@Article{         Krajcek:JSL:1997,
  Author        = "Kraj{\'\i}{\v c}ek, Jan",
  Abstract      = "A proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuit-size is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: (1) Feasible interpolation theorems for the following proof systems: (a) resolution (b) a subsystem of LK corresponding to the bounded arithmetic theory S22({$\alpha$}) (c) linear equational calculus (d) cutting planes. (2) New proofs of the exponential lower bounds (for new formulas) (a) for resolution ([15]) (b) for the cutting planes proof system with coefficients written in unary ([4]). (3) An alternative proof of the independence result of [43] concerning the provability of circuit-size lower bounds in the bounded arithmetic theory S22({$\alpha$}). In the other direction we show that a depth 2 subsystem of LK does not admit feasible monotone interpolation theorem (the so called Lyndon theorem), and that a feasible monotone interpolation theorem for the depth 1 subsystem of LK would yield new exponential lower bounds for resolution proofs of the weak pigeonhole principle.",
  date-added    = "2019-10-09 22:00:59 +0200",
  date-modified = "2019-10-15 14:38:21 +0200",
  ISSN          = "00224812",
  Journal       = "The Journal of Symbolic Logic",
  Number        = "2",
  Pages         = "457--486",
  Publisher     = "Association for Symbolic Logic",
  Title         = "Interpolation Theorems, Lower Bounds for Proof Systems, and Independence Results for Bounded Arithmetic",
  URL           = "http://www.jstor.org/stable/2275541",
  Volume        = "62",
  Year          = "1997",
  bdsk-url-1    = "http://www.jstor.org/stable/2275541",
  File          = "Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic - 10.2307@2275541 (0) - a - a - p.pdf"
}

@Article{ Krajcek:JSL:1997, Author = "Kraj{\'\i}{\v c}ek, Jan", Abstract = "A proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuit-size is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: (1) Feasible interpolation theorems for the following proof systems: (a) resolution (b) a subsystem of LK corresponding to the bounded arithmetic theory S22({$\alpha$}) (c) linear equational calculus (d) cutting planes. (2) New proofs of the exponential lower bounds (for new formulas) (a) for resolution ([15]) (b) for the cutting planes proof system with coefficients written in unary ([4]). (3) An alternative proof of the independence result of [43] concerning the provability of circuit-size lower bounds in the bounded arithmetic theory S22({$\alpha$}). In the other direction we show that a depth 2 subsystem of LK does not admit feasible monotone interpolation theorem (the so called Lyndon theorem), and that a feasible monotone interpolation theorem for the depth 1 subsystem of LK would yield new exponential lower bounds for resolution proofs of the weak pigeonhole principle.", date-added = "2019-10-09 22:00:59 +0200", date-modified = "2019-10-15 14:38:21 +0200", ISSN = "00224812", Journal = "The Journal of Symbolic Logic", Number = "2", Pages = "457--486", Publisher = "Association for Symbolic Logic", Title = "Interpolation Theorems, Lower Bounds for Proof Systems, and Independence Results for Bounded Arithmetic", URL = "http://www.jstor.org/stable/2275541", Volume = "62", Year = "1997", bdsk-url-1 = "http://www.jstor.org/stable/2275541", File = "Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic - 10.2307@2275541 (0) - a - a - p.pdf" }