Skip to main content Accessibility help
×
Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-30T01:33:24.368Z Has data issue: false hasContentIssue false

Bibliography

Published online by Cambridge University Press:  25 March 2019

Jan Krajíček
Affiliation:
Charles University, Prague
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Proof Complexity , pp. 481 - 505
Publisher: Cambridge University Press
Print publication year: 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aehlig, K. and Beckmann, A., A remark on the induction needed to prove the Ramsey principle, unpublished manuscript (2006). p. 292Google Scholar
Aisenberg, J., Bonet, M. L., Buss, S. R., Craciun, A. and Istrate, G., Short proofs of the Kneser–Lovász coloring principle, in: Proc. 42nd International Colloq. on Automata, Languages and Programming (ICALP), Springer Lecture Notes in Computer Science 9135 (2015), 44–55. p. 230Google Scholar
Aisenberg, J., Bonet, M. L. and Buss, S. R., Quasipolynomial size Frege proofs of Frankl’s theorem on the trace of sets, J. Symbolic Logic 81 (2)(2016), 1–24. p. 231Google Scholar
Ajtai, M., -formulas on finite structures, Ann. Pure and Applied Logic 24 (1983), 1–48. pp. 29, 183, 449, 454CrossRefGoogle Scholar
Ajtai, M., The complexity of the pigeonhole principle, in: Proc. IEEE 29th Annual Symp. on Foundations of Computer Science (1988), 346–355. pp. 3, 163, 165, 179, 183, 306, 307, 318, 327, 334, 454Google Scholar
Ajtai, M., Parity and the pigeonhole principle, in: Feasible Mathematics, eds. Buss, S. R. and Scott, P. J., Birkhauser (1990), 1–24. pp. 230, 335Google Scholar
Ajtai, M., The independence of the modulo p counting principles, in: Proc. 26th Annual ACM Symposium on Theory of Computing, ACM Press (1994), 402–411. pp. 230, 326, 335, 349Google Scholar
Ajtai, M., Symmetric systems of linear equations modulo p, in: Proc. Electronic Colloquium on Computational Complexity (ECCC), TR94–015 (1994). p. 349Google Scholar
Ajtai, M., Generalizations of the compactness theorem and Gödel’s completeness theorem for nonstandard finite structures, in: Proc. 4th International Conf. on Theory and Applications of Models of Computation (2007), 13–33. p. 454Google Scholar
Ajtai, M., A generalization of Gödel’s completeness theorem for nonstandard finite structures, unpublished manuscript (2011). p. 454Google Scholar
Alekhnovich, M., Ben-Sasson, E., Razborov, A. A., and Wigderson, A., Space complexity in propositional calculus, SIAM J. Computing, 31 (4)(2002), 1184–1211. p. 113Google Scholar
Alekhnovich, M., Ben-Sasson, E., Razborov, A. A., and Wigderson, A., Pseudorandom generators in propositional proof complexity, SIAM J. Computing, 34 (1) (2004), 67–88. pp. 157, 278, 292, 350, 437, 438Google Scholar
Alekhnovich, M., Hirsch, E. A. and Itsykson, D., Exponential lower bounds for the running time of DPLL algorithms on satisfiable formulas, J. Automated Reasoning, 35 (1–3) (2005), 51–72. p. 295Google Scholar
Alekhnovich, M., Johannsen, J., Pitassi, T. and Urquhart, A., An exponential separation between regular and general resolution, in: Proc. 34th Annual ACM Symp on Theory of Computing (STOC) (2002), 448–456. p. 295Google Scholar
Alekhnovich, M. and Razborov, A. A., Lower bound for polynomial calculus: non-binomial case, Proc. the Steklov Institute of Mathematics, 242(2003), 18–35. p. 350Google Scholar
Alekhnovich, M. and Razborov, A. A., Resolution is not automatizable unless WP is tractable, SIAM J. Computing, 38 (4) (2008), 1347–1363. p. 470CrossRefGoogle Scholar
Alexi, W. B., Chor, B., Goldreich, O. and Schnorr, C. P., RSA and Rabin functions: certain parts are as hard as the whole, SIAM J. Computing, 17(1988), 194–209. p. 364CrossRefGoogle Scholar
Alon, N. and Boppana, R., The monotone circuit complexity of Boolean functions, Combinatorica, 7 (1) (1987), 1–22. pp. 38, 288, 382Google Scholar
Alon, N. and Spencer, J. H., The Probabilistic Method, 3rd edn, John Wiley and Sons, (2011). p. 351Google Scholar
Andreev, A. E., On a method for obtaining lower bounds for the complexity of individual monotone functions, Sov. Math. Dokl., 31 (1985), 530–534. p. 38Google Scholar
Arai, N., Relative efficiency of propositional proof systems: Resolution and cut-free LK, Ann. Pure and Applied Logic, 104 (2000), 3–16. p. 112Google Scholar
Arai, N., Pitassi, T. and Urquhart, A., The complexity of analytic tableaux, in: Proc. ACM Symp. on the Theory of Computing (STOC) (2001), 356–363. p. 112Google Scholar
Arai, T., A bounded arithmetic AID for Frege systems, Ann. Pure and Applied Logic, 103 (2000), 155–199. p. 209Google Scholar
Artin, E., Uber die zerlegung definiter Funktionen in Quadrate, in: Abhandlungen aus dem mathematischen Seminar der Universitat Hamburg, 5(1927), 100–115. Springer. p. 132Google Scholar
Asser, G., Das Reprasentenproblem in Pradikatenkalkul der ersten Stufe mit Identitat, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1 (1955), 252–263. p. 2Google Scholar
Atserias, A., The proof-search problem between bounded-width resolution and bounded-degree semi-algebraic proofs, in: Proc. Conf. on Theory and Applications of Satisfiability Testing (SAT 2013), eds. Jarvisalo, M. and Van Gelder, A., 7962, Lecture Notes in Computer Science (2013), 1–17. p. 470Google Scholar
Atserias, A., A note on semi-algebraic proofs and gaussian elimination over prime fields, unpublished preprint (2015). pp. 348, 351Google Scholar
Atserias, A., Bonacina, I., de Rezende, S. F., Lauria, M., Nordström, J. and Razborov, A. A., Clique is hard on average for regular resolution, to appear in: Proc. 50th ACM Symp. on Theory of Computing (STOC) (2018). p. 293Google Scholar
Atserias, A. and Bonet, M. L., On the automatizability of resolution and related propositional proof systems, Information and Computation, 189 (2)(2004), 182–201. pp. 467, 471Google Scholar
Atserias, A., Bonet, M. L. and Levy, J., On Chvátal Rank and Cutting Planes Proofs, in: Proc. Electronic Colloq. on Computational Complexity(ECCC), TR03–041 (2003). p. 352Google Scholar
Atserias, A. and Dalmau, V., A combinatorial characterization of resolution width, J. Computer and System Sciences, 74 (3) (2008), 323–334. pp. 105, 113Google Scholar
Atserias, A., Fichte, J. K., and Thurley, M., Clause-learning algorithms with many restarts and bounded-width resolution, J. Artificial Intelligence Research, 40 (2011), 353–373. p. 295Google Scholar
Atserias, A., Galesi, N. and Gavaldá, R., Monotone proofs of the pigeon hole principle, Mathematical Logic Quarterly, 47 (4) (2001), 461–474. p. 158Google Scholar
Atserias, A., Galesi, N. and Pudlák, P., Monotone simulations of nonmonotone proofs, J. Computer and System Sciences, 65 (2002), 626–638. pp. 146, 158Google Scholar
Atserias, A., Kolaitis, P. and Vardi, M. Y., Constraint propagation as a proof system, in: Proc. 10th International Conf. on Principles and Practice of Constraint Programming (CP), 3258, Lecture Notes in Computer Science (2004), 77–91. pp. 147, 159Google Scholar
Atserias, A., Lauria, M. and Nordström, J., Narrow proofs may be maximally long, ACM Trans. Computational Logic, 17 (2016), 19:1–19:30. pp. 112, 294Google Scholar
Atserias, A. and Maneva, E., Mean-payoff games and propositional proofs, Information and Computation, 209 (4) (2011), 664–691. p. 470Google Scholar
Atserias, A. and Maneva, E., Sherali–Adams relaxations and indistinguishability in counting logics, SIAM J. Computing, 42 (1) (2013), 112–137. pp. 133, 455Google Scholar
Atserias, A. and Müller, M., Partially definable forcing and bounded arithmetic, Archive for Mathematical Logic, 54 (1–2) (2015), 1–33. p. 454Google Scholar
Atserias, A., Müller, M. and Oliva, S., Lower bounds for DNF-refutations of a relativized weak pigeonhole principle, J. Symbolic Logic, 80 (2)(2015), 450–476. p. 292Google Scholar
Atserias, A. and Ochremiak, J., Definable ellipsoid method, sums-of-squares proofs, and the isomorphism problem, to appear in: Proc. 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), July 2018. pp. 351, 455Google Scholar
Avigad, J., Plausibly hard combinatorial tautologies, in: Proof Complexity and Feasible Arithmetics, eds Buss, S. R. and Beame, P., American Mathematical Society (1997), 1–12. pp. 63, 436Google Scholar
Babai, L., Gál, A., Kollár, J., Rónyai, L., Szabó, T. and Wigderson, A., Extremal bipartite graphs and superpolynomial lower bounds for monotone span programs, in: Proc. 28th Annual ACM Symp. on Theory of Computing (STOC) (1996), 603–611. p. 393Google Scholar
Baker, T., Gill, J. and Solovay, R., Relativizations of the P =? N P question, SIAM J. Comput., 4 (1975), 431–442. p. 463Google Scholar
Barak, B., Brandao, F. G. S. L., Harrow, A. W., Kelner, J. A., Steurer, D. and Zhou, Y., Hypercontractivity, sum-of-squares proofs, and their applications, in: Proc. 28th Annual ACM Symp. on Theory of Computing (STOC) (1996), 307–326. p. 351Google Scholar
Barak, B., Kelner, J. A., Steurer, D., Rounding sum of squares relaxations, in: Proc. 46th Annual ACM Symp. on Theory of Computing (STOC) (2014), 31–40. p. 351Google Scholar
Barak, B. and Steurer, D., Sum-of-squares proofs and the quest toward optimal algorithms, in: Proc. ICM Conf. (Seul) (2014). pp. 133, 351Google Scholar
Beame, P., Proof complexity, in: Computational Complexity Theory, eds. Rudich, S. and Wigderson, A., AMS, IAS/Park City Math. Ser., 10(2004), 199–246. p. 6Google Scholar
Beame, P., Beck, C. and Impagliazzo, R., Time–space tradeoffs in resolution: super-polynomial lower bounds for superlinear space, in: Proc. 44th Annual ACM Symp. on Theory of Computing (STOC) (2012), 212–232. p. 294Google Scholar
Beame, P., Cook, S. A., Edmonds, J., Impagliazzo, R. and Pitassi, T., The relative complexity of NP search problems, J. Computer Systems Sciences, 57 (1998), 3–19. pp. 426, 436Google Scholar
Beame, P., Culberson, J., Mitchell, D. and Moore, C., The resolution complexity of random graph k-colorability, Discrete Applied Mathematics, 153(2005), 25–47. p. 293Google Scholar
Beame, P., Impagliazzo, R., Krajíček, J., Pitassi, T. and Pudlák, P., Lower bounds on Hilbert’s Nullstellensatz and propositional proofs, Proc. London Mathematical Society, 73 (3) (1996), 1–26. pp. 131, 230, 326, 335, 338, 349, 350Google Scholar
Beame, P., Impagliazzo, R., Krajíček, J., Pitassi, T., Pudlák, P. and Woods, A., Exponential lower bounds for the pigeonhole principle, in: Proc. Annual ACM Symp. on Theory of Computing (STOC) (1992), 200–220. p. 334Google Scholar
Beame, P., Impagliazzo, R., Pitassi, T. and Segerlind, N., Formula Caching in DPLL, ACM Trans. on Computation Theory, 1 (3) (2010), 9:1–933. p. 295Google Scholar
Beame, P., Kautz, H. and Sabharwal, A., On the power of clause learning, in: Proc. 18th International Joint Conf. on Artificial Intelligence (IJCAI) (2003), 94–99. p. 295Google Scholar
Beame, P., and Pitassi, T., Exponential separation between the matching principles and the pigeonhole principle, in: Proc. 8th Annual IEEE Symposium on Logic in Computer Science (LICS) (1993), 308–319. p. 335Google Scholar
Beame, P. and Pitassi, T., Simplified and improved resolution lower bounds, in: Proc. 37th IEEE Symp. on Foundations of Computer Science(FOCS) (1996), 274–282. pp. 112, 470Google Scholar
Beame, P. and Pitassi, T., Propositional proof complexity: past, present, and future, in: Current Trends in Theoretical Computer Science: Entering the 21st Century, eds. Paun, G., Rozenberg, G. and Salomaa, A., World Scientific (2001), 42–70. pp. 6, 382Google Scholar
Beame, P., Pitassi, T. and Segerlind, N., Lower bounds for Lovász–Schrijver systems and beyond follow from multiparty communication complexity, in: Automata, Languages, and Programming: Proc. 32nd International Colloq. (2005), 1176–1188. pp. 395, 396, 407Google Scholar
Beckmann, A. and Buss, S. R., Improved witnessing and local improvement principles for second-order bounded arithmetic, ACM Trans. Computational Logic, 15 (1) (2014), article 2. p. 436Google Scholar
Beckmann, A. and Buss, S. R., The NP search problems of Frege and Extended Frege proofs, ACM Trans. on Computational Logic, 18 (2) (2017), Article 11. p. 436Google Scholar
Beckmann, A., Pudlák, P. and Thapen, N., Parity games and propositional proofs, ACM Trans. Computational Logic, 15 (2) (2014), article 17. pp. 210, 470Google Scholar
Bellantoni, S., Pitassi, T. and Urquhart, A., Approximation and small depth Frege proofs, SIAM J. Computing, 21 (6) (1992), 1161–1179. p. 334Google Scholar
Ben-David, S. and Gringauze, A., On the existence of propositional proof systems and oracle-relativized propositional logic, in: Proc. Electronic Colloq. on Computational Complexity (ECCC), TR98–021 (1998). pp. 91, 468Google Scholar
Bennett, J. H., On spectra, Ph.D. thesis, Princeton University (1962). pp. 2, 182, 183, 196, 257Google Scholar
Ben-Sasson, E., Expansion in proof complexity, Ph.D. thesis, Hebrew University, Jerusalem (2001). pp. 292, 471Google Scholar
Ben-Sasson, E., Hard examples for the bounded depth Frege proof system, Computational Complexity, 11 (3-4) (2002), 109–136. pp. 335, 437CrossRefGoogle Scholar
Ben-Sasson, E. and Harsha, P., Lower bounds for bounded depth Frege proofs via Buss–Pudlák games, ACM Trans. on Computational Logic, 11 (3) (2010), 1–17. p. 335Google Scholar
Ben-Sasson, E. and Impagliazzo, R., Random CNF’s are hard for the polynomial calculus, Computational Complexity, 19 (4) (2010), 501–519. pp. 131, 350, 434Google Scholar
Ben-Sasson, E., Impagliazzo, R. and Wigderson, A., Near-optimal separation of general and tree-like resolution, Combinatorica, 24 (4) (2004), 585–604. p. 291Google Scholar
Ben-Sasson, E. and Nordström, J., Short proofs may be spacious: an optimal separation of space and length in resolution, in: Proc. 49th Annual IEEE Symp. on Foundations of Computer Science (FOCS) (2008), 709–718. p. 294Google Scholar
Ben-Sasson, E. and Nordström, J., Understanding space in proof complexity: separations and trade-offs via substitutions, in: Proc. 2nd Symp. on Innovations in Computer Science (ICS ‘11) (2011), 401–416. p. 290Google Scholar
Ben-Sasson, E. and Wigderson, A., Short proofs are narrow – resolution made simple, in: Proc. 31st ACM Symp. on Theory of Computation(STOC) (1999), 517–526. pp. 100, 101, 112, 277, 281, 282, 292, 293Google Scholar
Beth, E. W., On Padoa’s method in the theory of definition, Indag. Math., 15 (1953), 330–339. p. 80Google Scholar
Beth, E. W., The Foundations of Mathematics. North-Holland (1959). p. 80Google Scholar
Beyersdorff, O., Bonacina, I. and Chew, L., Lower bounds: from circuits to QBF proof systems, in: Proc. Conf. on Innovations in Theoretical Computer Science (ITCS) (2016), 249–260. p. 92Google Scholar
Beyersdorff, O., Galesi, N. and Lauria, M., A characterization of tree-like resolution size, Information Processing Letters, 113 (18) (2013), 666–671. p. 291Google Scholar
Beyersdorff, O., Köbler, J. and Messner, J., Nondeterministic functions and the existence of optimal proof systems, Theoretical Computer Science, 410 (38–40) (2009), 3839–3855. p. 468Google Scholar
Beyersdorff, O., Köbler, J. and Müller, S., Proof systems that take advice, Information and Computation, 209 (3) (2011), 320–332. p. 471Google Scholar
Beyersdorff, O. and Kullmann, O., Unified characterisations of resolution hardness measures, in: Proc. 17th International Conf. on Theory and Applications of Satisfiability Testing (SAT), Lecture Notes in Computer Science, 8561 (2014), 170–187. pp. 112, 114, 293Google Scholar
Beyersdorff, O. and Pich, J., Understanding Gentzen and Frege systems for QBF, in: Proc. Conf. on Logic in Computer Science (LICS) (2016), 146–155. p. 92Google Scholar
Beyersdorff, O. and Sadowski, Z., Characterizing the existence of optimal proof systems and complete sets for promise classes, in: Proc. Conf on Computer Science – Theory and Applications (CSR), Lecture Notes in Computer Science, 5675 (2009), 47–58. p. 469Google Scholar
Beyersdorff, O. and Sadowski, Z., Do there exist complete sets for promise classes? Mathematical Logic Quarterly, 57 (6) (2011), 535–550. p. 469Google Scholar
Birget, J.-C., Reductions and functors from problems to word problems, Theoretical Computer Science, 237 (2000), 81–104. p. 160Google Scholar
Birget, J.-C., Ol’shanskii, A. Yu., Rips, E. and Sapir, M. V., Isoperimetric functions of groups and computational complexity of the word problem, Ann. Mathematics, 156 (2) (2002), 467–518. p. 153Google Scholar
Blake, A., Canonical expressions in boolean algebra, Ph.D. thesis, University of Chicago (1937). p. 112Google Scholar
Bochnak, J., Coste, M. and Roy, M.-F., Real Algebraic Geometry, Springer (1999). p. 132Google Scholar
Bonacina, I., Total space in resolution is at least width squared, in: Proc. 43rd International Colloq. on Automata, Languages, and Programming(ICALP) 55 (2016), 56:1–56:13. pp. 113, 294Google Scholar
Bonacina, I., Galesi, N. and Thapen, N., Total space in resolution, SIAM J. on Computing, 45 (5) (2016), 1894–1909. p. 294Google Scholar
Bonacina, I. and Talebanfard, N., Strong ETH and resolution via games and the multiplicity of strategies, Algorithmica (2016), 1–13. p. 295Google Scholar
Bonet, M. L. and Buss, S. R., On the deduction rule and the number of proof lines, in: Proc. 6th Annual IEEE Symp. on Logic in Computer Science(LICS 91), IEEE Computer Society Press (1991), 286–297. p. 62Google Scholar
Bonet, M. L. and Buss, S. R., The deduction rule and linear and near-linear proof simulations, J. Symbolic Logic, 58 (1993), 688–709. p. 62Google Scholar
Bonet, M. L., Domingo, C., Gavaldá, R., Maciel, A. and Pitassi, T., Non-automatizability of bounded-depth Frege proofs, Computational Complexity, 13 (2004), 47–68. p. 405Google Scholar
Bonet, M. L., Esteban, J. L., Galesi, N. and Johannsen, J., On the relative complexity of resolution refinements and cutting planes proof systems, SIAM J. Computing, 30 (5) (2000), 1462–1484. p. 406Google Scholar
Bonet, M. L., Pitassi, T. and Raz, R., Lower bounds for cutting planes proofs with small coefficients, in: Proc. 27th Annual ACM Symp. on the Theory of Computing (STOC) (1995), 575–584. pp. 157, 381, 382Google Scholar
Bonet, M. L., Pitassi, T. and Raz, R., Lower bounds for cutting planes proofs with small coefficients, J. Symbolic Logic, 62 (1997), 708–728. pp. 157, 381Google Scholar
Bonet, M. L., Pitassi, T., and Raz, R., On interpolation and automatization for Frege proof systems, SIAM J. Computing, 29 (6) (2000), 1939–1967. pp. 365, 382, 404, 410Google Scholar
Boone, W., The word problem, Proc. Nat. Acad. Sci. USA, 44 (1958), 265–269. p. 153Google Scholar
Boone, W., The word problem, Ann. Mathematics, 70 (1959), 207–265. p. 153Google Scholar
Boole, G., The Mathematical Analysis of Logic, Barclay and Macmillan (1847). p. 116Google Scholar
Boppana, R. and Sipser, M., The complexity of finite functions, in: Handbook of Theoretical Computer Science, Elsevier Science Publishers (1991), 759–804. p. 38Google Scholar
Bridson, M., The geometry of the word problem, in: Invitations to Geometry and Topology, Oxford University Press (2002). p. 160Google Scholar
Bryant, R. E., Graph-based algorithms for Boolean function manipulation, IEEE Trans. on Computing, C- 35 (1986), 677–691. p. 146Google Scholar
Bryant, R. E., Syntactic Boolean manipulation with ordered binary decision diagrams, ACM Computing Surveys, 2493 (1992), 293–318. p. 146Google Scholar
Buresh-Oppenheim, J., Galesi, N., Hoory, S., Magen, A. and Pitassi, T., Rank bounds and integrality gaps for cutting planes procedures, Theory of Computing, 2 (1) (2006), 65–90. p. 352Google Scholar
Buss, S. R., Bounded Arithmetic. Bibliopolis (1986). pp. 79, 164, 166, 183, 192, 193, 196, 236, 436, 478Google Scholar
Buss, S. R., The Boolean formula value problem is in ALOGTIME, in: Proc. 19th Annual ACM Symp. on Theory of Computing (STOC) (1987), 123–131. p. 209Google Scholar
Buss, S. R., Polynomial size proofs of the propositional pigeonhole principle, J. Symbolic Logic, 52 (1987), 916–927. pp. 53, 183, 213, 215, 230Google Scholar
Buss, S. R., Axiomatizations and conservation results for fragments of bounded arithmetic, in: Logic and Computation, Contemporary Mathematics, 106 (1990), 57–84. p. 193Google Scholar
Buss, S. R., Relating the bounded arithmetic and polynomial-time hierarchies, Ann. Pure and Applied Logic, 75 (1995), 67–77. p. 183Google Scholar
Buss, S. R., On Gödel’s theorems on lengths of proofs II: lower bounds for recognizing k symbol provability, in: Feasible Mathematics II, eds. Clote, P. and Remmel, J., Birkhauser (1995), 57–90. p. 2Google Scholar
Buss, S. R., Some remarks on the lengths of propositional proofs, Archive for Mathematical Logic, 34 (1995) 377–394. p. 62Google Scholar
Buss, S. R., Bounded arithmetic and propositional proof complexity, in: Logic of Computation, ed. Schwichtenberg, H., Springer (1997), 67–122. pp. 183, 196Google Scholar
Buss, S. R., Lower bounds on Nullstellensatz proofs via designs, in: Proof Complexity and Feasible Arithmetics, eds. Buss, S. and Beame, P., American Mathematical Society (1998), 59–71. p. 349Google Scholar
Buss, S. R., An Introduction to Proof Theory, in: Handbook of Proof Theory, ed. Buss, S. R., Elsevier (1998), 1–78. p. 6Google Scholar
Buss, S. R., First-order proof theory of arithmetic, in: Handbook of Proof Theory, ed. Buss, S. R., Elsevier (1998), 79–147. pp. 6, 79, 196Google Scholar
Buss, S. R., Propositional proof complexity: an introduction, in: Computational Logic, eds. Berger, U. and Schwichtenberg, H., Springer (1999), 127–178. p. 6Google Scholar
Buss, S. R., Bounded arithmetic, proof complexity and two papers of Parikh, Ann. Pure and Applied Logic, 96 (1999), 43–55. p. 182Google Scholar
Buss, S. R., Bounded arithmetic and constant depth Frege proofs, in: Complexity of Computations and Proofs, ed. Krajíček, J., Quaderni di Matematica, 13 (2004), 153–174. pp. 183, 196Google Scholar
Buss, S. R., Towards NP-P via proof complexity and search, Ann. Pure and Applied Logic, 163 (7) (2012), 906–917. p. 6Google Scholar
Buss, S. R. and Clote, P., Cutting planes, connectivity, and threshold logic, Archive for Mathematical Logic, 3591 (1996), 33–62. pp. 124, 132, 209, 231Google Scholar
Buss, S. R., Grigoriev, D., Impagliazzo, R. and Pitassi, T., Linear gaps between degrees for the polynomial calculus modulo distinct primes, in: Proc. 31st Annual ACM Symp. on Theory of Computing (STOC) (1999), 547–556. pp. 131, 350, 351Google Scholar
Buss, S. R., Hoffmann, J. and Johannsen, J., Resolution trees with lemmas – resolution refinements that characterize DLL-algorithms with clause learning, Logical Methods in Computer Science, 4 (2008), 4:13. p. 295Google Scholar
Buss, S. R., Impagliazzo, R., Krajíček, J., Pudlák, P., Razborov, A. A. and Sgall, J., Proof complexity in algebraic systems and bounded depth Frege systems with modular counting, Computational Complexity, 6 (3) (1996/1997), 256–298. pp. 131, 159, 230, 326, 333, 335, 336, 338, 350Google Scholar
Buss, S. R. and Johannsen, J., On linear resolution, J. Satisfiability, Boolean Modeling and Computation, 10 (2016), 23–35. p. 291Google Scholar
Buss, S. R., Kabanets, V., Kolokolova, A. and Koucký, M., Expanders in VNC1, in: Proc. Conf. Innovations in Theoretical Computer Science (ITCS 2017), Leibniz International Proceedings in Informatics (LIPIcs), 67 (2017). pp. 146, 158, 479Google Scholar
Buss, S. R. and Kolodziejczyk, L. A., Small stone in pool, Logical Methods in Computer Science, 10 (2) (2014), paper 16. p. 295Google Scholar
Buss, S. R., Kolodziejczyk, L. and Thapen, N., Fragments of approximate counting, J. Symbolic Logic, 79 (2) (2014), 496–525. pp. 160, 209, 293Google Scholar
Buss, S. R., Kolodziejczyk, L. and Zdanowski, K., Collapsing modular counting in bounded arithmetic and constant depth propositional proofs, Trans. AMS, 367 (2015), 7517–7563. pp. 334, 479Google Scholar
Buss, S. R. and Krajíček, J., An application of boolean complexity to separation problems in bounded arithmetic, Proc. London Mathematical Society, 69 (3) (1994), 1–21. pp. 304, 305, 435Google Scholar
Buss, S. R., Krajíček, J., and Takeuti, G., On provably total functions in bounded arithmetic theories and in: Arithmetic Proof Theory and Computational Complexity, eds. Clote, P. and Krajíček, J. (1993), 116161, Oxford Press.Google Scholar
Buss, S. R. and Pudlák, P., How to lie without being (easily) convicted and the lengths of proofs in propositional calculus, in: Computer Science Logic 94, eds. Pacholski, and Tiuryn, , Springer Lecture Notes in Computer Science, 933 (1995), 151–162. pp. 62, 335Google Scholar
Buss, S. R. and Turan, G., Resolution proofs of generalized pigeonhole principles, Theoretical Computer Science, 62 (3) (1988), 311–317. p. 292Google Scholar
Cavagnetto, S., Propositional proof complexity and rewriting, Ph.D. thesis, Charles University in Prague (2008). p. 160Google Scholar
Chen, Y. and Flum, J., A logic for PTIME and a parameterized halting problem, in: Fields of Logic and Computation, eds. Blass, A., Dershowitz, N. and Reisig, W., Lecture Notes in Computer Science, 6300 (2010), 251–276. p. 471Google Scholar
Chen, Y. and Flum, J., On p-optimal proof systems and logics for PTIME, in: Proc. ICALP: Automata, Languages and Programming, eds. Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F. and Spirakis, P. G., Lecture Notes in Computer Science, 6199 (2010), 321–332. p. 471Google Scholar
Chang, C. L. and Lee, R. C.-T., Symbolic Logical and Mechanical Theorem Proving, Academic Press (1973). p. 62Google Scholar
Chiari, M. and Krajíček, J., Witnessing functions in bounded arithmetic and search problems, J. Symbolic Logic, 63 (3) (1998), 1095–1115. pp. 293, 305, 350, 436, 437Google Scholar
Chiari, M. and Krajíček, J., Lifting independence results in bounded arithmetic, Archive for Mathematical Logic, 38 (2) (1999), 123–138. pp. 303, 305, 436Google Scholar
Chvátal, V., Edmonds polytopes and a hierarchy of combinatorial problems, Discrete Mathematics, 4 (1973), 305–337. pp. 115, 121, 130Google Scholar
Chvátal, V. and E. Szemerédi, , Many hard examples for resolution, J. ACM, 35 (4) (1988), 759–768. pp. 282, 434Google Scholar
Clegg, M., Edmonds, J. and Impagliazzo, R., Using the Groebner basis algorithm to find proofs of unsatisfiability, in: Proc. 28th Annual ACM Symp. on Theory of Computing (STOC) (1996), 174–183. pp. 112, 131, 350, 406Google Scholar
Church, A., A note on the Entscheidungsproblem, J. Symbolic Logic, 1 (1936), 40–41. pp. 1, 32Google Scholar
Clote, P. and Krajíček, J., eds., Arithmetic Proof Theory and Computational Complexity, Oxford University Press (1993). p. 2Google Scholar
Clote, P. and Kranakis, E., Boolean Functions and Models of Computation, Springer (2002). pp. 6, 79, 112, 132, 157, 209, 231, 350, 406Google Scholar
Cobham, A., The intrinsic computational difficulty of functions, in: Proc. Conf. on Logic, Methodology and Philosophy of Science, ed. Y. Bar-Hillel (1965), 24–30. pp. 233, 234, 235, 257Google Scholar
Cook, S. A., The complexity of theorem proving procedures, in: Proc. 3rd Annual ACM Symp. on Theory of Computing (STOC) (1971), 151–158. pp. 26, 37Google Scholar
Cook, S. A., A hierarchy for nondeterministic time complexity, J. Computational Systems Science, 7 (4) (1973), 343–353. p. 25Google Scholar
Cook, S. A., Feasibly constructive proofs and the propositional calculus, in: Proc. 7th Annual ACM Symp. on Theory of Computing (STOC) (1975), 83–97. pp. 3, 163, 165, 233, 235, 244, 258, 435, 468Google Scholar
Cook, S. A., Relativized propositional calculus, preprint at https://arxiv.org/abs/1203. 2168 (2012). p. 91Google Scholar
Cook, S. A. and Fontes, L., Formal theories for linear algebra, Logical Methods in Computer Science, 8 (2012), 1–31. p. 232Google Scholar
Cook, S. A. and Haken, A., An exponential lower bound for the size of monotone real circuits, J. Computer and System Science, 58 (2) (1999), 326–335. p. 387Google Scholar
Cook, S. A. and Krajíček, J., Consequences of the provability of NP ⊇ P/poly, J. Symbolic Logic, 72 (4) (2007), 1353–1371. pp. 157, 160, 471, 479Google Scholar
Cook, S. A. and Morioka, T., Quantified propositional calculus and a second-order theory for NC1, Archive for Mathematical Logic, 44 (6) (2005), 711–749. p. 91Google Scholar
Cook, S. A., and Nguyen, P., Logical Foundations of Proof Complexity, Cambridge University Press (2009). pp. 6, 79, 87, 183, 196, 209, 210, 230, 259, 404, 436, 478Google Scholar
Cook, S. A. and Reckhow, R. A., The relative efficiency of propositional proof systems, J. Symbolic Logic, 44 (1) (1979), 36–50. pp. 33, 34, 38, 39, 62, 79, 114, 183, 231Google Scholar
Cook, S. A. and Soltys, M., Boolean programs and quantified propositional proof systems, Bull. Logic Section, University of Lodz, 28 (3) (1999), 119–129. pp. 85, 86, 90, 92Google Scholar
Cook, S. A. and Soltys, M., The proof complexity of linear algebra, Ann. Pure and Applied Logic, 130 (2004), 277–323. pp. 232, 434Google Scholar
Cook, S. A. and Thapen, N., The strength of replacement in weak arithmetic, ACM Trans. on Computational Logic, 7 (4) (2006). p. 259Google Scholar
Cook, S. A. and Tzameret, I., Uniform, integral and efficient proofs for the determinant identities, in: Logic in Computer Science (LICS), Proc. 32nd Annual ACM/IEEE Symp., 32 (2017), 1–12. p. 435Google Scholar
Cook, W., Coullard, C. R. and Turán, G., On the complexity of cutting plane proofs, Discrete Applied Mathematics, 18 (1987), 25–38. p. 132Google Scholar
Cox, D., Little, J. and O’Shea, D., Ideals, Varieties, and Algorithms, Springer (2007). p. 131Google Scholar
Craig, W., Three uses of the Herbrand–Gentzen theorem in relating model theory and proof theory, J. Symbolic Logic, 22 (3) (1957), 269–285. pp. 14, 37Google Scholar
Dantchev, S. and Riis, S., On relativisation and complexity gap for resolution-based proof systems, in: Proc. 12th Annual Conf. of the EACSL Computer Science Logic, Springer (2003). pp. 270, 291Google Scholar
Dash, S., Exponential lower bounds on the lengths of some classes of branch-and-cut proofs, Mathematics of Operations Research, 30 (3) (2005), 678–700. p. 407Google Scholar
Davis, M., Logemann, G. and Loveland, D., A machine program for theorem proving, Commun. ACM, 5 (7) (1962), 394–397. pp. 96, 112Google Scholar
Davis, M., Matiyasevich, Y. and Robinson, J., Hilbert’s tenth problem: diophantine equations: positive aspects of a negative solution, in: Mathematical Developments Arising from Hilbert Problems, Proc. Symp. in Pure Mathematics, ed. F. E. Browder (1976), 323–378. p. 38Google Scholar
Davis, M. and Putnam, H., A computing procedure for quantification theory, J. ACM, 7 (3) (1960), 210–215. pp. 96, 112Google Scholar
Diffie, W. and Hellman, M., New directions in cryptography, IEEE Trans. Information Theory, 22 (1976), 423–439. p. 404Google Scholar
Dimitracopoulos, C. and Paris, J., Truth definitions for Δ0 formulae, in: Logic and Algorithmic, Monographie No. 30 de L’Enseignement Mathematique, (1982), 318–329. p. 183Google Scholar
Dimitracopoulos, C. and Paris, J., The pigeonhole principle and fragments of arithmetic, Zeitschrift f. Mathematikal Logik u. Grundlagen d. Mathematik, 32 (1986), 73–80. p. 182Google Scholar
Dowd, M., Propositional representations of arithmetic proofs, Ph.D. thesis, University of Toronto (1979). pp. 62, 231, 258Google Scholar
Edmonds, J., Paths, trees, and flowers, Canad. J. Math., 17 (1965), 449–467. p. 37Google Scholar
Enderton, H. B., A Mathematical Introduction to Logic, Academic Press (2001). p. 37Google Scholar
Erdös, P., Some remarks on the theory of graphs, Bull. AMS, 53 (1947), 292–294. pp. 275, 320Google Scholar
Esteban, J. L. and Toran, J., Space bounds for resolution, in: Proc. 16th Symp. on Theoretical Aspects of Computer Science (STACS) (1999), 551–561. pp. 113, 157Google Scholar
Esteban, J. L. and Toran, J., A combinatorial characterization of treelike resolution space, Information Processes Letters, 87 (6) (2003), 295–300. p. 113Google Scholar
Fagin, R., Generalized first-order spectra and polynomial-time recognizable sets, in: Complexity of Computation, ed. R. Karp, SIAM–AMS Proc., 7 (1974), 27–41. p. 21Google Scholar
Feige, U., Kim, J. H. and Ofek, E., Witnesses for nonsatisfiability of dense random 3CNF formulas, in: Proc. 47th IEEE Annual Symp. on Foundations of Computer Science (FOCS) (2006), 497–508. pp. 160, 471Google Scholar
Fernandes, A. and Ferreira, F., Groundwork for weak analysis, J. Symbolic Logic, 67 (2002), 557–578. p. 232Google Scholar
Fernandes, A., Ferreira, F. and Ferreira, G., Analysis in weak systems, in: Logic and Computation: Essays in Honour of Amilcar Sernadas, College Publications (2017), 231–261. p. 232Google Scholar
Ferreira, F. and Ferreira, G., The Riemann integral in weak systems of analysis, J. Universal Computer Science, 14 (2008), 908–937. p. 232Google Scholar
Filmus, Y., Hrubeš, P. and Lauria, M., Semantic versus syntactic cutting planes, in: Proc. 33rd Symp. on Theoretical Aspects of Computer Science(STACS) (2016). p. 405Google Scholar
Filmus, Y., Lauria, M., Mikša, M., Nordström, J. and Vinyals, M., From small space to small width in resolution, ACM Trans. Computational Logic, 16(4) article 28. p. 113Google Scholar
Filmus, Y., Lauria, M., Nordström, J., Thapen, N. and Ron-Zevi, N., Space complexity in polynomial calculus, SIAM J. Computing, 44 (4) (2015), 1119–1153. p. 293Google Scholar
Filmus, Y., Pitassi, T. and Santhanam, R., Exponential lower bounds for AC0-Frege imply super-polynomial Frege lower bounds, in: Proc. 38th International Colloq. on Automata, Languages and Programming (ICALP), eds. Aceto, L., Henzinger, M. and Sgall, J., Lecture Notes in Computer Science, 6755 (2011), 618–629. pp. 61, 62Google Scholar
Fleming, N., Pankratov, D., Pitassi, T. and Robere, R., Random (log n)-CNFs are hard for cutting planes, in: Proc. 58th Annual Symp. on Foundations of Computer Science (FOCS) (2017), 109–120. p. 434Google Scholar
Frege, G., Begriffsschrift: Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle (1879). pp. 1, 37, 41, 52, 62Google Scholar
Friedman, H., The complexity of explicit definitions, Adv. Mathematics, 20 (1976), 18–29. p. 381Google Scholar
Friedman, H., On the consistency, completeness, and correctness problems, unpublished preprint (1979). p. 461Google Scholar
Friedgut, E., Sharp thresholds of graph properties and the k-SAT problem, J. American Matheamtical Society, 12 (4) (1999), 1017–1054. p. 282Google Scholar
Furst, M., Saxe, J. B. and Sipser, M., Parity, circuits and the polynomial-time hierarchy, Math. Systems Theory, 17 (1984), 13–27. pp. 29, 183Google Scholar
Galesi, N. and Thapen, N., Resolution and pebbling games, in: Proc. 8th International Conf. on Theory and Applications of Satisfiability Testing(SAT), Lecture Notes in Computer Science, 3569 (2005), 76–90. pp. 113, 293Google Scholar
Garey, M. R., Johnson, D. S. and Stockmeyer, L., Some simplifed NP-complete graphs problems, Theoretical Computer Science, 1 (1976), 237–267. p. 160Google Scholar
Garlík, M., A new proof of Ajtai’s completeness theorem for nonstandard finite structures, Archive for Mathematical Logic, 54 (3–4) (2015), 413–424. p. 454Google Scholar
Garlík, M., Construction of models of bounded arithmetic by restricted reduced powers, Archive for Mathematical Logic, 55 (5) (2016), 625648.Google Scholar
Garlík, M. and Kolodziejczyk, L., Some subsystems of constant-depth Frege with parity, preprint (2016). pp. 157, 336Google Scholar
Gentzen, G., Die Widerspruchsfreiheit der reinen Zahlentheorie, Mathematische Annalen, 112 (1936), 493–565. p. 64Google Scholar
Ghasemloo, K., Uniformity and nonuniformity in proof complexity, Ph.D. thesis, University of Toronto (2016). pp. 63, 184Google Scholar
Gödel, K., Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte fur Mathematik und Physik, 38(1931), 173–198. pp. 1, 469Google Scholar
Gödel, K., a letter to John von Neumann from 1956, reprinted in: Arithmetic, Proof Theory and Computational Complexity, eds. Clote, P. and Krajíček, J., Oxford University Press (1993). p. 2Google Scholar
Goerdt, A., Cutting plane versus Frege proof systems, in: Proc. Conf. Computer Science Logic (CSL 1990), eds. Börger, E., Kleine Büning, H., Richter, M. M. , and Schonfeld, W., Lecture Notes in Computer Science, 533, Springer (1991), 174–194. p. 231Google Scholar
Goerdt, A., Davis–Putnam resolution versus unrestricted resolution, Ann. Mathematics and Artificial Intelligence, 6 (1992), 169–184. p. 295Google Scholar
Goldreich, O. and Levin, L., Hard-core predicates for any one-way function, in: Proc. 21st ACM Symp. on Theory of Computing (STOC) (1989), 25–32. p. 363Google Scholar
Gomory, R. E., An algorithm for integer solutions of linear programs, in: Recent Advances in Mathematical Programming, eds. R. L. Graves and P. Wolfe (1963), 269–302. pp. 115, 121Google Scholar
Grigoriev, D., Tseitin’s tautologies and lower bounds for Nullstellensatz proofs, in: Proc. IEEE 39th Annual Symp. on Foundations of Computer Science (FOCS) (1998), 648–652. pp. 350, 351Google Scholar
Grigoriev, D., Linear lower bound on degrees of Positivstellensatz calculus proofs for the parity, Theoretial Computer Science, 259 (1-2) (2001), 613–622. pp. 132, 347Google Scholar
Grigoriev, D., Complexity of Positivstellensatz proof for the knapsack, Computational Complexity, 10 (2) (2001), 139–154. p. 351Google Scholar
Grigoriev, D., Hirsch, E. A. and Pasechnik, D. V., Complexity of semi-algebraic proofs, Moscow Mathematical J., 2 (4) (2002), 647–679. pp. 131, 132, 351, 352, 408Google Scholar
Grigoriev, D. and Vorobjov, N., Complexity of Null and Positivstellensatz proofs, Ann. Pure and Applied Logic, 113 (1) (2001), 153–160. pp. 132, 351Google Scholar
Grochow, J. and Pitassi, T., Circuit complexity, proof complexity and polynomial identity testing, in: Conf. on Proc. Foundations of Computer Science (FOCS) (2014). pp. 152, 159, 160Google Scholar
Gurevich, Y., Towards logic tailored for computational complexity, in: Proc. Logic Colloq. 1983, Lecture Notes in Mathematics, 1104 (1984), 175–216. p. 381Google Scholar
Gutfreund, D., Shaltiel, R. and Ta-Shma, A., If NP languages are hard on the worst-case, then it is easy to find their hard instances, Computational Complexity, 16 (4) (2007), 412–441. p. 479Google Scholar
Hájek, P. and Pudlák, P., Metamathematics of first-order arithmetic, in: Perspectives in Mathematical Logic, Springer (1993). pp. 159, 434, 478Google Scholar
Hajós, G., Über eine Konstruktion nicht n-färbbarer Graphen, Wiss. Zeitschr. Martin Luther Univ. Halle-Wittenberg, A 10 (1961). p. 154Google Scholar
Haken, A., The intractability of resolution, Theoretical Computer Science, 39 (1985), 297–308. pp. 3, 292, 294Google Scholar
Haken, A., Counting bottlenecks to show monotone P = NP, in: Proc. 36th IEEE Symp. on Foundations of Computer Science (FOCS) (1995), 36–40. pp. 359, 382Google Scholar
Hanika, J., Search problems and bounded arithmetic, Ph.D. thesis, Charles University, Prague (2004). p. 435Google Scholar
Hanika, J., Herbrandizing search problems in bounded arithmetic, Mathematical Logic Quarterly, 50 (6) (2004), 577–586. p. 435Google Scholar
Hartmanis, J., Lewis, P. M. and Stearns, R. E., Hierarchies of memory limited computations, in: Proc. IEEE Conf. on Switching Circuit Theory and Logic Design, Ann Arbor (1965), 179–190. pp. 26, 38Google Scholar
Hartmanis, J. and Stearns, R. E., On the computational complexity of algorithms, Trans. AMS, 117 (1965), 285–306. pp. 25, 38Google Scholar
Hastad, J., Almost optimal lower bounds for small depth circuits. in: Randomness and Computation, ed. S. Micali, (1989), 143–170. pp. 38, 299, 300, 304Google Scholar
Hastad, J., On small-depth Frege proofs for Tseitin for grids, in: Proc. the IEEE 58th Annual Symp. on Foundations of Computer Science (2017), 97–108. pp. 335, 437Google Scholar
Van Heijenoort, J., ed., From Frege to Gödel (A Source Book in Mathematical Logic, 1879–1931), Harvard University Press (1977). p. 1Google Scholar
Henzl, J., Weak formal systems, M.Sc. thesis, Charles University, Prague (2003). p. 159Google Scholar
Herbrand, J., Recherches sur la théorie de la démonstration, Travaux Soc. Sciences et Lettres de Varsovie, Class III, Sciences Mathematiques et Physiques, 33 (1930). p. 258Google Scholar
Higman, G., Subgroups of finitely presented groups, Proc. Royal Soc. London, A 262 (1961), 455–475. p. 153Google Scholar
Hilbert, D. and Ackermann, W., Principles of Mathematical Logic, Chelsea (1950). Translation of 1938 German edition. p. 39Google Scholar
Hilbert, D. and Bernays, P., Grundlagen der Mathematik. I, in: Die Grundlehren der mathematischen Wissenschaften, 40, Springer (1934). pp. 37, 39Google Scholar
Hilbert, D. and Bernays, P., Grundlagen der Mathematik. II, in: Die Grundlehren der mathematischen Wissenschaften, 50, Springer (1939). pp. 37, 39Google Scholar
Hirsch, E. A. and Kojevnikov, A., Several notes on the power of Gomory–Chvátal cuts, Ann. Pure and Applied Logic, 141 (3) (2006), 429–436. p. 406Google Scholar
Hodges, W., Building Models by Games, Dover (2006). pp. 289, 454Google Scholar
Hrubeš, P., Lower bounds for modal logics, J. Symbolic Logic, 72 (3) (2007), 941–958. p. 410Google Scholar
Hrubeš, P., A lower bound for intuitionistic logic, Ann. Pure and Applied Logic, 146 (2007), 72–90. p. 410Google Scholar
Hrubeš, P., On lengths of proofs in non-classical logics, Ann. Pure and Applied Logic, 157 (2-3) (2009), 194–205. p. 410Google Scholar
Hrubeš, P. and Pudlák, P., A note on monotone real ciruits, Information Processing Letters, 131 (2017), 15–19. p. 406Google Scholar
Hrubeš, P. and Pudlák, P., Random formulas, monotone circuits, and interpolation, in: Proc. 58th IEEE Symp. on Foundations of Computer Science(FOCS) (2017), 121–131. pp. 380, 382, 434Google Scholar
Hrubeš, P. and Tzameret, I., Short proofs for the determinant identities, SIAM J. Computing, 44 (2) (2015), 340–383. p. 435Google Scholar
Huang, L. and Pitassi, T., Automatizability and simple stochastic games, In: Proc. 38th International Colloq. on Automata, Languages and Programming (ICALP), Aceto, L., Henzinger, M. and Sgall Zurich, J., Lecture Notes in Computer Science, 6755 (2011), 605–617. p. 470Google Scholar
Immerman, N., Nondeterministic space is closed under complementation, SIAM J. Computing, 17 (5) (1988), 935–938. p. 26Google Scholar
Impagliazzo, R., Kabanets, V., and Wigderson, A., In search of an easy witness: exponential time vs. probabilistic polynomial time, J.Computer Systems Science, 65 (4) (2002), 672–694. p. 433Google Scholar
Impagliazzo, R. and Krajíček, J., A note on conservativity relations among bounded arithmetic theories, Mathematical Logic Quarterly, 48 (3)(2002), 375–377. pp. 231, 304, 305Google Scholar
Impagliazzo, R. and Naor, M., Efficient cryptographic schemes provably as secure as subset sum, J. Cryptology, 9 (4) (1996), 199216.CrossRefGoogle Scholar
Impagliazzo, R., Pitassi, T. and Urquhart, A., Upper and lower bounds for tree-like cutting planes proofs, in: Proc. Conf. on Logic in Computer Science (LICS) (1994), 220–228. pp. 382, 389, 406Google Scholar
Impagliazzo, R. and Pudlák, P., A lower bound for DLL algorithms for SAT, in: Proc. 11th Symp. on Discrete Algorithms (2000), 128–136. pp. 113, 293Google Scholar
Impagliazzo, R., Pudlák, P., and Sgall, J., Lower bounds for the polynomial calculus and the Groebner basis algorithm, Computational Complexity, 8 (2) (1999), 127–144. pp. 346, 350, 352Google Scholar
Impagliazzo, R. and Segerlind, N., Counting axioms do not polynomially simulate counting gates, in: Proc. IEEE 42nd Annual Symp. on Foundation of Computer Science (FOCS) (2001), 200–209. pp. 335, 336Google Scholar
Impagliazzo, R. and Segerlind, N., Constant-depth Frege systems with counting axioms polynomially simulate Nullstellensatz refutations, ACM Trans. Computational Logic, 7 (2) (2006), 199–218. p. 231Google Scholar
Impagliazzo, R. and Wigderson, A., P = BPP unless E has sub-exponential circuits: derandomizing the XOR lemma, in: Proc. 29th Annual ACM Symp. on Theory of Computing (STOC) (1997), 220–229. pp. 432, 433Google Scholar
Itsykson, D., Knop, A., Romashchenko, A. and Sokolov, D., On OBDD based algorithms and proof systems that dynamically change order of variables, in: Leibniz International Proc. in Informatics, 66 (2017), 43:1–43:14. pp. 159, 408Google Scholar
Itsykson, D. and Kojevnikov, A., Lower bounds of static Lovász–Schrijver calculus proofs for Tseitin tautologies, J. Math. Sci., 145 (2007), 4942–4952. p. 351Google Scholar
Itsykson, D. and Sokolov, D., Lower bounds for splittings by linear combinations, in: Proc. Conf. on Mathematical Foundations of Computer Science (2014), 372–383. pp. 157, 402, 409Google Scholar
James, G. D., The module orthogonal to the specht module, J. Algebra, 46 (2) (1977), 451–456. p. 349Google Scholar
James, G. D., The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics, 682 (1978). p. 349Google Scholar
Jeřábek, E., Dual weak pigeonhole principle, Boolean complexity, and derandomization, Ann. Pure and Applied Logic, 129 (2004), 1–37. p. 478Google Scholar
Jeřábek, E., Weak pigeonhole principle, and randomized computation, Ph.D. thesis, Charles University, Prague (2005). pp. 157, 159, 258, 336, 478Google Scholar
Jeřábek, E., Approximate counting in bounded arithmetic, J. Symbolic Logic, 72 (3) (2007), 959–993. pp. 250, 258, 478Google Scholar
Jeřábek, E., On independence of variants of the weak pigeonhole principle, J. Logic and Computation, 17 (3) (2007), 587–604. p. 258Google Scholar
Jeřábek, E., Approximate counting by hashing in bounded arithmetic, J. Symbolic Logic, 7493 (2009), 829–860. pp. 250, 258, 478Google Scholar
Jeřábek, E., A sorting network in bounded arithmetic, Annals of Pure and Applied Logic, 162 (4) (2011), 341–355. pp. 158, 478Google Scholar
Jeřábek, E. and Nguyen, P., Simulating non-prenex cuts in quantified propositional calculus, Mathematical Logic Quarterly, 57 (5) (2011), 524–532. p. 91Google Scholar
Jones, J. P. and Matiyasevich, Y., Basis for the polynomial time computable functions, in: Proc. Conf. on Number Theory, Banff, Alberta (1988), 255–270. p. 257Google Scholar
Jukna, S., Boolean Function Complexity, Springer (2012). p. 38Google Scholar
Karchmer, M. and Wigderson, A., Monotone circuits for connectivity require super-logarithmic depth, in: Proc. 20th Annual ACM Symp. on Theory of Computing (STOC) (1988), 539–550. p. 367Google Scholar
Karchmer, M. and Wigderson, A., On span programs, in: Proc. Conf. on 8th Structure in Complexity Theory, IEEE (1993), 102–111. p. 407Google Scholar
Karp, R. M., Reducibility among combinatorial problems, in: Complexity of Computer Computations, eds. Miller, R. E. and Thatcher, J. W., Plenum (1972), 85–103. pp. 37, 121Google Scholar
Karp, R. M. and Lipton, R. J., Some connections between nonuniform and uniform complexity classes, in: Proc. 12th Annual ACM Symp. on Theory of Computing (STOC) (1980), 302–309. p. 38Google Scholar
Kleene, S., Introduction to Metamathematics, North Holland (1952). p. 62Google Scholar
Köbler, J., Messner, J., Complete problems for promise classes by optimal proof systems for test sets, in: Proc. 13th Annual IEEE Conf. on Computational Complexity (CCC 98) (1998), 132–140. p. 468Google Scholar
Köbler, J., Messner, J. and Torán, J., Optimal proof systems imply complete sets for promise classes, Infinite Computation, 184 (1) (2003), 71–92. p. 469Google Scholar
Kolodziejczyk, L., Nguyen, P. and Thapen, N., The provably total NP search problems of weak second-order bounded arithmetic, Ann. Pure and Applied Logic, 162 (2011), 419–446. p. 436Google Scholar
Krajíček, J., On the number of steps in proofs, Ann. Pure and Applied Logic, 41 (1989), 153–178. pp. 60, 62Google Scholar
Krajíček, J., Speed-up for propositional Frege systems via generalizations of proofs, Comments. Mathematicae Universitas Carolinae, 30 (1)(1989), 137–140. p. 62Google Scholar
Krajíček, J., Exponentiation and second-order bounded arithmetic, Ann. Pure and Applied Logic, 48 (3) (1990), 261–276. pp. 196, 258Google Scholar
Krajíček, J., Fragments of bounded arithmetic and bounded query classes, Trans. AMS, 338 (2) (1993), 587–598. pp. 304, 305Google Scholar
Krajíček, J., Lower bounds to the size of constant-depth propositional proofs, J. Symbolic Logic, 59 (1) (1994), 73–86. pp. 3, 43, 80, 209, 210, 292, 300, 301, 304, 334, 381Google Scholar
Krajíček, J., On Frege and extended Frege proof Systems. in: Feasible Mathematics II. eds. Clote, P. and Remmel, J., Birkhauser (1995), 284–319. pp. 209, 335, 454Google Scholar
Krajíček, J., Bounded Arithmetic, Propositional Logic, and Complexity Theory, Encyclopedia of Mathematics and Its Applications, 60, Cambridge University Press (1995). pp. 6, 60, 61, 62, 80, 87, 112, 114, 170, 179, 183, 196, 203, 205, 209, 210, 230, 231, 236, 257, 258, 291, 292, 301, 304, 335, 336, 441, 454, 478Google Scholar
Krajíček, J., A fundamental problem of mathematical logic, Ann. Kurt Gödel Society, Collegium Logicum, 2 (1996), 56–64. p. 6Google Scholar
Krajíček, J., Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic, J. Symbolic Logic, 62 (2)(1997), 457–486. pp. 80, 107, 113, 157, 293, 371, 381, 382, 409Google Scholar
Krajíček, J., On methods for proving lower bounds in propositional logic, in: Logic and Scientific Methods, eds. Dalla Chiara, M. L. et al., Kluwer Academic (1997), 69–83. p. 6Google Scholar
Krajíček, J., Lower bounds for a proof system with an exponential speed-up over constant-depth Frege systems and over polynomial calculus, in: Proc. 22nd International Symp. on Mathematical Foundations of Computer Science, eds. Prívara, I. and Růžička, P., Lecture Notes in Computer Science, 1295, Springer (1997), 85–90. pp. 157, 335, 336Google Scholar
Krajíček, J., Interpolation by a game, Mathematical Logic Quarterly, 44 (4) (1998), 450–458. pp. 405, 406Google Scholar
Krajíček, J., Discretely ordered modules as a first-order extension of the cutting planes proof system, J. Symbolic Logic, 63 (4) (1998), 1582–1596. pp. 157, 158, 406, 408, 478Google Scholar
Krajíček, J., Extensions of models of PV, in: Proc: Logic Colloq.’95, eds. Makowsky, J. A. and Ravve, E. V., ASL/Springer Series Lecture Notes in Logic, 11 (1998), 104–114. p. 454Google Scholar
Krajíček, J., On the degree of ideal membership proofs from uniform families of polynomials over a finite field, Illinois J. Mathematics, 45 (1)(2001), 41–73. p. 349Google Scholar
Krajíček, J., Uniform families of polynomial equations over a finite field and structures admitting an Euler characteristic of definable sets, Proc. London Mathematical Society, 3 (81) (2000), 257–284. pp. 349, 455Google Scholar
Krajíček, J., On the weak pigeonhole principle, Fundamenta Mathematicae, 170 (1–3) (2001), 123–140. pp. 108, 113, 209, 291, 335, 437Google Scholar
Krajíček, J., Tautologies from pseudo-random generators, Bull. Symbolic Logic, 7 (2) (2001), 197–212. pp. 437, 439, 480Google Scholar
Krajíček, J., Dehn function and length of proofs, International J. Algebra and Computation, 13 (5) (2003), 527–542. p. 160Google Scholar
Krajíček, J., Dual weak pigeonhole principle, pseudo-surjective functions, and provability of circuit lower bounds, J. Symbolic Logic, 69 (1)(2004), 265–286. pp. 157, 258, 292, 293, 433, 437, 439, 440Google Scholar
Krajíček, J., Diagonalization in proof complexity, Fundamenta Mathematicae, 182 (2004), 181–192. pp. 158, 439, 464, 470Google Scholar
Krajíček, J., Implicit proofs, J. Symbolic Logic, 69 (2) (2004), 387–397. pp. 92, 158, 258, 474Google Scholar
Krajíček, J., Combinatorics of first-order structures and propositional proof systems, Archive for Mathematical Logic, 43 (4) (2004), 427–441. pp. 159, 291, 455Google Scholar
Krajíček, J., Hardness assumptions in the foundations of theoretical computer science, Archive for Mathematical Logic, 44 (6) (2005), 667–675. p. 480Google Scholar
Krajíček, J., Structured pigeonhole principle, search problems and hard tautologies, J. Symbolic Logic, 70 (2) (2005), 619–630. pp. 291, 335, 436, 439Google Scholar
Krajíček, J., Proof complexity, in: Proc. European Congress of Mathematics (ECM), ed. Laptev, A., European Mathematical Society (2005), 221–231. p. 6Google Scholar
Krajíček, J., Substitutions into propositional tautologies, Information Processing Letters, 101 (4) (2007), 163–167. pp. 439, 441Google Scholar
Krajíček, J., An exponential lower bound for a constraint propagation proof system based on ordered binary decision diagrams, J. Symbolic Logic, 73 (1) (2008), 227–237. pp. 159, 408Google Scholar
Krajíček, J., A proof complexity generator, in: Proc. 13th International Congress of Logic, Methodology and Philosophy of Science (Beijing, 2007), eds. C. Glymour, W. Wang, and D. Westerstahl (2009), 185–190. pp. 430, 437, 439Google Scholar
Krajíček, J., A form of feasible interpolation for constant depth Frege systems, J. Symbolic Logic, 75 (2) (2010), 774–784. p. 410Google Scholar
Krajíček, J., On the proof complexity of the Nisan–Wigderson generator based on a hard NP ∩ coNP function, J. Mathematical Logic, 11 (1)(2011), 11–27. pp. 383, 439Google Scholar
Krajíček, J., A note on propositional proof complexity of some Ramsey-type statements, Archive for Mathematical Logic, 50 (1–2) (2011), 245–255. pp. 291, 292, 335Google Scholar
Krajíček, J., Forcing with Random Variables and proof Complexity, London Mathematical Society Lecture Note Series, 382, Cambridge University Press (2011). pp. 6, 161, 210, 336, 437, 439, 452, 453, 454, 480Google Scholar
Krajíček, J., A note on SAT algorithms and proof complexity, Information Processing Letters, 112 (2012), 490–493. p. 478Google Scholar
Krajíček, J., Pseudo-finite hard instances for a student–teacher game with a Nisan–Wigderson generator, Logical Methods in Computer Science, 8 (3) (2012), 1–8. pp. 439, 454Google Scholar
Krajíček, J., A saturation property of structures obtained by forcing with a compact family of random variables, Archive for Mathematical Logic, 52 (1) (2013), 1928.Google Scholar
Krajíček, J., On the computational complexity of finding hard tautologies, Bulletin of the London Mathematical Society, 46 (1) (2014), 111–125. pp. 468, 479Google Scholar
Krajíček, J., A reduction of proof complexity to computational complexity for AC0p Frege systems, Proceedings of the AMS, 143 (11) (2015), 4951–4965. p. 336Google Scholar
Krajíček, J., Consistency of circuit evaluation, extended resolution and total NP search problems, Forum of Mathematics, Sigma, 4 (2016), e15. DOI: 10.1017/fms.2016.13. pp. 158, 436, 437Google Scholar
Krajíček, J., Expansions of pseudofinite structures and circuit and proof complexity, in: Liber Amicorum Alberti, eds. van Eijck, J., Iemhoff, R. and Joosten, J. J., Tributes Series 30, College Publications, London (2016), 195–203. p. 454Google Scholar
Krajíček, J., Randomized feasible interpolation and monotone circuits with a local oracle, preprint (2016). pp. 157, 402, 408, 409Google Scholar
Krajíček, J. and Oliveira, I. C., Unprovability of circuit upper bounds in Cook’s theory PV, Logical Methods in Computer Science, 13 (1) (2017). p. 479Google Scholar
Krajíček, J. and Oliveira, I. C., On monotone circuits with local oracles and clique lower bounds, Chicago J. Theoretical Computer Science, to appear. p. 409Google Scholar
Krajíček, J. and Pudlák, P., The number of proof lines and the size of proofs in first order logic, Archive for Mathematical Logic, 27 (1988), 69–84. p. 62Google Scholar
Krajíček, J. and Pudlák, P., On the structure of initial segments of models of arithmetic, Archive for Mathematical Logic, 28 (2) (1989), 91–98. pp. 163, 259Google Scholar
Krajíček, J. and Pudlák, P., Propositional proof systems, the consistency of first-order theories and the complexity of computations, J. Symbolic Logic, 54 (3) (1989), 1063–1079. pp. 34, 35, 38, 62, 435, 462, 468Google Scholar
Krajíček, J. and Pudlák, P., Quantified propositional calculi and fragments of bounded arithmetic, Zeitschr. f. Mathematikal Logik u. Grundlagen d. Mathematik, 36 (1) (1990), 29–46. pp. 248, 258Google Scholar
Krajíček, J. and Pudlák, P., Propositional provability in models of weak arithmetic, in: Proc. Conf. on Computer Science Logic (Kaiserlautern ‘89), eds. Boerger, E., Kleine-Bunning, H. and Richter, M. M., Lecture Notes in Computer Science, 440, Springer (1990), 193–210. p. 454Google Scholar
Krajíček, J. and Pudlák, P., Some consequences of cryptographical conjectures for and EF, in: Proc. Meeting on Logic and Computational Complexity (Indianapolis, 1994), ed. Leivant, D., Lecture Notes in Computer Science, 960, Springer (1995), 210–220. p. 382Google Scholar
Krajíček, J. and Pudlák, P., Some consequences of cryptographical conjectures for and EF, Information and Computation, 140 (1) (1998), 82–94. pp. 382, 404, 409Google Scholar
Krajíček, J., Pudlák, P., and Sgall, J., Interactive computations of optimal solutions, in: Mathematical Foundations of Computer Science, ed. Rovan, B., Lecture Notes in Computer Science, 452, Springer (1990), 48–60. p. 258Google Scholar
Krajíček, J., Pudlák, P. and Takeuti, G., Bounded arithmetic and the polynomial hierarchy, Ann. Pure and Applied Logic, 52 (1991), 143–153. pp. 235, 239, 257, 258, 479Google Scholar
Krajíček, J., Pudlák, P., and Woods, A., An exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle, Random Structures and Algorithms, 7 (1) (1995), 15–39. pp. 318, 334, 335Google Scholar
Krajíček, J., Skelley, A. and Thapen, N., NP search problems in low fragments of bounded arithmetic, J. Symbolic Logic, 72 (2) (2007), 649–672. pp. 292, 436, 437Google Scholar
Krajíček, J. and Takeuti, G., On bounded -polynomial induction, in: Feasible Mathematics, eds. Buss, S. R. and Scott, P. J., Birkhauser (1990), 259–280. pp. 248, 258Google Scholar
Krajíček, J. and Takeuti, G., On induction-free provability, Ann. Mathematics and Artificial Intelligence, 6 (1992), 107–126. p. 435Google Scholar
Kreisel, G., Technical report NO. 3, Applied Mathematics and Statistics Laboratories, Stanford University, unpublished (1961). p. 381Google Scholar
Krishnamurthy, B. and Moll, R. N., Examples of hard tautologies in the propositional calculus, in: Proc. 13th Annual ACM Symp. on Theory of Computing (STOC) (1981), 28–37. pp. 112, 291Google Scholar
Krivine, J. L., Anneaux préordonnés, J. d’Analyse Mathématique, 12 (1964), 307–326. pp. 115, 132Google Scholar
Kullmann, O., On a generalization of extended resolution, Discrete Applied Mathematics, 96 -97 (1999), 149–176. p. 295Google Scholar
Kullmann, O., Upper and lower bounds on the complexity of generalized resolution and generalized constraint satisfaction problems, Ann. Mathematics and Artificial Intelligence, 40 (3–4) (2004), 303–352. p. 295Google Scholar
Lagarias, J. C., An elementary problem equivalent to the Riemann hypothesis, American Mathematical Monthly, 109 (6) (2002), 534–543. p. 38Google Scholar
Lasserre, J. B., Global optimization with polynomials and the problem of moments, SIAM J. Optimization, 11 (3) (2001), 796–817. p. 131Google Scholar
Laurent, M., A comparison of the Sherali–Adams, Lovász–Schrijver and Lasserre relaxations for 0-1 programming, Mathematics of Operations Research, 28 (2001), 470–496. p. 132Google Scholar
Lauria, M., Short Res*(polylog) refutations if and only if narrow Res refutations, unpublished notes available at www.dsi.uniroma1.it/ ~ lauria, (2011). pp. 114, 209Google Scholar
Lauria, M. and Nordström, J., Tight size–degree bounds for sums-of-squares proofs, in: Proc. 30th Conf. on Computational Complexity (CCC’15), ed. D. Zuckerman (2015), 448–466. p. 352Google Scholar
Tri Man Le, Dai, Bounded Arithmetic and Formalizing Probabilistic Proofs, Ph.D. thesis, University of Toronto (2014). p. 479Google Scholar
Lee, T. and Shraibman, A., Disjointness is hard in the multi-party number-on-the-forehead model, Computational Complexity, 18 (2) (2009), 309–336. p. 407Google Scholar
Levin, L. A., Universal sequential search problems, Problems of Information Transmission (translated from Problemy Peredachi Informatsii (Russian)), 9 (3) (1973), 115–116. p. 38Google Scholar
Lovász, L., Naor, M., Newman, I. and Wigderson, A., Search problems in the decision tree model, in: Proc. 32nd IEEE Symp. on Foundations of Computer Science (FOCS), (1991), 576–585. p. 112Google Scholar
Lovász, L. and Schrijver, A., Cones of matrices and set-functions and 0-1 optimization, SIAM J. Optimization, 1 (1991), 166–190. pp. 115, 130, 131Google Scholar
Lukasiewicz, J., Elements of Mathematical Logic, Pergamon Press (1963). p. 41Google Scholar
Lupanov, O. B., The synthesis of contact circuits, Dokl. Akad. Nauk SSSR (NS), 119 (1958), 23–26. p. 38Google Scholar
Lyndon, R., An interpolation theorem in the predicate calculus, Pacific J. Mathematics, 9 (1) (1959), 129–142. pp. 14, 37Google Scholar
Lyndon, R. and Schupp, P., Combinatorial Group Theory, Springer (1997). p. 160Google Scholar
Maciel, A. and Pitassi, T., Towards lower bounds for bounded-depth Frege proofs with modular connectives, in: Proof Complexity and Feasible Arithmetics, eds. P. Beame and S. Buss, DIMACS Series, 39 (1998), 195–227. p. 335Google Scholar
Maciel, A., Pitassi, T. and Woods, A., A new proof of the weak pigeonhole principle, J. Computer Systems Sciences, 64 (2002), 843–872. pp. 220, 231Google Scholar
Maly, J., Jan Krajíček’s Forcing Construction and Pseudo Proof Systems, M.Sc. Thesis, University of Vienna (2016). p. 161Google Scholar
Maly, J. and Müller, M., Pseudo proof systems and hard instances of SAT, preprint 2017. p. 161Google Scholar
Maté, A., Nondeterministic polynomial-time computations and models of arithmetic, J. ACM, 37 (1) (1990), 175–193. p. 454Google Scholar
Messner, J., On optimal algorithms and optimal proof systems, in: Proc. Symp. on Theoretical Aspects of Computer Science (STACS 1999), eds. Meinel, C. and Tison, S., Lecture Notes in Computer Science, 1563 (1999), 541–550. p. 468Google Scholar
Messner, J. and Torán, J., Optimal proof systems for propositional logic and complete sets, in: Proc. Symp. on Theoretical Aspects of Computer Science (STACS 1998), eds. Morvan, M., Meinel, C., and Krob, D. , Lecture Notes in Computer Science, 1373 (1998), 477–487. p. 468Google Scholar
Mikle-Barát, O., Strong proof systems, M.Sc. thesis, Charles University, Prague (2007). pp. 159, 408Google Scholar
Mikša, M. and Nordström, J., A generalized method for proving polynomial calculus degree lower bounds, in: Proc. 30th Annual Computational Complexity Conference (CCC 2015), Leibniz International Proceedings in Informatics, 33 (2015), 467–487. pp. 350, 351Google Scholar
Morioka, T., Logical approaches to the complexity of search problems: proof complexity, quantified propositional calculus, and bounded arithmetic, Ph.D. thesis, University of Toronto (2005). p. 91Google Scholar
Muchnik, A. A., On two approaches to the classification of recursive functions (in Russian), in: Problems of Mathematical Logic, Complexity of Algorithms and Classes of Computable Functions, eds. V. A. Kozmidiadi and A. A. Muchnik (1970), 123–138. p. 257Google Scholar
Müller, M. and Pich, J., Feasibly constructive proofs of succinct weak circuit lower bounds, in: Proc. Electronic Colloq. on Computational Complexity (ECCC), TR17–144 (2017). pp. 441, 474, 478Google Scholar
Müller, S., Polylogarithmic cuts in models of V0, Logical Methods in Computer Science, 9 (1:16) (2013). p. 231Google Scholar
Müller, S. and Tzameret, I., Short propositional refutations for dense random 3CNF formulas, Ann. Pure and Applied Logic, 165 (12) (2014), 1864–1918. pp. 160, 471Google Scholar
Mundici, D., Complexity of Craig’s interpolation, Fundamenta Informaticae, 5 (1982), 261–278. p. 380Google Scholar
Mundici, D., A lower bound for the complexity of Craig’s interpolants in sentential logic, Archiv fur Math. Logik, 23 (1983), 27–36. p. 380Google Scholar
Mundici, D., Tautologies with a unique Craig interpolant, uniform vs. non-uniform complexity, Ann. Pure and Applied Logic, 27 (1984), 265–273. p. 380Google Scholar
Mundici, D., NP and Craig’s interpolation theorem, in: Proc. Logic Colloq. 1982, North-Holland (1984), 345–358. p. 380Google Scholar
Nepomnjascij, V., Rudimentary predicates and Turing calculations, Soviet Math. Dokl., 6 (1970), 1462–1465. p. 182Google Scholar
Nisan, N., The communication complexity of the threshold gates, in: Combinatorics, P. Erdös is Eighty, 1, eds. Miklós, et. al., Bolyai Mathematical Society (1993), 301–315. p. 388Google Scholar
Nisan, N. and Wigderson, A., Hardness vs. randomness, J. Computer System Sciences, 49 (1994), 149–167. pp. 432, 433, 438, 439Google Scholar
Nordström, J., Short proofs may be spacious: understanding space in resolution, Ph.D. thesis, The Royal Institute of Technology, Stockholm (2008). p. 294Google Scholar
Nordström, J., Narrow proofs may be spacious: separating space and width in resolution, SIAM J. Computing, 39 (1) (2009), 59–121. pp. 113, 294Google Scholar
Nordström, J., On the relative strength of pebbling and resolution, ACM Trans. Computational Logic, 13 (2), article 16 (2012). p. 294Google Scholar
Nordström, J., Pebble games, proof complexity and time-space trade-offs, Logical Methods in Computer Science, 9 (15) (2013), 1–63. p. 294Google Scholar
Nordström, J., On the interplay between proof complexity and SAT solving, ACM SIGLOG News, 2 (3) (2015), 19–44. p. 294Google Scholar
Nordström, J. and Hastad, J., Towards an optimal separation of space and length in resolution, Theory of Computing, 9, article 14 (2013), 471–557. p. 294Google Scholar
Novikov, P. S., On the algorithmic unsolvability of the word problem in group theory, Trudy Mat. Inst. Steklova, 44 (1955), 143. p. 153Google Scholar
O’Donnell, R., SOS is not obviously automatizable, even approximately, in: Proc. Conf. on Innovations in Theoretical Computer Science (ITCS) (2017), 59:1–59:10. p. 471Google Scholar
de Oliveira, M. and Pudlák, P., Representations of monotone Boolean functions by linear programs, in: Proc. 32nd Computational Complexity Conf. (CCC 2017), Leibniz International Proceedings in Informatics, 79 (2017), 3:1–3:14. pp. 395, 407Google Scholar
Papadimitriou, C., Computational Complexity, Addison Wesley (1994). p. 37Google Scholar
Papadimitriou, C., The complexity of the parity argument and other inefficient proofs of existence, J. Computer and System Sciences, 48 (3)(1994), 498–532. p. 436Google Scholar
Parikh, R., Existence and feasibility in arithmetic, J. Symbolic Logic, 36 (1971), 494–508. pp. 163, 165, 166, 182Google Scholar
Paris, J. B., O struktuře modelu omezené E1 indukce (in Czech), Časopis pěstování matematiky, 109 (1984), 372–379. p. 182Google Scholar
Paris, J. and Dimitracopoulos, C., Truth definitions for Δ0 formulas, in: Logic and Algorithmic, l’Enseignement Mathematique, 30 (1982), 318–329. p. 182Google Scholar
Paris, J. and Dimitracopoulos, C., A note on undefinability of cuts, J. Symbolic Logic, 48 (1983), 564–569. p. 182Google Scholar
Paris, J. B., Handley, W. G. and Wilkie, A. J., Characterizing some low arithmetic classes, Colloquia Mathematica Soc. J. Bolyai, 44 (1984), 353–364. p. 182Google Scholar
Paris, J. B. and Harrington, L., A mathematical incompleteness in Peano Arithmetic, in: Handbook of Mathematical Logic, ed. Barwise, J., North-Holland (1977). p. 414Google Scholar
Paris, J. B. and Kirby, L. Σn-collection schemes in arithmetic, in: Proc. Logic Colloq. ‘77, North-Holland (1978), 199–209. p. 182Google Scholar
Paris, J. B. and Wilkie, A. J., Models of arithmetic and rudimentary sets, Bull. Soc. Mathem. Belg., B 33 (1981), 157–169. pp. 182, 478Google Scholar
Paris, J. B. and Wilkie, A. J., Δ0 sets and induction, in: Proc. Jadwisin Logic Conf., (1983), 237–248. pp. 182, 183, 478Google Scholar
Paris, J. B. and Wilkie, A. J., Some results on bounded induction, in: Proc. 2nd Easter Conf. on Model Theory (1984), 223–228. pp. 182, 478Google Scholar
Paris, J. and Wilkie, A. J., Counting problems in bounded arithmetic, in: Methods in Mathematical Logic, Lecture Notes in Mathematics, 1130(1985), 317–340. pp. 3, 61, 163, 165, 167, 169, 170, 182, 183, 209, 210, 221, 231, 258, 307, 336, 454, 478Google Scholar
Paris, J. B. and Wilkie, A. J., Counting Δ0 sets, Fundamenta Mathematica, 127 (1987), 67–76. pp. 183, 259, 478Google Scholar
Paris, J. B. and Wilkie, A. J., On the scheme of induction for bounded arithmetic formulas, Ann. Pure and Applied Logic, 35 (1987), 261–302. pp. 166, 183, 435, 478, 480Google Scholar
Paris, J., Wilkie, A. J. and Woods, A., Provability of the pigeonhole principle and the existence of infinitely many primes, J. Symbolic Logic, 53 (4)(1988), 1235–1244. pp. 38, 183, 218, 220, 231, 259, 336Google Scholar
Perron, S., Examining fragments of the quantified propositional calculus J. Symbolic Logic, 73 (3) (2008), 1051–1080. p. 91Google Scholar
Perron, S., Power of non-uniformity in proof complexity, Ph.D. thesis, University of Toronto (2009). p. 91Google Scholar
Pich, J., Hard tautologies, M.Sc. thesis, Charles University, Prague (2011). p. 439Google Scholar
Pich, J., Nisan–Wigderson generators in proof systems with forms of interpolation, Mathematical Logic Quarterly, 57 (4) (2011), 379–383. p. 439Google Scholar
Pich, J., Complexity Theory in Feasible Mathematics, Ph.D. thesis, Charles University, Prague (2014). p. 478Google Scholar
Pich, J., Circuit lower bounds in bounded arithmetic, Ann. Pure and Applied Logic, 166 (1) (2015), 29–45. p. 479Google Scholar
Pich, J., Logical strength of complexity theory and a formalization of the PCP theorem in bounded arithmetic, Logical Methods in Computer Science, 11 (2) (2015). p. 478Google Scholar
Pitassi, T., Beame, P., and Impagliazzo, R., Exponential lower bounds for the pigeonhole principle, Computational Complexity, 3 (1993), 97–308. pp. 318, 334, 335Google Scholar
Pitassi, T. and Raz, R., Regular resolution lower bounds for the weak pigeonhole principle, Combinatorica, 24 (3) (2004), 503–524. p. 293Google Scholar
Pitassi, T. and Santhanam, R., Effective polynomial simulations, in: Proc. 1st Symp. on Innovations in CS (2010), 370–382. pp. 38, 469Google Scholar
Pitassi, T. and Segerlind, N., Exponential lower bounds and integrality gaps for tree-like Lovász–Schrijver procedures, SIAM J. Computing, 41 (1)(2012), 128–159. p. 349Google Scholar
Pitassi, T. and Urquhart, A., The complexity of the Hajos calculus, SIAM J. Discrete Mathematics, 8 (3) (1995), 464–483. pp. 155, 160Google Scholar
Pratt, V. R., Every prime has a succinct certificate, SIAM J. Computing, 4 (1975), 214–220. p. 409Google Scholar
Prawitz, D., Natural Deduction. A Proof-Theoretic Study, Stockholm (1965). p. 79Google Scholar
Pudlák, P., A definition of exponentiation by bounded arithematic formula, Comment. Mathematicae Universitas Carolinae, 24 (4) (1983), 667–671. p. 183Google Scholar
Pudlák, P., Cuts, consistency statements and interpretations, J. Symbolic Logic, 50 (1985), 423–441. pp. 259, 470Google Scholar
Pudlák, P., On the length of proofs of finitistic consistency statements in first-order theories, in: Proc. Logic Colloquium 84, North Holland (1986), 165–196. pp. 259, 461, 462Google Scholar
Pudlák, P., Improved bounds to the length of proofs of finitistic consistency statements, Contemporary Mathematics, 65 (1987), 309–331. pp. 259, 469, 470Google Scholar
Pudlák, P., Ramsey’s theorem in bounded arithmetic, in: Proc. Computer Science Logic ‘90, eds. Borger, E., Kleine Buning, H., Richter, M. M. and Schonfeld, W., Lecture Notes in Computer Science, 533 (1991), 308–317. p. 292Google Scholar
Pudlák, P., Lower bounds for resolution and cutting planes proofs and monotone computations, J. Symbolic Logic, 62 (3) (1997), 981–998. pp. 378, 382, 386, 387, 405Google Scholar
Pudlák, P., The lengths of proofs, in: Handbook of Proof Theory, ed. Buss, S. R., Elsevier (1998), 547–637. pp. 6, 382Google Scholar
Pudlák, P., On the complexity of propositional calculus, in: Sets and Proofs, Proc. Logic Colloq. ‘97, Cambridge University Press (1999), 197–218. pp. 132, 157, 158, 395, 407Google Scholar
Pudlák, P., Proofs as games, American Math. Monthly (June–July 2000), 541–550. p. 293Google Scholar
Pudlák, P., On reducibility and symmetry of disjoint NP-pairs, Theoretical Computer Science, 295 (2003), 323–339. pp. 383, 468, 471Google Scholar
Pudlák, P., Consistency and games – in search of new combinatorial principles, in: Proc. Logic Colloq. ‘03, eds. Stoltenberg-Hansen, V. and Vaananen, J., Association for Symbolic Logic (2006), 244–281. p. 436Google Scholar
Pudlák, P., Twelve problems in proof complexity, in: Proc. 3rd International Computer Science Symp. in Russia (CSR) (2008), 13–27. p. 6Google Scholar
Pudlák, P., Quantum deduction rules, Ann. Pure and Applied Logic, 157 (2009), 16–29. p. 160Google Scholar
Pudlák, P., A lower bound on the size of resolution proofs of the Ramsey theorem, Information Processing Letters, 112 (14–15) (2012), 610–611. p. 292Google Scholar
Pudlák, P., Logical Foundations of Mathematics and Computational Complexity, A Gentle Introduction, Springer (2013). pp. 6, 480Google Scholar
Pudlák, P., Incompleteness in the finite domain, Bulletin of Symbolic Logic, 23 (4) (2017), 405–441. p. 469Google Scholar
Pudlák, P. and Sgall, J., Algebraic models of computation, and interpolation for algebraic proof systems, in: Proof Complexity and Feasible Arithmetic, ed. S. Buss, DIMACS Series 39 (1998), 279–295. pp. 392, 407Google Scholar
Pudlák, P. and Thapen, N., Random resolution refutations, preprint, in: Proc. 32nd Computational Complexity Conf. (CCC 2017), Leibniz International Proceedings in Informatics, 79 (2017), 1:1–1:10. pp. 156, 160, 293Google Scholar
Raghavendra, P. and Weitz, B., On the bit-complexity of sum-of-squares proofs, in: Proc. 44th International Colloq. on Automata, Languages, and Programming (ICALP) (2017), 80:1–80:13. p. 471Google Scholar
Raz, R., Resolution lower bounds for the weak pigeonhole principle, J. Association for Computing Machinery, 51 (2) (2004), 115–138. p. 292Google Scholar
Raz, R. and Tzameret, I., Resolution over linear equations and multilinear proofs, Ann. Pure and Applied Logic, 155 (3) (2008), 194–224. p. 157Google Scholar
Raz, R. and Wigderson, A., Probabilistic communication complexity of Boolean relations, in: Proc. IEEE 30th Annual Symp. on Foundations of Computer Science (FOCS) (1989), 562–567. pp. 389, 408Google Scholar
Raz, R. and Wigderson, A., Monotone circuits for matching require linear depth, J. ACM, 39 (3) (1992), 736–744. pp. 360, 382, 389Google Scholar
Razborov, A. A., Lower bounds for the monotone complexity of some Boolean functions, Doklady Akademii Nauk SSSR, 281 (4) (1985), 798–801. English translation in Soviet Math. Doklady, 31 (1985), 354–357. pp. 30, 288, 382Google Scholar
Razborov, A. A., Lower bounds on the size of bounded depth networks over a complete basis with logical addition, Matem. Zametki, 41 (4) (1987), 598–607. pp. 30, 329Google Scholar
Razborov, A. A., An equivalence between second-order bounded domain bounded arithmetic and first order bounded arithmetic, in: Arithmetic, Proof Theory and Computational Complexity, eds. Clote, P. and Krajíček, J., Oxford University Press (1993), 247–277. p. 196Google Scholar
Razborov, A. A., On provably disjoint NP-pairs, Basic Research in Computer Science Center, Aarhus, RS-94-36 (1994), unpublished report. p. 382Google Scholar
Razborov, A. A., Bounded arithmetic and lower bounds in Boolean complexity, in: Feasible Mathematics II, eds. Clote, P. and Remmel, J., Birkhauser (1995), 344–386. p. 478Google Scholar
Razborov, A. A., Unprovability of lower bounds on the circuit size in certain fragments of bounded arithmetic, Izvestiya RAN., 59 (1) (1995), 201–224. pp. 368, 381, 382, 440, 478Google Scholar
Razborov, A. A., Lower bounds for the polynomial calculus, Computational Complexity, 7 (4) (1998), 291–324. pp. 343, 350Google Scholar
Razborov, A. A., Improved resolution lower bounds for the weak pigeonhole principle, in: Proc. Electronic Colloq. on Computational Complexity, TR01–055 (2001). p. 293Google Scholar
Razborov, A. A., Resolution lower bounds for the weak functional pigeonhole principle, Theoretical Computer Science, 303 (1) (2003), 233–243. p. 293Google Scholar
Razborov, A. A., Resolution lower bounds for perfect matching principles, J. Computer and System Sciences, 69 (1) (2004), 3–27. pp. 293, 438, 439Google Scholar
Razborov, A. A., Pseudorandom generators hard for k-DNF resolution polynomial calculus resolution, Ann. Mathematics, 181 (2) (2015), 415–472. pp. 438, 439, 440Google Scholar
Razborov, A. A., On space and depth in resolution, preprint (2016). p. 113Google Scholar
Razborov, A. A., Proof Complexity and Beyond, SIGACT News, 47 (2), (2016), 66–86. p. 6Google Scholar
Razborov, A. A., On the width of semi-algebraic proofs and algorithms, Mathematics of Operations Research, 42 (4) (2017), 1106–1134. p. 352Google Scholar
Razborov, A. A. and Rudich, S., Natural proofs, J. Computer System Sciences, 55 (1) (1997), 24–35. pp. 381, 433, 478Google Scholar
Reckhow, R. A., On the lengths of proofs in the propositional calculus, Ph.D. thesis, University of Toronto (1976). pp. 52, 62, 79Google Scholar
Riis, S., Independence in bounded arithmetic, D.Phil. thesis, Oxford University (1993). pp. 291, 454Google Scholar
Riis, S., Count(q) does not imply Count(p), Ann. Pure and Applied Logic, 90 (1997), 1–56. p. 230Google Scholar
Riis, S., A complexity gap for tree-resolution, Computational Complexity, 10 (3) (2001), 179–209. p. 291Google Scholar
Ritchie, R., Classes of predictably computable functions. Trans. AMS, 106 (1963), 139–173. p. 182Google Scholar
Rivest, M., Shamir, A. and Adleman, L., A method of obtaining digital signatures and public-key cryptosystems, ACM Communications, 21 (1978), 120–126. p. 364Google Scholar
Robere, R., Pitassi, T., Rossman, B. and Cook, S. A., Exponential lower bounds for monotone span programs, in: Proc. 57th IEEE Symp. on Foundations of Computer Science (FOCS) (2016), 406–415. pp. 393, 407Google Scholar
Robinson, A., On ordered fields and definite functions, Math. Ann., 130 (1955), 257–271. p. 132Google Scholar
Robinson, J. A., A machine-oriented logic based on the resolution principle, J. ACM (1965), 12 (1), 2341. p. 112Google Scholar
Rudich, S., Super-bits, demi-bits, and NP/qpoly-natural proofs, in: Proc. 1st Int. Symp. on Randomization and Approximation Techniques in Computer Science, Lecture Notes in Computer Science, 1269 (1997), 85–93. p. 437Google Scholar
Sadowski, Z., On an optimal deterministic algorithm for SAT, in: Computer Science Logic, Gottlob, G. Grandjean, E. and Seyr, K., Lecture Notes in Computer Science, 1584 (1998), 179–187. p. 468Google Scholar
Sadowski, Z., Optimal proof systems, optimal acceptors and recursive presentability, Fundam. Inform., 79 (1–2) (2007), 169–185. p. 469Google Scholar
Šanda, P., Implicit propositional proofs, M.Sc. thesis, Charles University, Prague (2006). p. 158Google Scholar
Savage, J. E., Computational work and time on finite machines, J. ACM, 19 (4) (1972), 660–674. p. 28Google Scholar
Savitch, W. J., Relationships between nondeterministic and deterministic tape complexities, J. Computer and System Sciences, 4 (1970), 177–192. p. 26Google Scholar
Scheiderer, C., Sums of squares of polynomials with rational coefficients, J. EMS, 18 (7) (2016), 1495–1513. p. 231Google Scholar
Schoenebeck, G., Linear level Lasserre lower bounds for certain k-CSPs, in: Proc. 49th IEEE Symp. on Foundations of Computer Science (FOCS) (2008), 593–602. pp. 347, 351Google Scholar
Scholz, H., Ein ungel ostes Problem in der symbolischen Logik, J. Symbolic Logic, 17 (1952), 160. p. 2Google Scholar
Scott, D., A proof of the independence of the continuum hypothesis, Mathematical Systems Theory, 1 (1967), 89–111. p. 196Google Scholar
Segerlind, N., New separations in propositional proof complexity, Ph.D. thesis, University of California, San Diego (2003). pp. 336, 408Google Scholar
Segerlind, N., Nearly exponential size lower bounds for symbolic quantifier elimination algorithms and OBDD-based proofs of unsatisfiability, in: Proc. Electronic Colloq. on Computational Complexity, TR07–009 (2007). p. 408Google Scholar
Shannon, C. E., The synthesis of two-terminal switching circuits, Bell System Technology J., 28 (1949), 59–98. p. 28Google Scholar
Sherali, H. D. and Adams, W. P., A hierarchy of relaxations and convex hull characterizations for mixed-integer 0–1 programming problems, Discrete Applied Mathematics, 52 (1) (1994), 83–106. p. 131Google Scholar
Sipser, M., The history and status of the P versus NP question, in: Proc. 24th Annual ACM Symp. on Theory of Computing (STOC) (1992), 603–618. pp. 2, 38Google Scholar
Sipser, M., Introduction to the Theory of Computation, Cengage Learning, 3rd ed. (2005). p. 37Google Scholar
Shoenfield, J. R., Mathematical Logic, Association for Symbolic Logic (1967). pp. 37, 159, 258Google Scholar
Skelley, A., Propositional PSPACE reasoning with Boolean programs versus quantified Boolean formulas, in: Proc. 31st International Colloq. on Automata, Languages and Programming (ICALP), Springer Lecture Notes in Computer Science, 3142 (2004), 1163–1175. p. 92Google Scholar
Skelley, A., Theories and proof systems for PSPACE and the EXP-time hierarchy, Ph.D. thesis, University of Toronto (2005). p. 92Google Scholar
Skelley, A. and Thapen, N., The provably total search problems of bounded arithmetic, Proc. London Mathematical Society, 103 (1) (2011), 106–138. p. 305Google Scholar
Smolensky, R., Algebraic methods in the theory of lower bounds for Boolean circuit complexity, in: Proc. 19th Annnal ACM Symp. on Theory of Computing (STOC) (1987), 77–82. pp. 30, 329Google Scholar
Smorynski, C., The incompleteness theorem, in: Handbook of Mathematical Logic, ed. Barwise, J., Studies in Logic and the Foundations of Mathematics, North Holland (1989), 821–866. pp. 259, 469Google Scholar
Smullyan, R. M., Theory of Formal Systems, Annals of Mathematical Studies, 47, Princeton University Press (1961). pp. 2, 182, 196Google Scholar
Smullyan, R. M., First-Order Logic, Springer (1968). p. 113Google Scholar
Sokolov, D., Dag-like communication and its applications, in: Proc. Electronic Colloquium in Computational Complexity, TR16–202 (2016). p. 409Google Scholar
Soltys, M., The complexity of derivations of matrix identities, Ph.D. thesis, University of Toronto (2001). p. 232Google Scholar
Soltys, M. and Thapen, N., Weak theories of linear algebra, Archive for Mathematical Logic, 44 (2) (2005), 195–208. p. 232Google Scholar
Spira, P. M., On time–hardware complexity of tradeoffs for Boolean functions, in: Proc. 4th Hawaii Symp. on System Sciences (1971), 525–527. pp. 17, 37Google Scholar
Statman, R., Complexity of derivations from quantifier-free Horn formulae, mechanical introduction of explicit definitions, and refinement of completeness theorems, in: Proc. Logic Colloquium ‘76, North-Holland (1977), 505–517. p. 79Google Scholar
Statman, R., Bounds for proof-search and speed-up in the predicate calculus, Ann. Mathematical Logic, 15 (1978), 225–287. p. 79Google Scholar
Stengle, G., A Nullstellensatz and a Positivstellensatz in Semialgebraic Geometry, Mathematische Annalen, 207 (2) (1974), 87–97. pp. 115, 132Google Scholar
Szelepcsényi, R., The method of forcing for nondeterministic automata, Bull. European Association for Theoretical Computer Science, 33 (1987), 96–100. p. 26Google Scholar
Takeuti, G., Proof Theory, Dover (1975); 2nd edn. (2003). pp. 79, 158, 159Google Scholar
Takeuti, G., RSUV isomorphism, in: Arithmetic, Proof Theory and Computational Complexity, eds. Clote, P. and Krajíček, J., Oxford University Press (1993), 364–386. p. 196Google Scholar
Thapen, N., Higher complexity search problems for bounded arithmetic and a formalized no-gap theorem, Archive for Mathematical Logic, 50 (7–8) (2011), 665–680. pp. 305, 436Google Scholar
Thapen, N., A tradeoff between length and width in resolution, Theory of Computing, 12 (5) (2016), 1–14. p. 294Google Scholar
Trakhtenbrot, B. A., The impossibility of an algorithm for the decidability problem on finite classes, Proc. USSR Academy of Sciences (in Russian), 70 (4) (1950), 569–572. pp. 233, 447Google Scholar
Trakhtenbrot, B. A., A survey of Russian approaches to Perebor (brute-force searches) algorithms, J. IEEE Ann. History of Computing, 6 (4)(1984), 384–400. p. 38Google Scholar
Tseitin, G. C., On the complexity of derivations in propositional calculus, in: Studies in Mathematics and Mathematical Logic, Part II, ed. A. O. Slisenko (1968), 115–125. pp. 3, 62, 94, 97, 111, 112, 280Google Scholar
Tseitin, G. C. and Choubarian, A. A., On some bounds to the lengths of logical proofs in classical propositional calculus (in Russian), Trudy Vyčisl Centra AN Arm SSR i Erevanskovo Univ., 8 (1975), 57–64. p. 62Google Scholar
Turing, A., On computable numbers, with an application to the Entscheidungsproblem, Proc. London Mathematical Society, Series 2, 42 (1936–1937), 230–265. pp. 1, 32, 37Google Scholar
Tzameret, I., Algebraic proofs over noncommutative formulas, Information and Computation, 209 (10) (2011), 1269–1292. p. 160Google Scholar
Urquhart, A., Hard examples for resolution, J. ACM, 34 (1) (1987), 209–219. pp. 281, 293Google Scholar
Urquhart, A., The complexity of propositional proofs, Bull. Symbolic Logic, 194 (1995), 425–467. pp. 6, 79, 112Google Scholar
Urquhart, A., The depth of resolution proofs, Studia Logica, 99 (2011), 349–364. p. 113Google Scholar
Vaananen, J., Pseudo-finite model theory, Matematica Contemporanea, 24 (2003), 169–183. p. 454Google Scholar
van den Dries, L., Tame Topology and o-Minimal Structures, Cambridge University Press (1998). p. 132Google Scholar
von Neumannn, J., Collected Works, Pergamon Press (1963). p. 230Google Scholar
Wagner, K. W., Bounded query classes, SIAM J. Computing, 19 (5) (1990), 833846.Google Scholar
Wang, Z., Implicit resolution, Logical Methods in Computer Science, 9 (4–7) (2013), 1–10. p. 158Google Scholar
Wegener, I., The Complexity of Boolean Functions, Wiley-Teubner Series in Computer Science (1987). p. 38Google Scholar
Wegener, I., Branching Programs and Binary Decision Diagrams – Theory and Applications, SIAM Monographs in Discrete Mathematics and Its Applications (2000). p. 146Google Scholar
Williams, R., Improving exhaustive search implies superpolynomial lower bounds, SIAM J. Computing, 42 (3) (2013), 1218–1244. p. 480Google Scholar
Woods, A., Some problems in logic and number theory, and their connections, Ph.D. Thesis, University of Manchester (1981). pp. 38, 259Google Scholar
Wrathall, C., Rudimentary predicates and relative computation, SIAM J. Computing, 7 (1978), 194–209. p. 182Google Scholar
Yao, A. C.-C., Theory and applications of trapdoor functions, in: Proc. 23rd Annual IEEE Symp. on Foundations of Computational Science(FOCS) (1982), 80–91. p. 429Google Scholar
Yao, A. C.-C., Separating the polynomial-time hierarchy by oracles, in: Proc. 26th Annual IEEE Symp. on Foundations of Computational Science(FOCS) (1985), 1–10. pp. 38, 304Google Scholar
Žák, S., A Turing machine hierarchy, Theoretical Computer Science, 26 (1983), 327–333. p. 38Google Scholar
Zambella, D., Notes on polynomially bounded arithmetic, J. Symbolic Logic, 61 (3) (1996), 942–966. p. 183Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Bibliography
  • Jan Krajíček, Charles University, Prague
  • Book: Proof Complexity
  • Online publication: 25 March 2019
  • Chapter DOI: https://doi.org/10.1017/9781108242066.030
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • Bibliography
  • Jan Krajíček, Charles University, Prague
  • Book: Proof Complexity
  • Online publication: 25 March 2019
  • Chapter DOI: https://doi.org/10.1017/9781108242066.030
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Bibliography
  • Jan Krajíček, Charles University, Prague
  • Book: Proof Complexity
  • Online publication: 25 March 2019
  • Chapter DOI: https://doi.org/10.1017/9781108242066.030
Available formats
×