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15
Linear Time Computable Problems and FirstOrder Descriptions
, 1996
"... this article is a proof that each FO problem can be solved in linear time if only relational structures of bounded degree are considered. The basic idea of the proof is a localization technique based on a method that was originally developed by Hanf (Hanf 1965) to show that the elementary theories o ..."
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this article is a proof that each FO problem can be solved in linear time if only relational structures of bounded degree are considered. The basic idea of the proof is a localization technique based on a method that was originally developed by Hanf (Hanf 1965) to show that the elementary theories of two structures are equal under certain conditions, i.e., that two structures agree on all firstorder sentences. Fagin, Stockmeyer and Vardi (Fagin et al. 1993) developed a variant of this technique, which is applicable in descriptive complexity theory to classes of finite relational structures of uniformly bounded degree. Variants of this result can also be found in Gaifman (1982) (see also Thomas (1991)). The essential content of this result, which is also called the HanfSphere Lemma, is that two relational structures of bounded degree satisfy the same firstorder sentences of a certain quantifierrank if both contain, up to a certain number m, the same number of isomorphism types of substructures of a bounded radius r. In addition, a technique of model interpretability from Rabin (1965) (see also Arnborg et al. (1991), Seese (1992), Compton and Henson (1987) and Baudisch et al. (1982)) is adapted to descriptive complexity classes, and proved to be useful for reducing the case of an arbitrary class of relational structures to a class of structures consisting only of the domain and one binary irreflexive and symmetric relation, i.e., the class of simple graphs. It is shown that the class of simple graphs is lintimeuniversal with respect to firstorder logic, which shows that many problems on descriptive complexity classes, described in languages extending firstorder logic for arbitrary structures, can be reduced to problems on simple graphs. This paper is organized as f...
A NATURAL AXIOMATIZATION OF COMPUTABILITY AND PROOF OF CHURCH’S THESIS
, 2008
"... Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent ..."
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Cited by 42 (17 self)
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Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church’s Thesis, as Gödel and others suggested may be possible. In a similar way, but with a different set of basic operations, one can prove Turing’s Thesis, characterizing the effective string functions, and—in particular—the effectivelycomputable functions on string representations of numbers.
Gödel on computability
"... Around 1950, both Gödel and Turing wrote papers for broader audiences. 1 Gödel drew in his 1951 dramatic philosophical conclusions from the general formulation of his second incompleteness theorem. These conclusions concerned the nature of mathematics and the human mind. The general formulation of t ..."
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Around 1950, both Gödel and Turing wrote papers for broader audiences. 1 Gödel drew in his 1951 dramatic philosophical conclusions from the general formulation of his second incompleteness theorem. These conclusions concerned the nature of mathematics and the human mind. The general formulation of the second theorem was explicitly based on Turing’s 1936 reduction of finite procedures to machine computations. Turing gave in his 1954 an understated analysis of finite procedures in terms of Post production systems. This analysis, prima facie quite different from that given in 1936, served as the basis for an exposition of various unsolvable problems. Turing had addressed issues of mentality and intelligence in contemporaneous essays, the best known of which is of course Computing machinery and intelligence. Gödel’s and Turing’s considerations from this period intersect through their attempt, on the one hand, to analyze finite, mechanical procedures and, on the other hand, to approach mental phenomena in a scientific way. Neuroscience or brain science was an important component of the latter for both: Gödel’s remarks in the Gibbs Lecture as well as in his later conversations with Wang and Turing’s Intelligent Machinery can serve as clear evidence for that. 2 Both men were convinced that some mental processes are not mechanical, in the sense that Turing machines cannot mimic them. For Gödel, such processes were to be found in mathematical experience and he was led to the conclusion that mind is separate from matter. Turing simply noted that for a machine or a brain it is not enough to be converted into a universal (Turing) machine in order to become intelligent: “discipline”, the characteristic
Modal Logics with a Hierarchy of Local Propositional Quantifiers (Extended Abstract)
 Advances in Modal Logic
, 1999
"... Local propositions arise in the context of the semantics for logics of knowledge in multiagent systems. A proposition is local to an agent when it depends only on that agent's local state. We consider a logic that extends S5, the modal logic of necessity (in which the modality refers to truth ..."
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Cited by 6 (4 self)
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Local propositions arise in the context of the semantics for logics of knowledge in multiagent systems. A proposition is local to an agent when it depends only on that agent's local state. We consider a logic that extends S5, the modal logic of necessity (in which the modality refers to truth at all worlds) by adding a quantifier ranging over the set of all propositions and, for each agent, a propositional quantifier ranging over the agent's local propositions. This logic is able to express a large variety of epistemic modalities, including knowledge, common knowledge and distributed knowledge. However, this expressiveness comes at a cost: the logic is undecidable when two independent agents are present [Engelhardt, van der Meyden and Moses, TARK'98]. We identify a natural condition on the multiagent case, that the agents' information has the structure of a linear hierarchy, with respect to which the multiagent case becomes decidable. We also establish completeness of an axiomatization for the hierarchical case. The single agent case is hierarchical and EXPTIMEhard. Connections to graded modal logic are also discussed.
Predicative Fragments of Frege Arithmetic
, 2003
"... Frege Arithmetic (FA) is the secondorder theory whose sole nonlogical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be onetoone correlated. According to Frege’s Theorem, FA and some natural definitions imply al ..."
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Frege Arithmetic (FA) is the secondorder theory whose sole nonlogical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be onetoone correlated. According to Frege’s Theorem, FA and some natural definitions imply all of secondorder Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume’s Principle, the other, with the underlying secondorder logic—and investigates how much of Frege’s Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension. 1
A natural axiomatization of Church’s thesis
, 2007
"... The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requ ..."
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Cited by 2 (0 self)
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The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requirement regarding basic operations implies Church’s Thesis, namely, that the only numeric functions that can be calculated by effective means are the recursive ones (which are the same, extensionally, as the Turingcomputable numeric functions). In particular, this gives a natural axiomatization of Church’s Thesis, as Gödel and others suggested may be possible.
Modal Logics with a Linear Hierarchy of Local Propositional Quantifiers
 Advances in Modal Logic
, 2002
"... This paper identi es a class of multiagent S5 structures, hierarchical structures, in which the agents' information has the structure of a linear hierarchy. All systems with just a single agent are hierarchical. It is shown that LLP becomes decidable with respect to hierarchical systems. ..."
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This paper identi es a class of multiagent S5 structures, hierarchical structures, in which the agents' information has the structure of a linear hierarchy. All systems with just a single agent are hierarchical. It is shown that LLP becomes decidable with respect to hierarchical systems. The main result of the paper is the completeness of an axiomatization for the hierarchical case
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.