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The Undecidability of kProvability
 Annals of Pure and Applied Logic
, 1989
"... The kprovability problem is, given a first order formula φ and an integer k, to determine if φ has a proof consisting of k or fewer lines (i.e., formulas or sequents). This paper shows that the kprovability problem for the sequent calculus is undecidable. Indeed, for every r.e. srt ..."
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The kprovability problem is, given a first order formula &phi; and an integer k, to determine if &phi; has a proof consisting of k or fewer lines (i.e., formulas or sequents). This paper shows that the kprovability problem for the sequent calculus is undecidable. Indeed, for every r.e. srt X...
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
On Frege and Extended Frege Proof Systems
, 1993
"... We propose a framework for proving lower bounds to the size of EF  proofs (equivalently, to the number of proofsteps in Fproofs) in terms of boolean valuations . The concept is motivated by properties of propositional provability in models of bounded arithmetic and it is a finitisation of a parti ..."
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We propose a framework for proving lower bounds to the size of EF  proofs (equivalently, to the number of proofsteps in Fproofs) in terms of boolean valuations . The concept is motivated by properties of propositional provability in models of bounded arithmetic and it is a finitisation of a particular forcing construction explained also in the paper. It reduces the question of proving a lower bound to the question of constructing a partial boolean algebra and a map of formulas into that algebra with particular properties. We show that in principle one can obtain via this method optimal lower bounds (up to a polynomial increase). Introduction A propositional proof system is any polynomial time function P whose range is exactly the set of tautologies TAUT, cf. [17]. For ø a tautology any string ß such that P (ß) = ø is called a P proof of ø . Any usual propositional calculus, be it resolution or extended resolution, a Hilbert style system based on finitely many axiom schemes and inf...
How to Lie Without Being (easily) Convicted and the Lengths of Proofs in Propositional Calculus
"... We shall describe two general methods for proving lower bounds on the lengths of proofs in propositional calculus and give examples of such lower bounds. One of the methods is based on interactive proofs where one player is claiming that he has a falsifying assignment for a tautology and the sec ..."
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We shall describe two general methods for proving lower bounds on the lengths of proofs in propositional calculus and give examples of such lower bounds. One of the methods is based on interactive proofs where one player is claiming that he has a falsifying assignment for a tautology and the second player is trying to convict him of a lie.
Methods of CutElimination
 PROJECTION, LECTURE
"... This short report presents the main topics of methods of cutelimination which will be presented in the course at the ESSLLI'99. It gives a short introduction addressing the problem of cutelimination in general. Furthermore we give a brief description of several methods and refer to other pape ..."
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Cited by 13 (11 self)
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This short report presents the main topics of methods of cutelimination which will be presented in the course at the ESSLLI'99. It gives a short introduction addressing the problem of cutelimination in general. Furthermore we give a brief description of several methods and refer to other papers added to the course material.
The Complexity of ResourceBounded FirstOrder Classical Logic
 11th Symposium on Theoretical Aspects of Computer Science
, 1994
"... . We give a finer analysis of the difficulty of proof search in classical firstorder logic, other than just saying that it is undecidable. To do this, we identify several measures of difficulty of theorems, which we use as resource bounds to prune infinite proof search trees. In classical firstord ..."
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. We give a finer analysis of the difficulty of proof search in classical firstorder logic, other than just saying that it is undecidable. To do this, we identify several measures of difficulty of theorems, which we use as resource bounds to prune infinite proof search trees. In classical firstorder logic without interpreted symbols, we prove that for all these measures, the search for a proof of bounded difficulty (i.e, for a simple proof) is \Sigma p 2 complete. We also show that the same problem when the initial formula is a set of Horn clauses is only NPcomplete, and examine the case of firstorder logic modulo an equational theory. These results allow us not only to give estimations of the inherent difficulty of automated theorem proving problems, but to gain some insight into the computational relevance of several automated theorem proving methods. Topics: computational complexity, logics, computational issues in AI (automated theorem proving). 1 Introduction Firstorder ...
On Gödel's Theorems on Lengths of Proofs I: Number of Lines and Speedup for Arithmetics
"... 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 ar ..."
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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 Hilbertstyle calculi with + and as function symbols or with the language of PRA. The same results are established for any weakly schematic formalization of higherorder logic; this allows all tautologies as axioms and allows all generalizations of axioms as axioms.
The Craig Interpolation Theorem for Schematic Systems
, 1996
"... The notion of Schematic System has been introduced by Parikh in the early seventies. It is a metamathematical notion describing the concept of deduction system and the operation of substitution of terms and formulas in it. We show a generalization of the Craig Interpolation Theorem for a natural ..."
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The notion of Schematic System has been introduced by Parikh in the early seventies. It is a metamathematical notion describing the concept of deduction system and the operation of substitution of terms and formulas in it. We show a generalization of the Craig Interpolation Theorem for a natural class of schematic systems while we determine sufficient conditions for a schematic system to enjoy Interpolation.
Unbounded prooflength speedup in deduction modulo
 CSL 2007, VOLUME 4646 OF LNCS
, 2007
"... In 1973, Parikh proved a speedup theorem conjectured by Gödel 37 years before: there exist arithmetical formulæ that are provable in first order arithmetic, but whose shorter proof in second order arithmetic is arbitrarily smaller than any proof in first order. On the other hand, resolution for h ..."
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Cited by 7 (3 self)
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In 1973, Parikh proved a speedup theorem conjectured by Gödel 37 years before: there exist arithmetical formulæ that are provable in first order arithmetic, but whose shorter proof in second order arithmetic is arbitrarily smaller than any proof in first order. On the other hand, resolution for higher order logic can be simulated step by step in a first order narrowing and resolution method based on deduction modulo, whose paradigm is to separate deduction and computation to make proofs clearer and shorter. We prove that i+1th order arithmetic can be linearly simulated into ith order arithmetic modulo some confluent and terminating rewrite system. We also show that there exists a speedup between ith order arithmetic modulo this system and ith order arithmetic without modulo. All this allows us to prove that the speedup conjectured by Gödel does not come from the deductive part of the proofs, but can be expressed as simple computation, therefore justifying the use of deduction modulo as an efficient first order setting simulating higher order.
Efficiently Simulating HigherOrder Arithmetic by a FirstOrder Theory Modulo
"... Deduction modulo is a paradigm which consists in applying the inference rules of a deductive system—such as for instance natural deduction—modulo a rewrite system over terms and propositions. It has been shown that higherorder logic can be simulated into the firstorder natural deduction modulo. Ho ..."
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Deduction modulo is a paradigm which consists in applying the inference rules of a deductive system—such as for instance natural deduction—modulo a rewrite system over terms and propositions. It has been shown that higherorder logic can be simulated into the firstorder natural deduction modulo. However, a theorem stated by Gödel and proved by Parikh expresses that proofs in secondorder arithmetic may be unboundedly shorter than proofs in firstorder arithmetic, even when considering only formulæ provable in firstorder arithmetic. We investigate how deduction modulo can be used to translate proofs of higherorder arithmetic into firstorder proofs without inflating their length. First we show how higher orders can be encoded through a quite simple (finite, terminating, confluent, leftlinear) rewrite system. Then, a proof in higherorder arithmetic can be linearly translated into a proof in firstorder arithmetic modulo this system. Second, in the continuation of a work of Dowek and Werner, we show how to express the whole higherorder arithmetic as a rewrite system. Then, proofs of higherorder arithmetic can be linearly translated into proofs in the empty theory modulo this rewrite system. These results show that the speedup between firstand secondorder arithmetic, and more generally between i th and i +1 storder arithmetic, can in fact be expressed as computation, and does not lie in the really deductive part of the proofs.