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26 results in Computational Complexity

Page 2 of 2

  • 9 - Probabilistic Proof Systems

  • Oded Goldreich, Weizmann Institute of Science, Israel
  • Book:
    Computational Complexity
    Published online:
    05 June 2012
    Print publication:
    28 April 2008, pp 349-415

  • Summary

    A proof is whatever convinces me.

    Shimon Even (1935–2004)

    The glory attached to the creativity involved in finding proofs makes us forget that it is the less glorified process of verification that gives proofs their value. Conceptually speaking, proofs are secondary to the verification process, whereas technically speaking, proof systems are defined in terms of their verification procedures.

    The notion of a verification procedure presumes the notion of computation and furthermore the notion of efficient computation. This implicit stipulation is made explicit in the definition of NP, where efficient computation is associated with deterministic polynomial-time algorithms. However, as argued next, we can gain a lot if we are willing to take a somewhat non-traditional step and allow probabilistic verification procedures.

    In this chapter, we shall study three types of probabilistic proof systems, called interactive proofs, zero-knowledge proofs, and probabilistic checkable proofs. In each of these three cases, we shall present fascinating results that cannot be obtained when considering the analogous deterministic proof systems.

    Summary: The association of efficient procedures with deterministic polynomial-time procedures is the basis for viewing NP-proof systems as the canonical formulation of proof systems (with efficient verification procedures). Allowing probabilistic verification procedures and, moreover, ruling by statistical evidence gives rise to various types of probabilistic proof systems. Indeed, these probabilistic proof systems carry a probability of error (which is explicitly bounded and can be reduced by successive applications of the proof system), yet they offer various advantages over the traditional (deterministic and errorless) proof systems. […]

  • Preface

  • Oded Goldreich, Weizmann Institute of Science, Israel
  • Book:
    Computational Complexity
    Published online:
    05 June 2012
    Print publication:
    28 April 2008, pp xv-xvi

  • Summary

    The quest for efficiency is ancient and universal, as time and other resources are always in shortage. Thus, the question of which tasks can be performed efficiently is central to the human experience.

    A key step toward the systematic study of the aforementioned question is a rigorous definition of the notion of a task and of procedures for solving tasks. These definitions were provided by computability theory, which emerged in the 1930s. This theory focuses on computational tasks, and considers automated procedures (i.e., computing devices and algorithms) that may solve such tasks.

    In focusing attention on computational tasks and algorithms, computability theory has set the stage for the study of the computational resources (like time) that are required by such algorithms. When this study focuses on the resources that are necessary for any algorithm that solves a particular task (or a task of a particular type), the study becomes part of the theory of Computational Complexity (also known as Complexity Theory).

    Complexity Theory is a central field of the theoretical foundations of computer science. It is concerned with the study of the intrinsic complexity of computational tasks. That is, a typical complexity theoretic study refers to the computational resources required to solve a computational task (or a class of such tasks), rather than referring to a specific algorithm or an algorithmic schema. Actually, research in Complexity Theory tends to start with and focus on the computational resources themselves, and addresses the effect of limiting these resources on the class of tasks that can be solved.

  • Appendix B - On the Quest for Lower Bounds

  • Oded Goldreich, Weizmann Institute of Science, Israel
  • Book:
    Computational Complexity
    Published online:
    05 June 2012
    Print publication:
    28 April 2008, pp 469-481

  • Summary

    Alas, Philosophy, Medicine, Law, and unfortunately also Theology, have I studied in detail, and still remained a fool, not a bit wiser than before. Magister and even Doctor am I called, and for a decade am I sick and tired of pulling my pupils by the nose and understanding that we can know nothing.

    J. W. Goethe, Faust, lines 354–64

    Summary: This appendix briefly surveys some attempts at proving lower bounds on the complexity of natural computational problems. In the first part, devoted to circuit complexity, we describe lower bounds on the size of (restricted) circuits that solve natural computational problems. This can be viewed as a program whose long-term goal is proving that P ≠ NP. In the second part, devoted to proof complexity, we describe lower bounds on the length of (restricted) propositional proofs of natural tautologies. This can be viewed as a program whose long-term goal is proving that NP ≠ coNP.

    We comment that while the activity in these areas is aimed toward developing proof techniques that may be applied to the resolution of the “big problems” (such as P versus NP), the current achievements (though very impressive) seem very far from reaching this goal. Current crown-jewel achievements in these areas take the form of tight (or strong) lower bounds on the complexity of computing (resp., proving) “relatively simple” functions (resp., claims) in restricted models of computation (resp., proof systems).

  • 2 - P, NP, and NP-Completeness

  • Oded Goldreich, Weizmann Institute of Science, Israel
  • Book:
    Computational Complexity
    Published online:
    05 June 2012
    Print publication:
    28 April 2008, pp 44-107

  • Summary

    For as much as many have taken in hand to set forth in order a declaration of those things which are most surely believed among us; Even as they delivered them unto us, who from the beginning were eyewitnesses, and ministers of the word; It seems good to me also, having had perfect understanding of all things from the very first, to write unto thee in order, most excellent Theophilus; That thou mightest know the certainty of those things, wherein thou hast been instructed.

    Luke, 1:1–4

    The main focus of this chapter is the P-vs-NP Question and the theory of NP-completeness. Additional topics covered in this chapter include the general notion of a polynomial-time reduction (with a special emphasis on self-reducibility), the existence of problems in NP that are neither NP-complete nor in P, the class coNP, optimal search algorithms, and promise problems.

    Summary: Loosely speaking, the P-vs-NP Question refers to search problems for which the correctness of solutions can be efficiently checked (i.e., if there is an efficient algorithm that given a solution to a given instance determines whether or not the solution is correct). Such search problems correspond to the class NP, and the question is whether or not all these search problems can be solved efficiently (i.e., if there is an efficient algorithm that given an instance finds a correct solution).

    Thus, the P-vs-NP Question can be phrased as asking whether or not finding solutions is harder than checking the correctness of solutions. […]

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