The Complexity Theory Companion

Invitation S e c r e t 1 Algorith~r~s are at the heart of comple~ty theory. T h a t is the dark secret of complexity theory. It is recognized by complexity theorists, but would be literally incredible to most others. In this book, we hope to make this secret credible. In fact, t h e real secret is even more dramatic. S e c r e t 2 ~imple algorithms are a~ the heart o/comple.citll theory. A corollary of Secret 2 is that every practitioner of computer science or student of computer science already possesses the ability required to understand, enjoy, and employ complexity theory. We realize that these secrets fly in the face of conventional wisdom. Most people view complexity theory as an arcane realm populated by pointy-hatted (ff not indeed pointy-headed) sorcerers stirring cauldrons of recursion theory with wands of combinatorics, while chanting incantations involving complexity classes whose very names contain hundreds of characters and sear the tongues of mere mortals. This stereotype has sprung up in part due to the small amount of esoteric research t h a t fits this bill, but the stereotype is more strongly attributable to the failuze of complexity theorists to commlmlcate in expository forums the central role t h a t algorithms play in complexity theory. T h r o u g h o u t this book-f~om the tree-pruning and interval-pruning algorithms that shape the first chapter to the query simulation procedures t h a t d o m i u a t e the last chapter-we will see t h a t proofs in complexity theory usually employ algorithms as their central tools. In fact, to more clearly highlight t h e role of algorithmic techniques in complexity theory, thgs book i8 organized by technique rather than by topic. T h a t is, in contrast to the organization of other books on complexity theory, each chapter of this book focuses on one te~hnlque what it is, and what results and applications it has yielded. The most thrilling times in complexity theory are when a new technique is introd~!ced and sweeps like fire over the field. In addition to highlighting the centrality of algorithm~ in t~e proof arsenal of complexity theory, we feel t h a t our technique-based approach more vividly conveys to the reader the flavor and excitement of such conflagrations. We invite t h e …

[1]  A. Church.  The calculi of lambda-conversion , 1941 .

[2]  Martin D. Davis,et al.  Computability and Unsolvability , 1959, McGraw-Hill Series in Information Processing and Computers.

[3]  Richard Edwin Stearns,et al.  Memory bounds for recognition of context-free and context-sensitive languages , 1965, SWCT.

[4]  J. Hartmanis,et al.  On the Computational Complexity of Algorithms , 1965 .

[5]  Stephen A. Cook,et al.  Review: Alan Cobham, Yehoshua Bar-Hillel, The Intrinsic Computational Difficulty of Functions , 1969 .

[6]  Stephen A. Cook,et al.  A hierarchy for nondeterministic time complexity , 1972, J. Comput. Syst. Sci..

[7]  Ronald V. Book,et al.  On Languages Accepted in Polynomial Time , 1972, SIAM J. Comput..

[8]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[9]  Ronald V. Book,et al.  Comparing Complexity Classes , 1974, J. Comput. Syst. Sci..

[10]  Ronald V. Book,et al.  Tally Languages and Complexity Classes , 1974, Inf. Control..

[11]  R.E. Ladner,et al.  A Comparison of Polynomial Time Reducibilities , 1975, Theor. Comput. Sci..

[12]  R. Ladner.  The circuit value problem is log space complete for P , 1975, SIGA.

[13]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[14]  Celia Wrathall,et al.  Complete Sets and the Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[15]  Leslie G. Valiant,et al.  Relative Complexity of Checking and Evaluating , 1976, Inf. Process. Lett..

[16]  Alan J. Demers,et al.  Some Comments on Functional Self-Reducibility and the NP Hierarchy , 1976 .

[17]  L. Berman.  Polynomial reducibilities and complete sets. , 1977 .

[18]  Istvan Simon.  On some subrecursive reducibilities , 1977 .

[19]  Richard J. Lipton,et al.  A Probabilistic Remark on Algebraic Program Testing , 1978, Inf. Process. Lett..

[20]  Piotr Berman.  Relationship Between Density and Deterministic Complexity of NP-Complete Languages , 1978, ICALP.

[21]  Ivan Hal Sudborough,et al.  On the Tape Complexity of Deterministic Context-Free Languages , 1978, JACM.

[22]  John E. Hopcroft,et al.  A Note on Cryptography and NP$\cap$ CoNP-P , 1978 .

[23]  Leonard M. Adleman,et al.  Two theorems on random polynomial time , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[24]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[25]  Steven Fortune,et al.  A Note on Sparse Complete Sets , 1978, SIAM J. Comput..

[26]  Richard E. Ladner,et al.  Propositional Dynamic Logic of Regular Programs , 1979, J. Comput. Syst. Sci..

[27]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[28]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[29]  Gilles Brassard,et al.  A note on the complexity of cryptography (Corresp.) , 1979, IEEE Trans. Inf. Theory.

[30]  Patrick C. Fischer,et al.  Refining Nondeterminism in Relativized Polynomial-Time Bounded Computations , 1980, SIAM J. Comput..

[31]  Leonard M. Adleman,et al.  On distinguishing prime numbers from composite numbers , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[32]  John E. Hopcroft,et al.  Polynomial-time algorithms for permutation groups , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[33]  Joel I. Seiferas,et al.  Limitations on Separating Nondeterministic Complexity Classes , 1981, SIAM J. Comput..

[34]  Michael Sipser,et al.  Parity, circuits, and the polynomial-time hierarchy , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[35]  John E. Hopcroft,et al.  Recent Directions in Algorithmic Research , 1981, Theoretical Computer Science.

[36]  Leslie G. Valiant,et al.  A complexity theory based on Boolean algebra , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[37]  Michael Sipser,et al.  On Relativization and the Existence of Complete Sets , 1982, ICALP.

[38]  Timothy J. Long.  Strong Nondeterministic Polynomial-Time Reducibilities , 1982, Theor. Comput. Sci..

[39]  Zoltán Füredi,et al.  Families of Finite Sets in Which No Set Is Covered by the Union of Two Others , 1982, J. Comb. Theory, Ser. A.

[40]  Stathis Zachos,et al.  Robustness of Probabilistic Computational Complexity Classes under Definitional Perturbations , 1982, Inf. Control..

[41]  Joseph Y. Halpern,et al.  Decision procedures and expressiveness in the temporal logic of branching time , 1982, STOC '82.

[42]  Michael Sipser,et al.  Borel sets and circuit complexity , 1983, STOC.

[43]  Ker-I Ko.  On Self-Reducibility and Weak P-Selectivity , 1983, J. Comput. Syst. Sci..

[44]  Yaacov Yesha,et al.  On Certain Polynomial-Time Truth-Table Reducibilities of Complete Sets to Sparse Sets , 1983, SIAM J. Comput..

[45]  Christopher B. Wilson.  Relativized circuit complexity , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[46]  Chee-Keng Yap,et al.  Some Consequences of Non-Uniform Conditions on Uniform Classes , 1983, Theor. Comput. Sci..

[47]  Allan Borodin,et al.  Bounds for width two branching programs , 1983, SIAM J. Comput..

[48]  C. Papadimitriou,et al.  Two remarks on the power of counting , 1983 .

[49]  Michael Ben-Or,et al.  A theorem on probabilistic constant depth Computations , 1984, STOC '84.

[50]  Uzi Vishkin,et al.  Constant Depth Reducibility , 1984, SIAM J. Comput..

[51]  Timothy J. Long,et al.  Quantitative Relativizations of Complexity Classes , 1984, SIAM J. Comput..

[52]  Alan L. Selman,et al.  Qualitative Relativizations of Complexity Classes , 1985, J. Comput. Syst. Sci..

[53]  Silvio Micali,et al.  The knowledge complexity of interactive proof-systems , 1985, STOC '85.

[54]  Ker-I Ko,et al.  On Circuit-Size Complexity and the Low Hierarchy in NP , 1985, SIAM J. Comput..

[55]  Neil Immerman,et al.  On Complete Problems for NP$\cap$CoNP , 1985, ICALP.

[56]  David A. Russo.  Structural properties of complexity classes , 1985, Research in computer science.

[57]  Paul Young,et al.  Reductions Among Polynomial Isomorphism Types , 1985, Theor. Comput. Sci..

[58]  Vijay V. Vazirani,et al.  Random polynomial time is equal to slightly-random polynomial time , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[59]  Leslie G. Valiant,et al.  NP is as easy as detecting unique solutions , 1985, STOC '85.

[60]  A. Yao.  Separating the polynomial-time hierarchy by oracles , 1985 .

[61]  Stathis Zachos,et al.  A Decisive Characterization of BPP , 1986, Inf. Control..

[62]  Juris Hartmanis,et al.  On Sparse Oracles Separating Feasible Complexity Classes , 1986, STACS.

[63]  Mark W. Krentel.  The Complexity of Optimization Problems , 1986, Computational Complexity Conference.

[64]  Juris Hartmanis,et al.  Complexity Classes without Machines: On Complete Languages for UP , 1986, Theor. Comput. Sci..

[65]  Ding-Zhu Du,et al.  On One-Way Functions and Polynomial-Time Isomorphisms , 1986, Theor. Comput. Sci..

[66]  Eric Allender.  Isomorphisms and 1-L Reductions , 1986, Computational Complexity Conference.

[67]  Ian Parberry,et al.  On the Construction of Parallel Computers from Various Bases of Boolean Functions , 1986, Theor. Comput. Sci..

[68]  H. Venkateswaran,et al.  Properties that characterize LOGCFL , 1987, J. Comput. Syst. Sci..

[69]  Vijay V. Vazirani,et al.  Matching is as easy as matrix inversion , 1987, STOC.

[70]  Jim Kadin,et al.  P^(NP[O(log n)]) and Sparse Turing-Complete Sets for NP , 1989, J. Comput. Syst. Sci..

[71]  Stathis Zachos,et al.  Probabalistic Quantifiers vs. Distrustful Adversaries , 1987, FSTTCS.

[72]  Denis Thérien,et al.  Non-Uniform Automata Over Groups , 1987, Inf. Comput..

[73]  Avi Wigderson,et al.  Multi-prover interactive proofs: how to remove intractability assumptions , 2019, STOC '88.

[74]  Osamu Watanabe,et al.  On Hardness of One-Way Functions , 1988, Inf. Process. Lett..

[75]  Stathis Zachos,et al.  Probabilistic Quantifiers and Games , 1988, J. Comput. Syst. Sci..

[76]  Ker-I Ko.  Relativized polynomial time hierarchies having exactly K levels , 1988, STOC '88.

[77]  José L. Balcázar,et al.  Structural Complexity I , 1988, EATCS Monographs on Theoretical Computer Science Series.

[78]  Richard Beigel,et al.  On the relativized power of additional accepting paths , 1989, [1989] Proceedings. Structure in Complexity Theory Fourth Annual Conference.

[79]  Sanjay Jain,et al.  On the Limitations of Locally Robust Positive Reductions , 1989, FSTTCS.

[80]  Jin-Yi Cai,et al.  On the Power of Parity Polynomial Time , 1989, STACS.

[81]  Richard J. Lipton,et al.  New Directions In Testing , 1989, Distributed Computing And Cryptography.

[82]  Juris Hartmanis,et al.  Robust Machines Accept Easy Sets , 1990, Theor. Comput. Sci..

[83]  Joan Feigenbaum,et al.  Hiding Instances in Multioracle Queries , 1990, STACS.

[84]  Manuel Blum,et al.  Self-testing/correcting with applications to numerical problems , 1990, STOC '90.

[85]  Lane A. Hemaspaandra,et al.  On the Complexity of Ranking , 1990, J. Comput. Syst. Sci..

[86]  Lane A. Hemaspaandra,et al.  On sets with efficient implicit membership tests , 1990, Proceedings Fifth Annual Structure in Complexity Theory Conference.

[87]  Ronitt Rubinfeld,et al.  Self-testing/correcting for polynomials and for approximate functions , 1991, STOC '91.

[88]  Daniel A. Spielman,et al.  The perceptron strikes back , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[89]  Stuart A. Kurtz,et al.  Gap-definable counting classes , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[90]  Osamu Watanabe,et al.  On the computational complexity of small descriptions , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[91]  Edith Hemaspaandra,et al.  Bounded Reductions , 1991, Complexity Theory: Current Research.

[92]  Lane A. Hemaspaandra,et al.  On Sets Polynomially Enumerable by Iteration , 1991, Theor. Comput. Sci..

[93]  Silvio Micali,et al.  Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems , 1991, JACM.

[94]  Christoph Meinel,et al.  Structure and Importance of Logspace-MOD-Classes , 1991, STACS.

[95]  R. Beigel,et al.  Bounded Queries to SAT and the Boolean Hierarchy , 1991, Theor. Comput. Sci..

[96]  Richard Beigel,et al.  Relativized Counting Classes: Relations among Thresholds, Parity, and Mods , 1991, J. Comput. Syst. Sci..

[97]  Jürgen Kämper,et al.  Nonuniform Proof Systems: A New Framework to Describe Nonuniform and Probabilistic Complexity Classes , 1991, Theor. Comput. Sci..

[98]  Viktória Zankó,et al.  #P-Completeness via Many-One Reductions , 1990, Int. J. Found. Comput. Sci..

[99]  Danilo Bruschi.  Strong Separations of the Polynomial Hierarchy with Oracles: Constructive Separations by Immune and Simple Sets , 1992, Theor. Comput. Sci..

[100]  Osamu Watanabe,et al.  How hard are sparse sets? , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[101]  Nikolai K. Vereshchagin,et al.  Banishing Robust Turing Completeness , 1992, LFCS.

[102]  Lane A. Hemaspaandra,et al.  Is #P Closed under Substraction? , 1992, Bull. EATCS.

[103]  Sanjay Gupta,et al.  On the Closure of Certain Function Classes Under Integer Division by Polynomially-Bounded Functions , 1992, Inf. Process. Lett..

[104]  Adi Shamir,et al.  IP = PSPACE , 1992, JACM.

[105]  Paul Young.  How reductions to sparse sets collapse the polynomial-time hierarchy: a primer; part I: polynomial-time Turing reductions , 1992, SIGA.

[106]  Michael Sipser,et al.  The history and status of the P versus NP question , 1992, STOC '92.

[107]  Michael Ben-Or,et al.  Computing Algebraic Formulas Using a Constant Number of Registers , 1992, SIAM J. Comput..

[108]  Richard Beigel,et al.  Perceptrons, PP, and the polynomial hierarchy , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[109]  Peter van Emde Boas,et al.  Twenty Questions to a P-Selector , 1993, Inf. Process. Lett..

[110]  Andrew Chi-Chih Yao,et al.  A Circuit-Based Proof of Toda's Theorem , 1993, Inf. Comput..

[111]  Edith Hemaspaandra,et al.  A modal perspective on the computational complexity of attribute value grammar , 1993, J. Log. Lang. Inf..

[112]  Mitsunori Ogihara,et al.  P-selective sets, and reducing search to decision vs. self-reducibility , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[113]  Lance Fortnow,et al.  Gap-Definability as a Closure Property , 1993, STACS.

[114]  Aravind Srinivasan,et al.  Randomness-optimal unique element isolation, with applications to perfect matching and related problems , 1993, SIAM J. Comput..

[115]  Lane A. Hemaspaandra,et al.  Threshold Computation and Cryptographic Security , 1993, ISAAC.

[116]  Luc Longpré,et al.  On Reductions of NP Sets to Sparse Sets , 1994, J. Comput. Syst. Sci..

[117]  Ronald V. Book.  On Collapsing the Polynomial-Time Hierarchy , 1994, Inf. Process. Lett..

[118]  Burchard von Braunmühl,et al.  The Alternation Hierarchy for Machines with Sublogarithmic Space is Infinite , 1994, STACS.

[119]  D. Sivakumar,et al.  On Quasilinear-Time Complexity Theory , 1995, Theor. Comput. Sci..

[120]  Bernd Borchert.  Predicate classes, promise classe, and the acceptance power of regular languages , 1994 .

[121]  P. Beame.  A switching lemma primer , 1994 .

[122]  Lane A. Hemaspaandra,et al.  Semi-membership algorithms: some recent advances , 1994, SIGA.

[123]  Pierluigi Crescenzi,et al.  A compendium of NP optimization problems , 1994, WWW Spring 1994.

[124]  Harry Buhrman,et al.  On the structure of complete sets , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[125]  H. James Hoover,et al.  Limits to Parallel Computation: P-Completeness Theory , 1995 .

[126]  Lane A. Hemaspaandra,et al.  Worlds to die for , 1995, SIGA.

[127]  Howard Straubing,et al.  The power of local self-reductions , 1995, Proceedings of Structure in Complexity Theory. Tenth Annual IEEE Conference.

[128]  Lane A. Hemaspaandra,et al.  Pseudorandom Generators and the Frequency of Simplicity , 1995, STACS.

[129]  Sanjay Gupta,et al.  Closure Properties and Witness Reduction , 1995, J. Comput. Syst. Sci..

[130]  Sanjeev Arora.  Probabilistic checking of proofs and hardness of approximation problems , 1995 .

[131]  Daniel A. Spielman,et al.  PP is closed under intersection , 1991, STOC '91.

[132]  Johannes Köbler,et al.  On the Structure of Low Sets , 1995, SCT.

[133]  Jie Wang.  Some Results on Selectivity and Self-Reducibility , 1995, Inf. Process. Lett..

[134]  Stuart A. Kurtz,et al.  The isomorphism conjecture fails relative to a random oracle , 1995, JACM.

[135]  John D. Rogers.  The isomorphism conjecture holds and one-way functions exist relative to an oracle , 1995, Proceedings of Structure in Complexity Theory. Tenth Annual IEEE Conference.

[136]  Lane A. Hemaspaandra,et al.  P-Selectivity: Intersections and Indices , 1995, Theor. Comput. Sci..

[137]  Lane A. Hemaspaandra,et al.  Reducibility Classes of P-Selective Sets , 1996, Theor. Comput. Sci..

[138]  Jin-Yi Cai,et al.  On the Existence of Hard Sparse Sets under Weak Reductions , 1996, STACS.

[139]  Carsten Lund,et al.  Hardness of approximations , 1996 .

[140]  László Lovász,et al.  Interactive proofs and the hardness of approximating cliques , 1996, JACM.

[141]  M. Bellare.  Proof Checking and Approximation: Towards Tight Results , 1996 .

[142]  Lane A. Hemaspaandra,et al.  Optimal Advice , 1996, Theor. Comput. Sci..

[143]  Vikraman Arvind,et al.  Quasi-Linear Truth-Table Reductions to p-Selective Sets , 1996, Theor. Comput. Sci..

[144]  Lance Fortnow,et al.  PP is Closed Under Truth-Table Reductions , 1996, Inf. Comput..

[145]  Heribert Vollmer,et al.  Nondeterministic NC 1 Computation. , 1996 .

[146]  Harry Buhrman,et al.  P-Selektive Self-Reducible Sets: A New Characterization of P , 1996, J. Comput. Syst. Sci..

[147]  Judy Goldsmith,et al.  Scalability and the Isomorphism Problem , 1996, Inf. Process. Lett..

[148]  Dieter van Melkebeek.  Reducing P to a Sparse Set using a Constant Number of Queries Collapses P to L , 1996, Computational Complexity Conference.

[149]  Richard Beigel.  Closure properties of GapP and #P , 1997, Proceedings of the Fifth Israeli Symposium on Theory of Computing and Systems.

[150]  Jörg Rothe,et al.  Exact analysis of Dodgson elections: Lewis Carroll's 1876 voting system is complete for parallel access to NP , 1997, JACM.

[151]  Bin Fu,et al.  Circuits over PP and PL , 1997, Proceedings of Computational Complexity. Twelfth Annual IEEE Conference.

[152]  Adi Shamir,et al.  Fully Parallelized Multi-Prover Protocols for NEXP-Time , 1997, J. Comput. Syst. Sci..

[153]  Jörg Rothe,et al.  Unambiguous Computation: Boolean Hierarchies and Sparse Turing-Complete Sets , 1997, SIAM J. Comput..

[154]  Edith Hemaspaandra,et al.  An Introduction to Query Order , 1997, Bull. EATCS.

[155]  Frank Stephan,et al.  Looking for an Analogue of Rice's Theorem in Circuit Complexity Theory , 1997, Kurt Gödel Colloquium.

[156]  Miklos Santha,et al.  Verifying the determinant in parallel , 1994, computational complexity.

[157]  A. Selman,et al.  Complexity theory retrospective II , 1998 .

[158]  Edith Hemaspaandra,et al.  Recognizing when Greed can Approximate Maximum Independent Sets is Complete for Parallel Access to NP , 1998, Inf. Process. Lett..

[159]  Edith Hemaspaandra,et al.  Downward Collapse from a Weaker Hypothesis , 1998, ArXiv.

[160]  Edith Hemaspaandra,et al.  A Downward Collapse within the Polynomial Hierarchy , 1999, SIAM J. Comput..

[161]  Marius Zimand.  On the Size of Classes with Weak Membership Properties , 1998, Theor. Comput. Sci..

[162]  Lance Fortnow.  Relativized Worlds with an Infinite Hierarchy , 1999, Inf. Process. Lett..

[163]  Jörg Rothe,et al.  Creating Strong, Total, Commutative, Associative One-Way Functions from Any One-Way Function in Complexity Theory , 1999, J. Comput. Syst. Sci..

[164]  Edith Hemaspaandra,et al.  Extending Downward Collapse from 1-versus-2 Queries to j-versus-j+1 Queries , 1999, STACS.

[165]  Leonid A. Levin,et al.  A Pseudorandom Generator from any One-way Function , 1999, SIAM J. Comput..

[166]  Richard Beigel,et al.  Circuit lower bounds collapse relativized complexity classes , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[167]  Joe Kilian,et al.  Primality testing using elliptic curves , 1999, JACM.

[168]  Oded Goldreich,et al.  Foundations of Cryptography: Basic Tools , 2000 .

[169]  Jörg Rothe,et al.  A second step towards complexity-theoretic analogs of Rice's Theorem , 2000, Theor. Comput. Sci..

[170]  Eric Allender,et al.  Making Nondeterminism Unambiguous , 2000, SIAM J. Comput..

[171]  Heribert Vollmer,et al.  A note on closure properties of logspace MOD classes , 2000, Information Processing Letters.

[172]  Thomas Thierauf,et al.  The complexity of verifying the characteristic polynomial and testing similarity , 2000, Proceedings 15th Annual IEEE Conference on Computational Complexity.

[173]  Eric Allender,et al.  Complexity of finite-horizon Markov decision process problems , 2000, JACM.

[174]  Christian Glasser,et al.  Consequences of the Existence of Sparse Sets Hard for NP under a Subclass of Truth-Table Reductions , 2000 .

[175]  Christopher Homan.  Low Ambiguity in Strong, Total, Associative, One-Way Functions , 2000, ArXiv.

[176]  Jörg Rothe,et al.  If P != NP Then Some Strongly Noninvertible Functions Are Invertible , 2001, FCT.

[177]  Vikraman Arvind,et al.  On pseudorandomness and resource-bounded measure , 2001, Theor. Comput. Sci..

[178]  Lane A. Hemaspaandra,et al.  Reducing the Number of Solutions of NP Functions , 2002, J. Comput. Syst. Sci..

[179]  Jacobo Torán,et al.  On counting and approximation , 1989, Acta Informatica.