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24
Interpretability logic
 Mathematical Logic, Proceedings of the 1988 Heyting Conference
, 1990
"... Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength ..."
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Cited by 42 (9 self)
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Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength of theories, or better to prove
Disjoint NPPairs
, 2003
"... We study the question of whether the class DisNP of disjoint pairs (A, B) of NPsets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NPsets that is N ..."
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Cited by 23 (8 self)
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We study the question of whether the class DisNP of disjoint pairs (A, B) of NPsets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NPsets that is NPhard. We show under reasonable hypotheses that nonsymmetric disjoint NPpairs exist, which provides additional evidence for the existence of Pinseparable disjoint NPpairs. We construct
A Complexity Gap for TreeResolution
 COMPUTATIONAL COMPLEXITY
, 1999
"... It is shown that any sequence #n of tautologies which expresses the validity of a fixed combinatorial principle either is "easy" i.e. has polynomial size treeresolution proofs or is "difficult" i.e requires exponential size treeresolution proofs. It is shown that the class o ..."
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Cited by 19 (3 self)
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It is shown that any sequence #n of tautologies which expresses the validity of a fixed combinatorial principle either is "easy" i.e. has polynomial size treeresolution proofs or is "difficult" i.e requires exponential size treeresolution proofs. It is shown that the class of tautologies which are hard (for treeresolution) is identical to the class of tautologies which are based on combinatorial principles which are violated for infinite sets. Actually it is
Formalizing forcing arguments in subsystems of secondorder arithmetic
 Annals of Pure and Applied Logic
, 1996
"... We show that certain modeltheoretic forcing arguments involving subsystems of secondorder arithmetic can be formalized in the base theory, thereby converting them to effective prooftheoretic arguments. We use this method to sharpen conservation theorems of Harrington and BrownSimpson, giving an ..."
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Cited by 18 (9 self)
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We show that certain modeltheoretic forcing arguments involving subsystems of secondorder arithmetic can be formalized in the base theory, thereby converting them to effective prooftheoretic arguments. We use this method to sharpen conservation theorems of Harrington and BrownSimpson, giving an effective proof that W KL+0 is conservative over RCA0 with no significant increase in the lengths of proofs. 1
ON THE NUMBER OF STEPS IN PROOFS
, 1989
"... In this paper we prove some results about the complexity of proofs. We consider proofs in Hilbertstyle formal systems such as in [17J. Thus a proof is a sequence of formulas satisfying certain conditions. We caD view the formulas as being strings of symbols; hence the whole proof is a string too. W ..."
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Cited by 15 (2 self)
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In this paper we prove some results about the complexity of proofs. We consider proofs in Hilbertstyle formal systems such as in [17J. Thus a proof is a sequence of formulas satisfying certain conditions. We caD view the formulas as being strings of symbols; hence the whole proof is a string too. We consider the following measures of complexity of proofs: length ( = the number of symbols in the proof), depth ( = the maximal depth of a formula in the proof) and number o! steps ( = the number of formulas in the proof). For a particular formaI system and a given formula A we consider the shortest length of a proof of A, the minimal depth ofa proof of A and the minimal number of steps in a proof of A. The main results are the following: (1) a bound on the depth in terms of the number of steps: Theorem 2.2, (2) a bound on the depth in terms of the length: Theorem 2.3, (3) a bound on the length in terms of the number of steps for restricted systems: Theorem 3.1. These results are applied to obtain several corollaries. In particular we show: (1) a bound on the number of steps in a cutfree proof, (2) some speedup results, (3) bounds on the number of steps in proofs of ParisHarrington sentences. Some paper
Survey of Disjoint NPPairs and Relations to Propositional Proof Systems
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 72
, 2005
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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.
Propositional Consistency Proofs
, 2002
"... Partial consistency statements can be expressed as polynomialsize propositional formulas. Frege proof systems have polynomialsize partial selfconsistency proofs. Frege proof systems have polynomialsize proofs of partial consistency of extended Frege proof systems if and only if Frege proof syste ..."
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Cited by 6 (0 self)
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Partial consistency statements can be expressed as polynomialsize propositional formulas. Frege proof systems have polynomialsize partial selfconsistency proofs. Frege proof systems have polynomialsize proofs of partial consistency of extended Frege proof systems if and only if Frege proof systems polynomially simulate extended Frege proof systems. We give a new proof of Reckhow's theorem that any two Frege proof systems psimulate each other. The proofs depend on polynomial size propositional formulas defining the truth of propositional formulas. These are already known to exist since the Boolean formula value problem is in alternating logarithmic time; this paper presents a proof of this fact based on a construction which is somewhat simpler than the prior proofs of Buss and of BussCookGuptaRamachandran.
Optimal proof systems for propositional logic and complete sets
 IN PROCEEDINGS 15TH SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE, LECTURE NOTES IN COMPUTER SCIENCE
, 1997
"... A polynomial time computable function h : \Sigma whose range is the set of tautologies in Propositional Logic (TAUT), is called a proof system. Cook and Reckhow defined this concept in [6] and in order to compare the relative strength of different proof systems, they considered the notion of p ..."
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Cited by 3 (0 self)
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A polynomial time computable function h : \Sigma whose range is the set of tautologies in Propositional Logic (TAUT), is called a proof system. Cook and Reckhow defined this concept in [6] and in order to compare the relative strength of different proof systems, they considered the notion of psimulation. Intuitively a proof system h psimulates a second one h if there is a polynomial time computable function fl translating proofs in h into proofs in h. A proof system is called optimal if it psimulates every other proof system. The question of whether poptimal proof systems exist is an important one in the field. Kraj'icek and Pudl'ak [14, 13] have given a sufficient condition for the existence of such optimal systems, showing that if the deterministic and nondeterministic exponential time classes coincide, then poptimal proof systems exist. They also give a condition implying the existence of optimal proof systems (a related concept to the one of poptimal systems) exist. In this paper we improve this result giving a weaker sufficient condition for this fact. We show that if a particular class of sets with low information content in nondeterministic double exponential time is included in the corresponding deterministic class, then poptimal proof systems exist. We also show some complexity theoretical consequences that follow from the assumption of the existence of poptimal systems. We prove that if poptimal systems exist the the class UP (an some other related complexity classes) have manyone complete languages, and that manyone complete sets for NP " SPARSE follow from the existence of optimal proof systems.
On the Existence of Optimal Propositional Proof Systems and OracleRelativized Propositional Logic
 Electronic Colloquium on Computational Complexity
, 1998
"... We investigate sufficient conditions for the existence of optimal propositional proof systems. We concentrate on conditions of the form CoNF = NF . We introduce a purely combinatorial property of complexity classes  the notions of slim vs. fat classes. These notions partition the collection of all ..."
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We investigate sufficient conditions for the existence of optimal propositional proof systems. We concentrate on conditions of the form CoNF = NF . We introduce a purely combinatorial property of complexity classes  the notions of slim vs. fat classes. These notions partition the collection of all previously studied timecomplexity classes into two complementary sets. We show that for every slim class an appropriate collapse entails the existence of an optimal propositional proof system. On the other hand, we introduce a notion of a propositional proof system relative to an oracle and show that for every fat class there exists an oracle relative to which such an entailment fails. As the classes P (polynomial functions), E (2 O(n) functions) and EE (2 O(2 n ) functions) are slim, this result includes all the previously known sufficiency conditions for the existence of optimal propositional proof systems. On the other hand, the classes NEXP, QP (the class of quasipolynomial functions) and EEE (2 O(2 2 n ) functions), as well as any other natural timecomplexity class which is not covered by our sufficiency result, are fat classes. As the proofs of all the known sufficiency conditions for the existence of optimal propositional proof systems carry over to the corresponding oraclerelativized notions, our oracle result shows that no extension of our sufficiency condition to nonslim classes can be obtained by the type of reasoning used so far in proofs on these issues.