1.2 Post Correspondence Problem : : : : : : : : : : : : : : : : : : 13 1.3 Periodic instances : : : : : : : : : : : : : : : : : : : : : : : : : 14 2 Prime Words: Basic Types 17 2.1 Primitive solutions : : : : : : : : : : : : : : : : : : : : : : : : 18 2.1.1 Primitive solutions in periodic instances : : : : : : : : 19 2.2 Ch
On the Positivity Problem for simple linear recurrence sequences
 In Proceedings of ICALP’14, 2014. CoRR, abs/1309.1550
"... Abstract. Given a linear recurrence sequence (LRS) over the integers, the Positivity Problem asks whether all terms of the sequence are positive. We show that, for simple LRS (those whose characteristic polynomial has no repeated roots) of order 9 or less, Positivity is decidable, with complexity ..."
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Cited by 2 (2 self)
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Abstract. Given a linear recurrence sequence (LRS) over the integers, the Positivity Problem asks whether all terms of the sequence are positive. We show that, for simple LRS (those whose characteristic polynomial has no repeated roots) of order 9 or less, Positivity is decidable, with complexity in the Counting Hierarchy. 1