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18
On Interpolation and Automatization for Frege Systems
, 2000
"... The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the ..."
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Cited by 52 (8 self)
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The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the proof system, sometimes assuming a (usually modest) complexitytheoretic assumption. In this paper, we show that this method cannot be used to obtain lower bounds for Frege systems, or even for TC 0 Frege systems. More specifically, we show that unless factoring (of Blum integers) is feasible, neither Frege nor TC 0 Frege has the feasible interpolation property. In order to carry out our argument, we show how to carry out proofs of many elementary axioms/theorems of arithmetic in polynomial size TC 0 Frege. As a corollary, we obtain that TC 0 Frege as well as any proof system that polynomially simulates it, is not automatizable (under the assumption that factoring of Blum integ...
Asymptotic cyclic expansion and bridge groups of formal proofs
 JOURNAL OF ALGEBRA
, 2001
"... Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify th ..."
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Cited by 5 (1 self)
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Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify them with the Baumslag–Solitar and Gersten groups. We observe that the distortion of cyclic subgroups in these groups equals the asymptotic growth of the procedure of elimination of lemmas from the proofs.
Interpolants, Cut Elimination and Flow Graphs . . .
, 1997
"... We analyse the structure of propositional proofs in the sequent calculus focusing on the wellknown procedures of Interpolation and Cut Elimination. We are motivated in part by the desire to understand why a tautology might be ‘hard to prove’. Given a proof we associate to it a logical graph tracing ..."
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Cited by 5 (3 self)
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We analyse the structure of propositional proofs in the sequent calculus focusing on the wellknown procedures of Interpolation and Cut Elimination. We are motivated in part by the desire to understand why a tautology might be ‘hard to prove’. Given a proof we associate to it a logical graph tracing the flow of formulas in it (Buss, 1991). We show some general facts about logical graphs such as acyclic @ of cutfree proofs and acyclic @ of contractionfree proofs (possibly containing cuts), and we give a proof of a strengthened version of the Craig Interpolation Theorem based on flows of formulas. We show that tautologies having minimal interpolants of nonlinear size (i.e. number of symbols) must have proofs with certain precise structural properties. We then show that given a proof ZI and a cutfree form Il ’ associated to it (obtained by a particular cut elimination procedure), certain subgraphs of II ’ which are logical graphs (i.e. graphs of proofs) correspond to subgraphs of Zl which are logical graphs for the same sequent. This locality property of cut elimination leads to new results on the complexity of interpolants, which cannot follow from the known constructions proving the Craig Interpolation Theorem.
ANNALS OF PURE AND APPLIED LOGIC
, 1997
"... In [ 131 Parikh proved the first mathematical result about concrete consistency of contradictory theories. In [6] it is shown that the bounds of concrete consistency given by Parikh are optimal. This was proved by noting that very large numbers can be actually constructed through very short proofs, ..."
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In [ 131 Parikh proved the first mathematical result about concrete consistency of contradictory theories. In [6] it is shown that the bounds of concrete consistency given by Parikh are optimal. This was proved by noting that very large numbers can be actually constructed through very short proofs, A more refined analysis of these short proofs reveals the presence of cyclic paths in their logical graphs, Indeed, in [6] it is shown that cycles need to exist for the proofs to be short. Here, we present a new sequent calculus for classical logic which is close to linear logic in spirit, enjoys cutelimination, is acyclic and its proofs are just &~errtar~ ~ larger than proofs in LK. The proofs in the new calculus can bc obtained by a srn~ll perturhntim of proofs in LK and they represent a geometrical alternative for studying structural properties of LKproofs. They satisfy the constructive disjunction property and most important. simpler geometrical properties of their logical graphs. The geometrical counterpart to a cycle in LK is represented in the new setting by a spiwl which is passing through sets of formulas logically grouped together by the
Identity of Proofs Based on Normalization and Generality
, 2003
"... Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz’s, bas ..."
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Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz’s, based on reduction to cutfree form in sequent systems, but he also suggested understanding identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally the rules involved in them they yield the same premises and conclusions up to renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic. The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it did not fare well. The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to lowdimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic.
Identity of Proofs Based on Normalization and Generality
, 2002
"... Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz’s, bas ..."
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Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz’s, based on reduction to cutfree form in sequent systems, but he also suggested to understand identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally the rules involved in them they yield the same premises and conclusions up to renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic. The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it didn’t fare well. The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to lowdimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic.
Identity of Proofs Based on Normalization and Generality
, 2002
"... Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz’s, bas ..."
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Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz’s, based on reduction to cutfree form in sequent systems, but he also suggested to understand identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally the rules involved in them they yield the same premises and conclusions up to renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic. The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it didn’t fare well. The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to lowdimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic.