We introduce martingales defined by probabilistic strategies, in which randomness is used to decide whether to bet. We show that different criteria for the success of computable probabilistic strategies can be used to characterize ML-randomness, computable randomness, and partial computable randomness. Our characterization of ML-randomness partially addresses a critique of Schnorr by formulating ML randomness in terms of a computable process rather than a computably enumerable function.
We present sharpened lower bounds on the size of cut free proofs for first-order logic. Prior lower bounds for eliminating cuts from a proof established superexponential lower bounds as a stack of exponentials, with the height of the stack proportional to the maximum depth d of the formulas in the original proof. Our results remove the constant of proportionality, giving an exponential stack of height equal to d — 0(1). The proof method is based on more efficiently expressing the Gentzen-Solovay (...) cut formulas as low depth formulas. (shrink)
We give the first systematic study of strong isomorphism reductions, a notion of reduction more appropriate than polynomial time reduction when, for example, comparing the computational complexity of the isomorphim problem for different classes of structures. We show that the partial ordering of its degrees is quite rich. We analyze its relationship to a further type of reduction between classes of structures based on purely comparing for every n the number of nonisomorphic structures of cardinality at most n in both (...) classes. Furthermore, in a more general setting we address the question of the existence of a maximal element in the partial ordering of the degrees. (shrink)
A pool resolution proof is a dag-like resolution proof which admits a depth-first traversal tree in which no variable is used as a resolution variable twice on any branch. The problem of determining whether a given dag-like resolution proof is a valid pool resolution proof is shown to be NP-complete.
We prove that the problem of determining the minimum propositional proof length is NP- hard to approximate within a factor of 2 log 1 - o(1) n . These results are very robust in that they hold for almost all natural proof systems, including: Frege systems, extended Frege systems, resolution, Horn resolution, the polynomial calculus, the sequent calculus, the cut-free sequent calculus, as well as the polynomial calculus. Our hardness of approximation results usually apply to proof length measured either by (...) number of symbols or by number of inferences, for tree-like or dag-like proofs. We introduce the Monotone Minimum (Circuit) Satisfying Assignment problem and reduce it to the problems of approximation of the length of proofs. (shrink)
The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
This volume contains articles covering a broad spectrum of proof theory, with an emphasis on its mathematical aspects. The articles should not only be interesting to specialists of proof theory, but should also be accessible to a diverse audience, including logicians, mathematicians, computer scientists and philosophers. Many of the central topics of proof theory have been included in a self-contained expository of articles, covered in great detail and depth. The chapters are arranged so that the two introductory articles come first; (...) these are then followed by articles from core classical areas of proof theory; the handbook concludes with articles that deal with topics closely related to computer science. (shrink)
Originating from work in operations research the cutting plane refutation systemCP is an extension of resolution, where unsatisfiable propositional logic formulas in conjunctive normal form are recognized by showing the non-existence of boolean solutions to associated families of linear inequalities. Polynomial sizeCP proofs are given for the undirecteds-t connectivity principle. The subsystemsCP q ofCP, forq≥2, are shown to be polynomially equivalent toCP, thus answering problem 19 from the list of open problems of . We present a normal form theorem forCP (...) 2-proofs and thereby for arbitraryCP-proofs. As a corollary, we show that the coefficients and constant terms in arbitrary cutting plane proofs may be exponentially bounded by the number of steps in the proof, at the cost of an at most polynomial increase in the number of steps in the proof. The extensionCPLE +, introduced in  and there shown top-simulate Frege systems, is proved to be polynomially equivalent to Frege systems. Lastly, since linear inequalities are related to threshold gates, we introduce a new threshold logic and prove a completeness theorem. (shrink)
We survey the best known lower bounds on symbols and lines in Frege and extended Frege proofs. We prove that in minimum length sequent calculus proofs, no formula is generated twice or used twice on any single branch of the proof. We prove that the number of distinct subformulas in a minimum length Frege proof is linearly bounded by the number of lines. Depthd Frege proofs ofm lines can be transformed into depthd proofs ofO(m d+1) symbols. We show that renaming (...) Frege proof systems are p-equivalent to extended Frege systems. Some open problems in propositional proof length and in logical flow graphs are discussed. (shrink)
This paper discusses lower bounds for proof length, especially as measured by number of steps (inferences). We give the first publicly known proof of Gödel's claim that there is superrecursive (in fact. unbounded) proof speedup of (i + 1)st-order arithmetic over ith-order arithmetic, where arithmetic is formalized in Hilbert-style calculi with + and · as function symbols or with the language of PRA. The same results are established for any weakly schematic formalization of higher-order logic: this allows all tautologies as (...) axioms and allows all generalizations of axioms as axioms. Our first proof of Gödel's claim is based on self-referential sentences: we give a second proof that avoids the use of self-reference based loosely on a method of Statman. (shrink)
We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (...) (or formulas) in the proof. A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by "nearly linear" is meant the ratio of proof lengths is O(α(n)) where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n · α(n)). Also we show that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n). (shrink)
Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounded arithmetic.