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Local stability of ergodic averages
 Transactions of the American Mathematical Society
"... We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesguemeasure preserving transformation of [0, 1] and a characteristic function f = χA such that the ergodic averages An ..."
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We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesguemeasure preserving transformation of [0, 1] and a characteristic function f = χA such that the ergodic averages Anf do not converge to a computable element of L2([0,1]). In particular, there is no computable bound on the rate of convergence for that sequence. On the other hand, we show that, for any nonexpansive linear operator T on a separable Hilbert space, and any element f, it is possible to compute a bound on the rate of convergence of (Anf) from T, f, and the norm ‖f ∗ ‖ of the limit. In particular, if T is the Koopman operator arising from a computable ergodic measure preserving transformation of a probability space X and f is any computable element of L2(X), then there is a computable bound on the rate of convergence of the sequence (Anf). The mean ergodic theorem is equivalent to the assertion that for every function K(n) and every ε> 0, there is an n with the property that the ergodic averages Amf are stable to within ε on the interval [n, K(n)]. Even in situations where the sequence (Anf) does not have a computable limit, one can give explicit bounds on such n in terms of K and ‖f‖/ε. This tells us how far one has to search to find an n so that the ergodic averages are “locally stable ” on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general prooftheoretic methods falling under the heading of “proof mining.” 1
Theories for Complexity Classes and their Propositional Translations
 Complexity of computations and proofs
, 2004
"... We present in a uniform manner simple twosorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus. ..."
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Cited by 32 (6 self)
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We present in a uniform manner simple twosorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus.
Optimal Proof Systems Imply Complete Sets For Promise Classes
 INFORMATION AND COMPUTATION
, 2001
"... A polynomial time computable function h : whose range is a set L is called a proof system for L. In this setting, an hproof for x 2 L is just a string w with h(w) = x. Cook and Reckhow de ned this concept in [11] and in order to compare the relative strength of dierent proof systems for ..."
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Cited by 25 (1 self)
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A polynomial time computable function h : whose range is a set L is called a proof system for L. In this setting, an hproof for x 2 L is just a string w with h(w) = x. Cook and Reckhow de ned this concept in [11] and in order to compare the relative strength of dierent proof systems for the set TAUT of tautologies in propositional logic, they considered the notion of psimulation. Intuitively, a proof system h psimulates h if any hproof w can be translated in polynomial time into an h for h(w). Krajcek and Pudlak [18] considered the related notion of simulation between proof systems where it is only required that for any hproof w there exists an h whose size is polynomially bounded in the size of w.
The Proof Complexity of Linear Algebra
 IN SEVENTEENTH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE (LICS
, 2002
"... We introduce three formal theories of increasing strength for linear algebra in order to study the complexity of the concepts needed to prove the basic theorems of the subject. We give what is apparently the rst feasible proofs of the CayleyHamilton theorem and other properties of the determina ..."
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Cited by 25 (9 self)
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We introduce three formal theories of increasing strength for linear algebra in order to study the complexity of the concepts needed to prove the basic theorems of the subject. We give what is apparently the rst feasible proofs of the CayleyHamilton theorem and other properties of the determinant, and study the propositional proof complexity of matrix identities such as AB = I ! BA = I .
Complete sequent calculi for induction and infinite descent
 Proceedings of LICS22
, 2007
"... This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing induct ..."
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This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing inductively defined predicates on the left of sequents. We show this system to be cutfree complete with respect to a natural class of Henkin models; the eliminability of cut follows as a corollary. The second system uses infinite (nonwellfounded) proofs to represent arguments by infinite descent. In this system, the left rules for inductively defined predicates are simple casesplit rules, and an infinitary, global condition on proof trees is required to ensure soundness. We show this system to be cutfree complete with respect to standard models, and again infer the eliminability of cut. The second infinitary system is unsuitable for formal reasoning. However, it has a natural restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs. This restricted “cyclic ” system subsumes the first system for proof by induction. We conjecture that the two systems are in fact equivalent, i.e., that proof by induction is equivalent to regular proof by infinite descent.
Towards a Semantic Characterization of cutelimination
, 2006
"... We introduce necessary and sufficient conditions for a (singleconclusion) sequent calculus to admit (reductive) cutelimination. Our conditions are formulated both syntactically and semantically. ..."
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Cited by 24 (5 self)
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We introduce necessary and sufficient conditions for a (singleconclusion) sequent calculus to admit (reductive) cutelimination. Our conditions are formulated both syntactically and semantically.
Narrow proofs may be spacious: Separating space and width in resolution (Extended Abstract)
 REVISION 02, ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY (ECCC
, 2005
"... The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of clauses kept in memory simultaneously if the proof is only allowed to infer new clauses from clauses currently in memory. Both of these measures have previously ..."
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Cited by 20 (7 self)
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The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of clauses kept in memory simultaneously if the proof is only allowed to infer new clauses from clauses currently in memory. Both of these measures have previously been studied and related to the resolution refutation size of unsatisfiable CNF formulas. Also, the refutation space of a formula has been proven to be at least as large as the refutation width, but it has been open whether space can be separated from width or the two measures coincide asymptotically. We prove that there is a family of kCNF formulas for which the refutation width in resolution is constant but the refutation space is nonconstant, thus solving a problem mentioned in several previous papers.
Lectures on proof theory
 in Proc. Summer School in Logic, Leeds 67
, 1968
"... This is a survey of some of the principal developments in proof theory from its inception in the 1920s, at the hands of David Hilbert, up to the 1960s. Hilbert's aim was to use this as a tool in his nitary consistency program to eliminate the \actual in nite " in mathematics from proofs of ..."
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Cited by 15 (5 self)
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This is a survey of some of the principal developments in proof theory from its inception in the 1920s, at the hands of David Hilbert, up to the 1960s. Hilbert's aim was to use this as a tool in his nitary consistency program to eliminate the \actual in nite " in mathematics from proofs of purely nitary statements. One of the main approaches that turned out to be the most useful in pursuit of this program was that due to Gerhard Gentzen, in the 1930s, via his calculi of \sequents" and his CutElimination Theorem for them. Following that we trace how and why prima facie in nitary concepts, such as ordinals, and in nitary methods, such as the use of in nitely long proofs, gradually came to dominate prooftheoretical developments. In this rst lecture I will give anoverview of the developments in proof theory since Hilbert's initiative in establishing the subject in the 1920s. For this purpose I am following the rst part of a series of expository lectures that I gave for the Logic Colloquium `94 held in ClermontFerrand 2123 July 1994, but haven't published. The theme of my lectures there was that although Hilbert established his theory of proofs as a part of his foundational program and, for philosophical reasons whichwe shall get into, aimed to have it developed in a completely nitistic way, the actual work in proof theory This is the rst of three lectures that I delivered at the conference, Proof Theory: History
A formal system for Euclid's Elements
, 2009
"... We present a formal system, E, which provides a faithful model of the proofs in Euclid’s Elements, including the use of diagrammatic reasoning. ..."
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We present a formal system, E, which provides a faithful model of the proofs in Euclid’s Elements, including the use of diagrammatic reasoning.
Quantified Propositional Calculus and a SecondOrder Theory for NC¹
, 2004
"... Let H be a proof system for the quantified propositional calculus (QPC). We j witnessing problem for H to be: given a prenex S j formula A, an Hproof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the S ..."
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Cited by 14 (3 self)
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Let H be a proof system for the quantified propositional calculus (QPC). We j witnessing problem for H to be: given a prenex S j formula A, an Hproof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the S witnessing problems for the systems G 1 and G 1 are complete for polynomial time and PLS (polynomial local search), respectively. We introduce