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Complexity Results for FirstOrder TwoVariable Logic with Counting
, 2000
"... Let C 2 p denote the class of first order sentences with two variables and with additional quantifiers "there exists exactly (at most, at least) i", for i p, and let C 2 be the union of C 2 p taken over all integers p. We prove that the satisfiability problem for C 2 1 sentences is NEXPTIMEcomplete ..."
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Let C 2 p denote the class of first order sentences with two variables and with additional quantifiers "there exists exactly (at most, at least) i", for i p, and let C 2 be the union of C 2 p taken over all integers p. We prove that the satisfiability problem for C 2 1 sentences is NEXPTIMEcomplete. This strengthens the results by E. Grädel, Ph. Kolaitis and M. Vardi [15] who showed that the satisfiability problem for the first order twovariable logic L 2 is NEXPTIMEcomplete and by E. Grädel, M. Otto and E. Rosen [16] who proved the decidability of C 2 . Our result easily implies that the satisfiability problem for C 2 is in nondeterministic, doubly exponential time. It is interesting that C 2 1 is in NEXPTIME in spite of the fact, that there are sentences whose minimal (and only) models are of doubly exponential size. It is worth noticing, that by a recent result of E. Gradel, M. Otto and E. Rosen [17], extensions of twovariables logic L 2 by a week access to car...
The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program
, 2001
"... . After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the ..."
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. After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for ever stronger and more comprehensive areas of mathematics and finitistic proofs of consistency of these systems. Early advances in these areas were made by Hilbert (and Bernays) in a series of lecture courses at the University of Gttingen between 1917 and 1923, and notably in Ackermann 's dissertation of 1924. The main innovation was the invention of the ecalculus, on which Hilbert's axiom systems were based, and the development of the esubstitution method as a basis for consistency proofs. The paper traces the development of the "simultaneous development of logic and mathematics" through the enotation and provides an analysis of Ackermann's consisten...
Relational properties expressible with one universal quantifier are testable
 Stochastic Algorithms: Foundations and Applications, 5th International Symposium, SAGA 2009
"... Abstract. In property testing a small, random sample of an object is taken and one wishes to distinguish with high probability between the case where it has a desired property and the case where it is far from having the property. Much of the recent work has focused on graphs. In the present paper t ..."
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Abstract. In property testing a small, random sample of an object is taken and one wishes to distinguish with high probability between the case where it has a desired property and the case where it is far from having the property. Much of the recent work has focused on graphs. In the present paper three generalized models for testing relational structures are introduced and relationships between these variations are shown. Furthermore, the logical classification problem for testability is considered and, as the main result, it is shown that Ackermann’s class with equality is testable. Key words: property testing, logic 1
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.
Review of: R. L. Epstein and W. A. Carnielli. Computability. Computable Functions, Logic, and the Foundations of Mathematics, 2nd Edition
"... rom some important papers on the subject. This includes, aside from `On the infinite', sections from Turing's `On computable numbers' (1936), Post's `Finite combinatory processes' (1936) in the part on computable functions, from G odel's `On formally undecidable propositions' (in a section discussin ..."
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rom some important papers on the subject. This includes, aside from `On the infinite', sections from Turing's `On computable numbers' (1936), Post's `Finite combinatory processes' (1936) in the part on computable functions, from G odel's `On formally undecidable propositions' (in a section discussing the relevance of the incompleteness theorems to Hilbert's programme in Chapter 24), and selections from Church, Turing, G odel, and Kalm ar in Chapter 25 (`Church's Thesis'). The final chapter (Chapter 26), `Constructivist Views of Mathematics' gives the book a nice ending by tying the discussion of computability and undecidability to the general philosophical issues of the nature of mathematics. Selections from Brouwer's `Intuitionism and formalism' (1913), Bishop's Foundations of Constructive Analysis' (1967), and a section on strict finitism (with readings from van Dantzig and David Isles) provide a basis for an informed discussion of these issues. The technical material is wellorgani
R. L. EPSTEIN and W. A. CARNIELLI. Computability. Computable Functions, Logic, and the Foundations of Mathematics, 2nd Edition, with ‘Computability
"... Epstein and Carnielli’s fine textbook on logic and computability is now in its second edition. The readers of this journal might be particularly interested in the timeline ‘Computability and Undecidability ’ added in this edition, and the included wallposter of the same title. The text itself, howe ..."
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Epstein and Carnielli’s fine textbook on logic and computability is now in its second edition. The readers of this journal might be particularly interested in the timeline ‘Computability and Undecidability ’ added in this edition, and the included wallposter of the same title. The text itself, however, has some aspects which are worth commenting on. Possibly the most distinguishing feature of Computability is the emphasis it puts on the historical and philosophical dimensions of logic and computability theory. The text is divided into four parts, ‘Fundamentals’, ‘Computable Functions’, ‘Logic and Arithmetic’, and ‘Church’s Thesis and Constructive Mathematics’. The inclusion of an entire part, not just a section or footnote, on the philosophical aspects of computability and the foundations of mathematics make the book not only unique among advanced logic textbooks, but, in particular, suitable for courses aimed at philosophy students. But already in the first part, ‘The Fundamentals’, we find sections on paradoxes and the nature of mathematical proofs as well as an entire chapter containing a long excerpt from Hilbert’s ‘On the infinite’.
The Classification Problem in Relational Property Testing
, 2009
"... In property testing, we desire to distinguish between objects that have a given property and objects that are far from the property by examining only a small, randomly selected portion of the objects. Property testing arose in the study of formal verification, however much of the recent work has bee ..."
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In property testing, we desire to distinguish between objects that have a given property and objects that are far from the property by examining only a small, randomly selected portion of the objects. Property testing arose in the study of formal verification, however much of the recent work has been focused on testing graph properties. In this thesis we introduce a generalization of property testing which we call relational property testing. Because property testers examine only a very small portion of their inputs, there are potential applications to efficiently testing properties of massive structures. Relational databases provide perhaps the most obvious example of such massive structures, and our framework is a natural way to characterize this problem. We introduce a number of variations of our generalization and prove the relationships between them. The second major topic of this thesis is the classification problem for testability. Using the general framework developed in previous chapters, we consider the testability of various syntactic fragments of firstorder logic. This problem is inspired by the classical problem for decidability. We compare the current classi cation for testability with
WWW:http://www.cl.cam.ac.uk/~lp15 Lectures on
, 2007
"... We give some lectures on the work on formal logic of Jacques Herbrand, and sketch his life and his influence on automated theorem proving. The intended audience ranges from students interested in logic over historians to logicians. Besides the wellknown correction of Herbrand’s False Lemma by Gödel ..."
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We give some lectures on the work on formal logic of Jacques Herbrand, and sketch his life and his influence on automated theorem proving. The intended audience ranges from students interested in logic over historians to logicians. Besides the wellknown correction of Herbrand’s False Lemma by Gödel and Dreben, we also present the hardly known unpublished correction of Heijenoort and its consequences on Herbrand’s Modus Ponens Elimination. Besides Herbrand’s Fundamental Theorem and its relation to the Löwenheim–Skolem Theorem, we carefully investigate Herbrand’s notion of intuitionism in connection with his notion of falsehood in an infinite domain. We sketch Herbrand’s two proofs of the consistency of arithmetic and his notion of a recursive function, and last but not least, present the
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, 2007
"... We give some lectures on the work on formal logic of Jacques Herbrand, and sketch his life and his influence on automated theorem proving. The intended audience ranges from students interested in logic over historians to logicians. Besides the wellknown correction of Herbrand’s False Lemma by Gödel ..."
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We give some lectures on the work on formal logic of Jacques Herbrand, and sketch his life and his influence on automated theorem proving. The intended audience ranges from students interested in logic over historians to logicians. Besides the wellknown correction of Herbrand’s False Lemma by Gödel and Dreben, we also present the hardly known unpublished correction of Heijenoort and its consequences on Herbrand’s Modus Ponens Elimination. Besides Herbrand’s Fundamental Theorem and its relation to the Löwenheim–Skolem Theorem, we carefully investigate Herbrand’s notion of intuitionism in connection with his notion of falsehood in an infinite domain. We sketch Herbrand’s two proofs of the consistency of arithmetic and his notion of a recursive function, and last but not least, present the