DOCSLIB.ORG

Probabilistically Checkable Debate Systems and Approximation Algorithms for PSPACE-Hard Functions (Extended Abstract)*

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Probabilistically Checkable Debate Systems and Approximation Algorithms for PSPACE-Hard Functions (Extended Abstract)* Probabilistically Checkable Debate Systems and Approximation Algorithms for PSPACE-Hard Functions (Extended Abstract)* Anne Condont Joan Feigenbaum$ Carsten Lund~ Peter Shorl~ Abstract 1 Introduction Suppose that two candidates, B and C, agree to a de- bate format. Voter V is too busy to catch more than We initiate an investigation of probabilistically check- a very small number of bits of the debate. How does able debate systems (PCDS’S), a natural generalization V decide which of B or C won the debate? In this pa- of the probabilistically checkable proof systems stud- per, we show that if B and C choose the right debate ied in [1, 2, 3, 8]. A PCDS for a language L consists format, V’s problem is solved. By listening to a few, of a probabilistic polynomial-time verifier V and a de- randomly chosen, sounds bites of the debate, V can bate between player 1, who claims that the input z is with near certainty figure out who won. in L, and player O, who claims that the input x is not Similarly, suppose that B or C is giving a speech in L. We show that there is a PCDS for L in which to a set of voters VI, . Vn, represented by finite au- V flips O(log n) random coins and reads O(1) bits of tomata. He would like to give the speech that results the debate if and only if L is in PSPACE. This char- in acceptance (votes) by the greatest number of Vi ‘s, acterization of PSPACE is used to show that certain PSPACE-hard functions are as hard to approximate We show that not only can he not compute this maxi- mum exactly, but he cannot come within an arbitrary as they are to compute exactly. constant factor, unless he has access to an oracle (po- litical consultant) with the full power of P!3PACE. Our work builds on the recent progress that has been made in the theory of probabilistically checkable t University of Wisconsin, Computer Sciences Depart- proof systems (PCPS ‘s). Results about the language.- ment, 1210 West Dayton Street, Madison, WI 57306 USA, recognition power of PCPS’S have led to lower bounds condon@cs. wise. edu. Supported in part by NSF grants CCR- on the difficulty of approximating NP-hardl functions. 9100886 and CCR-9257241. In this paper, we define probabilistically checkable de.. tAT&T Ben Laboratories, Room 2C473, 600 Mowt~n Avenue, P. O. Box 636, Murray Hill, NJ 07974-0636 USA, bate systems (PCDS’S). We prove several results about jf~research. att . corn. the language-recognition power of PCDS ‘~s and then ~AT&T Ben Laboratories, Room 2C324, 600 Momt& use them to obtain lower bounds on the difficulty of Avenue, P. O. Box 636, Murray Hill, NJ 07974-0636 USA, approximating PSPACE-hard functions. lrmd@research. att. coin. ~AT&T Bell Laboratory=, Room 2DI 49, 600 Momtain Let us describe the background for this work in more Avenue, P. O. Box 636, Murray Hill, NJ 07974-0636 USA, detail. Loosely speaking, a language L has a PCPS if, shor~research. att. corn. for every z c L, there is a string m such thi~t a proba- * The full version of this paper has been submitted for journal bilistic verifier V can be convinced with high probabil- publication and is available as DIMACS TR 93-1o. ity that z c L. The class PCP(r(n), g(n)) consists of Permission to copy without fee all or part of this material is those languages recognizable by PCPS’S in which the granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the verifier uses O(r(n)) coin flips and looks at O(q(n)) title of the publication and its date appear, and notice is given bits. It was recently shown that PCP(log.-, n, 1) = NP that copying is by permission of the Association for Computing (cf. [1, 2]). Machinery. To copy otherwise, or to republish, requires a fee Results on the power of classes PCP(r(n], g(n)) can and/or specific permission. 25th ACM STOC ‘93-5 /93/CA, USA be used to show that many important approximation 01993 ACM 0-S9791-591 -7/93 /0005 /0305 . ..$1.50 305 problems are hard, unless there is some unexpected The following is a technical building block, interest- collapse of complexity classes. The first result along ing in its own right, that is used in the proof that these lines showed that MAX-CLIQUE was difficult to PSPACE = PCD(log n, 1): If r(n) = fl(log n), then approximate [8]. The seminal result of [8] haa been im- PCD(r’(n), q(n)) contains the same languages if the proved several times, and it is now known that there is verifier reads O(q(n)) rounds of the debate as it does an c such that approximating MAX-CLIQUE within if the verifier reads O(q(n)) bits of the debate. a factor of ne is as difficult as solving NP-complete problems exactly [1]. Furthermore, there is a large We then use our main result about the language- class of natural optimization problems, those complete recognition power of PCDS’s to prove lower bounds for the class MAX-SNP defined in [15], that do not on the difficulty of approximating PSPACEhard func- have polynomial-time approximation schemes unless P tions. Let MAX Q3SAT be the following natural opti- = NP; that is, for each of these problems, there is an mization version of the canonical PSPACEcomplete e such that approximating the optimal solution within language QBF. Suppose @ = QIZIQZZZ . Qnxn ratio ~ is as hard as solving NP-complete problems ex- 4(zl, z2,..., zn ) is a quantified boolean formula, with actly [1]. This result on MAX-SNP shows that many Qi c {3, V}, and ~ in 3CNF. Suppose that the vari- well-known optimization problems are hard to approx- ables of the formula are chosen, in order of quantifica- imate closely, including Traveling Salesman with Tri- tion, by two players O and 1, where player O chooses the angle Inequality, MAX-SAT, and MAX-CUT. universally quantified variables and player 1 chooses A PCDS is a generalization of a PCPS. In a PCDS the existentially quantified variables. If player 1 can for L, there are two computationally powerful players, guarantee that k clauses of @ will be satisfied by 1 and O (called B and C at the beginning of this sec- the resulting assignment, regardless of what player O tion) and a probabilistic polynomial-time verifier V. chooses, we say that k clauses of @ are simultaneously Players 1 and O play a game in which they alternate satisfiable. We let MAX Q3SAT be the function that writing out strings on a debate tape iT. Player 1‘s goal maps a quantified 3CNF formula @ to the maximum is to convince V that an input a c L, and player O’s number of simultaneously satisfiable clauses. goal is to convince V that x @ L. When the debate is over, V looks at x and T and decides whether x c L (player 1 wins the debate) or z @ L (player O wins the Theorem: There is a constant O < c <1 such that ap- debate). Suppose V flips O(r(n)) random coins and proximating MAX Q3SAT within ratio c is PSPACE- reads O(q(n)) bits of ~. If, under the best strategies hard. Thus MAX Q3SAT is as hard to approximate of players 1 and O, V’s decision is correct with high as it is to compute exactly. probability, then we say that L is in PCD(r(n), q(n)). Specifically, we say that a language L is in We use reductions to prove that certain other PCD(r(n), q(n)) if it has a nonadaptive PCDS with PSPACEhard functions are PSPACE-hard to approx- one-sided error in which players 1 and O write on a imate in a stronger sense. These include maximiza- debate tape, and then V makes O(r(n)) coin flips and tion versions of the Finite Automata Intersection prob- queries O(q(n)) bits based on these coin flips. By non- lem, shown PSPACEcomplete by Kozen [12], and adaptive, we mean that the choice of bits queried by V the Generalized Geography problem, shown PSPACE- is based solely on the input and the coin flips. By one- complete by Schaefer [17]. We show that there is a con- sided error, we mean that whenever z E L, V must stant ~ such that approximating these problems within correctly decide that x E L, no matter which sequence ratio n’ is PSPACEhard. of O(r(n)) coins are flipped (assuming correct play on the part of player 1). When z # L, V is allowed to The rest of this paper is organized as follows. We conclude incorrectly that z ~ L with probability at define PCDS’S, and all of our other terms, precisely most c, for some fixed ~ < 1. in Section 2. Our results on the language-recognition With the above definition in hand, we can state our power of PCDS’S are given in Section 3. Those on main results about the language-recognition power of approximation of PSPACEhard functions are given PCDS’S. in Section 4. Section 5 contains open questions and a discussion of subsequent related results. Theorem: PSPACE ~ PCD(log n, 1). This re- We have omitted some proofs because of space limi- sult is best possible, because we can show that t ations; they can all be found in our journal submission PCD(log n, q(n)) is contained in PSPACE, for any (DIMACS TR 93-10).
Load more

Recommended publications
  • CS601 DTIME and DSPACE Lecture 5 Time and Space Functions: T, S
    CS601 DTIME and DSPACE Lecture 5 Time and Space functions: t, s : N → N+ Definition 5.1 A set A ⊆ U is in DTIME[t(n)] iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. A = L(M) ≡ w ∈ U M(w)=1 , and 2. ∀w ∈ U, M(w) halts within c · t(|w|) steps. Definition 5.2 A set A ⊆ U is in DSPACE[s(n)] iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. A = L(M), and 2. ∀w ∈ U, M(w) uses at most c · s(|w|) work-tape cells. (Input tape is “read-only” and not counted as space used.) Example: PALINDROMES ∈ DTIME[n], DSPACE[n]. In fact, PALINDROMES ∈ DSPACE[log n]. [Exercise] 1 CS601 F(DTIME) and F(DSPACE) Lecture 5 Definition 5.3 f : U → U is in F (DTIME[t(n)]) iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. f = M(·); 2. ∀w ∈ U, M(w) halts within c · t(|w|) steps; 3. |f(w)|≤|w|O(1), i.e., f is polynomially bounded. Definition 5.4 f : U → U is in F (DSPACE[s(n)]) iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. f = M(·); 2. ∀w ∈ U, M(w) uses at most c · s(|w|) work-tape cells; 3. |f(w)|≤|w|O(1), i.e., f is polynomially bounded. (Input tape is “read-only”; Output tape is “write-only”.
    [Show full text]
  • EXPSPACE-Hardness of Behavioural Equivalences of Succinct One
    EXPSPACE-hardness of behavioural equivalences of succinct one-counter nets Petr Janˇcar1 Petr Osiˇcka1 Zdenˇek Sawa2 1Dept of Comp. Sci., Faculty of Science, Palack´yUniv. Olomouc, Czech Rep. [email protected], [email protected] 2Dept of Comp. Sci., FEI, Techn. Univ. Ostrava, Czech Rep. [email protected] Abstract We note that the remarkable EXPSPACE-hardness result in [G¨oller, Haase, Ouaknine, Worrell, ICALP 2010] ([GHOW10] for short) allows us to answer an open complexity ques- tion for simulation preorder of succinct one counter nets (i.e., one counter automata with no zero tests where counter increments and decrements are integers written in binary). This problem, as well as bisimulation equivalence, turn out to be EXPSPACE-complete. The technique of [GHOW10] was referred to by Hunter [RP 2015] for deriving EXPSPACE-hardness of reachability games on succinct one-counter nets. We first give a direct self-contained EXPSPACE-hardness proof for such reachability games (by adjust- ing a known PSPACE-hardness proof for emptiness of alternating finite automata with one-letter alphabet); then we reduce reachability games to (bi)simulation games by using a standard “defender-choice” technique. 1 Introduction arXiv:1801.01073v1 [cs.LO] 3 Jan 2018 We concentrate on our contribution, without giving a broader overview of the area here. A remarkable result by G¨oller, Haase, Ouaknine, Worrell [2] shows that model checking a fixed CTL formula on succinct one-counter automata (where counter increments and decre- ments are integers written in binary) is EXPSPACE-hard. Their proof is interesting and nontrivial, and uses two involved results from complexity theory.
    [Show full text]
  • Probabilistic Proof Systems: a Primer
    Probabilistic Proof Systems: A Primer Oded Goldreich Department of Computer Science and Applied Mathematics Weizmann Institute of Science, Rehovot, Israel. June 30, 2008 Contents Preface 1 Conventions and Organization 3 1 Interactive Proof Systems 4 1.1 Motivation and Perspective ::::::::::::::::::::::: 4 1.1.1 A static object versus an interactive process :::::::::: 5 1.1.2 Prover and Veri¯er :::::::::::::::::::::::: 6 1.1.3 Completeness and Soundness :::::::::::::::::: 6 1.2 De¯nition ::::::::::::::::::::::::::::::::: 7 1.3 The Power of Interactive Proofs ::::::::::::::::::::: 9 1.3.1 A simple example :::::::::::::::::::::::: 9 1.3.2 The full power of interactive proofs ::::::::::::::: 11 1.4 Variants and ¯ner structure: an overview ::::::::::::::: 16 1.4.1 Arthur-Merlin games a.k.a public-coin proof systems ::::: 16 1.4.2 Interactive proof systems with two-sided error ::::::::: 16 1.4.3 A hierarchy of interactive proof systems :::::::::::: 17 1.4.4 Something completely di®erent ::::::::::::::::: 18 1.5 On computationally bounded provers: an overview :::::::::: 18 1.5.1 How powerful should the prover be? :::::::::::::: 19 1.5.2 Computational Soundness :::::::::::::::::::: 20 2 Zero-Knowledge Proof Systems 22 2.1 De¯nitional Issues :::::::::::::::::::::::::::: 23 2.1.1 A wider perspective: the simulation paradigm ::::::::: 23 2.1.2 The basic de¯nitions ::::::::::::::::::::::: 24 2.2 The Power of Zero-Knowledge :::::::::::::::::::::: 26 2.2.1 A simple example :::::::::::::::::::::::: 26 2.2.2 The full power of zero-knowledge proofs ::::::::::::
    [Show full text]
  • 26 Space Complexity
    CS:4330 Theory of Computation Spring 2018 Computability Theory Space Complexity Haniel Barbosa Readings for this lecture Chapter 8 of [Sipser 1996], 3rd edition. Sections 8.1, 8.2, and 8.3. Space Complexity B We now consider the complexity of computational problems in terms of the amount of space, or memory, they require B Time and space are two of the most important considerations when we seek practical solutions to most problems B Space complexity shares many of the features of time complexity B It serves a further way of classifying problems according to their computational difficulty 1 / 22 Space Complexity Definition Let M be a deterministic Turing machine, DTM, that halts on all inputs. The space complexity of M is the function f : N ! N, where f (n) is the maximum number of tape cells that M scans on any input of length n. Definition If M is a nondeterministic Turing machine, NTM, wherein all branches of its computation halt on all inputs, we define the space complexity of M, f (n), to be the maximum number of tape cells that M scans on any branch of its computation for any input of length n. 2 / 22 Estimation of space complexity Let f : N ! N be a function. The space complexity classes, SPACE(f (n)) and NSPACE(f (n)), are defined by: B SPACE(f (n)) = fL j L is a language decided by an O(f (n)) space DTMg B NSPACE(f (n)) = fL j L is a language decided by an O(f (n)) space NTMg 3 / 22 Example SAT can be solved with the linear space algorithm M1: M1 =“On input h'i, where ' is a Boolean formula: 1.
    [Show full text]
  • Computational Complexity
    Computational complexity Plan Complexity of a computation. Complexity classes DTIME(T (n)). Relations between complexity classes. Complete problems. Domino problems. NP-completeness. Complete problems for other classes Alternating machines. – p.1/17 Complexity of a computation A machine M is T (n) time bounded if for every n and word w of size n, every computation of M has length at most T (n). A machine M is S(n) space bounded if for every n and word w of size n, every computation of M uses at most S(n) cells of the working tape. Fact: If M is time or space bounded then L(M) is recursive. If L is recursive then there is a time and space bounded machine recognizing L. DTIME(T (n)) = fL(M) : M is det. and T (n) time boundedg NTIME(T (n)) = fL(M) : M is T (n) time boundedg DSPACE(S(n)) = fL(M) : M is det. and S(n) space boundedg NSPACE(S(n)) = fL(M) : M is S(n) space boundedg . – p.2/17 Hierarchy theorems A function S(n) is space constructible iff there is S(n)-bounded machine that given w writes aS(jwj) on the tape and stops. A function T (n) is time constructible iff there is a machine that on a word of size n makes exactly T (n) steps. Thm: Let S2(n) be a space-constructible function and let S1(n) ≥ log(n). If S1(n) lim infn!1 = 0 S2(n) then DSPACE(S2(n)) − DSPACE(S1(n)) is nonempty. Thm: Let T2(n) be a time-constructible function and let T1(n) log(T1(n)) lim infn!1 = 0 T2(n) then DTIME(T2(n)) − DTIME(T1(n)) is nonempty.
    [Show full text]
  • (Design And) Analysis of Algorithms NP and Complexity Classes
    Intro P and NP Hard problems CSE 548: (Design and) Analysis of Algorithms NP and Complexity Classes R. Sekar 1 / 38 Intro P and NP Hard problems Search and Optimization Problems Many problems of our interest are search problems with exponentially (or even infinitely) many solutions Shortest of the paths between two vertices Spanning tree with minimal cost Combination of variable values that minimize an objective We should be surprised we find ecient (i.e., polynomial-time) solutions to these problems It seems like these should be the exceptions rather than the norm! What do we do when we hit upon other search problems? 2 / 38 Intro P and NP Hard problems Hard Problems: Where you find yourself ... I can’t find an ecient algorithm, I guess I’m just too dumb. Images from “Computers and Intractability” by Garey and Johnson 3 / 38 Intro P and NP Hard problems Search and Optimization Problems What do we do when we hit upon hard search problems? Can we prove they can’t be solved eciently? 4 / 38 Intro P and NP Hard problems Hard Problems: Where you would like to be ... I can’t find an ecient algorithm, because no such algorithm is possible. Images from “Computers and Intractability” by Garey and Johnson 5 / 38 Intro P and NP Hard problems Search and Optimization Problems Unfortunately, it is very hard to prove that ecient algorithms are impossible Second best alternative: Show that the problem is as hard as many other problems that have been worked on by a host of brilliant scientists over a very long time Much of complexity theory is concerned with categorizing hard problems into such equivalence classes 6 / 38 Intro P and NP Hard problems P, NP, Co-NP, NP-hard and NP-complete 7 / 38 Intro P and NP Hard problems Nondeterminism and Search Problems Nondeterminism is an oft-used abstraction in language theory Non-deterministic FSA Non-deterministic PDA So, why not non-deterministic Turing machines? Acceptance criteria is analogous to NFA and NPDA if there is a sequence of transitions to an accepting state, an NDTM will take that path.
    [Show full text]
  • Advanced Complexity
    Advanced Complexity TD n◦3 Charlie Jacomme September 27, 2017 Exercise 1 : Space hierarchy theorem Using a diagonal argument, prove that for two space-constructible functions f and g such that f(n) = o(g(n)) (and as always f; g ≥ log) we have SPACE(f(n)) ( SPACE(g(n)). Solution: We dene a language which can be recognized using space O(g(n)) but not in f(n). L = f(M; w)jM reject w using space ≤ g(j(M; w)j andjΓj = 4g Where Γis the alphabet of the Turing Machine We show that L 2 SPACE(g(n)) by constructing the corresponding TM. This is where we need the fact that the alphabet is bounded. Indeed, we want to construct one xed machine that recognizes L for any TM M, and if we allow M to have an arbitrary size of alphabet, the xed machine might need a lot of space in order to represent the alphabet of M, and it might go over O(g(n)). On an input x, we compute g(x) and mark down an end of tape marker at position f(x), so that we reject if we use to much space. If x is not of the form (M; w), we reject, else we simulate M on w for at most jQj × 4g(x) × n steps. If we go over the timeout, we reject. Else, if w is accepted, we reject, and if w is rejected, we accept. This can be done in SPACE(O(g(n))), and we conclude with the speed up theorem.
    [Show full text]
  • PSPACE-Complete Languages
    CSCI 1590 Intro to Computational Complexity PSPACE-Complete Languages John E. Savage Brown University February 11, 2009 John E. Savage (Brown University) CSCI 1590 Intro to Computational Complexity February 11, 2009 1 / 10 Summary 1 Complexity Class Containment 2 Polynomial Time Hierarchy 3 PH-Complete Languages 4 Games and tqbf 5 tqbf is PSPACE-Complete John E. Savage (Brown University) CSCI 1590 Intro to Computational Complexity February 11, 2009 2 / 10 Complexity Classes from Last Lecture P, coNP, p p Πi and Σi PH PSPACE NP ∪ coNP PSPACE = NPSPACE NP ∩ coNP NP coNP L2 P NL L John E. Savage (Brown University) CSCI 1590 Intro to Computational Complexity February 11, 2009 3 / 10 Polynomial Time Hierarchy A language is in NP(coNP) if and only if it can be reduced in polynomial time to a statement of the form ∃x b(x) (∀x b(x)) By adding additional levels of quantification, as shown below, potentially new complexity classes are added. ∀x1 ∃x2 b(x1, x2) ∃x1 ∀x2 b(x1, x2) The sets of languages PTIME reducible to statements of this form are p p denoted Πi and Σi respectively, when there are i alternations of existential and universal quantifiers and the outermost quantifier is ∀ and ∃, respectively. Definition The Polynomial Hierarchy (PH) is defined as p PH = [ Σi i John E. Savage (Brown University) CSCI 1590 Intro to Computational Complexity February 11, 2009 4 / 10 PH and PSPACE Definition A language L is PH-complete if a) L ∈ PH and b) all languages in PH are ptime reducible to L. It is not hard to see that PH ⊆ PSPACE.
    [Show full text]
  • 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008
    MIT OpenCourseWare http://ocw.mit.edu 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 6.080/6.089 GITCS DATE Lecture 12 Lecturer: Scott Aaronson Scribe: Mergen Nachin 1 Review of last lecture • NP-completeness in practice. We discussed many of the approaches people use to cope with NP-complete problems in real life. These include brute-force search (for small instance sizes), cleverly-optimized backtrack search, fixed-parameter algorithms, approximation algorithms, and heuristic algorithms like simulated annealing (which don’t always work but often do pretty well in practice). • Factoring, as an intermediate problem between P and NP-complete. We saw how factoring, though believed to be hard for classical computers, has a special structure that makes it different from any known NP-complete problem. • coNP. 2 Space complexity Now we will categorize problems in terms of how much memory they use. Definition 1 L is in PSPACE if there exists a poly-space Turing machine M such that for all x, x is in L if and only if M(x) accepts. Just like with NP, we can define PSPACE-hard and PSPACE-complete problems. An interesting example of a PSPACE-complete problem is n-by-n chess: Given an arrangement of pieces on an n-by-n chessboard, and assuming a polynomial upper bound on the number of moves, decide whether White has a winning strategy. (Note that we need to generalize chess to an n-by-n board, since standard 8-by-8 chess is a finite game that’s completely solvable in O(1) time.) Let’s now define another complexity class called EXP, which is apparently even bigger than PSPACE.
    [Show full text]
  • Complexity Classes
    CHAPTER Complexity Classes In an ideal world, each computational problem would be classified at least approximately by its use of computational resources. Unfortunately, our ability to so classify some important prob- lems is limited. We must be content to show that such problems fall into general complexity classes, such as the polynomial-time problems P, problems whose running time on a determin- istic Turing machine is a polynomial in the length of its input, or NP, the polynomial-time problems on nondeterministic Turing machines. Many complexity classes contain “complete problems,” problems that are hardest in the class. If the complexity of one complete problem is known, that of all complete problems is known. Thus, it is very useful to know that a problem is complete for a particular complexity class. For example, the class of NP-complete problems, the hardest problems in NP, contains many hundreds of important combinatorial problems such as the Traveling Salesperson Prob- lem. It is known that each NP-complete problem can be solved in time exponential in the size of the problem, but it is not known whether they can be solved in polynomial time. Whether ? P and NP are equal or not is known as the P = NP question. Decades of research have been devoted to this question without success. As a consequence, knowing that a problem is NP- complete is good evidence that it is an exponential-time problem. On the other hand, if one such problem were shown to be in P, all such problems would be been shown to be in P,a result that would be most important.
    [Show full text]
  • 1 Agenda 2 Provably Intractable Problems 3 Variants of Bounded
    CS 221: Computational Complexity Prof. Salil Vadhan Lecture Notes 8 February 19, 2010 Scribe: Kevin Lee 1 Agenda 1. Provably Intractable Problems 2 Provably Intractable Problems Some lower bounds that we proved so far: 1. Hierarchy theorems give us unnatural problems that do not exist in certain complexity classes. These give us "unnatural\ problems in EXPnDTIME(2n) or PSPACEnSPACE(n). 2. 9 > 0 such that TQBF 2= SPACE(n). 3. 8 > 0 we have that SAT 2= TISP(n1+o(1); n1−). We want to find some more natural problems that require superpolynomial time. We can do this by finding \natural" complete problems for classes at the exponential level (like EXP, NEXP, EXPSPACE). Often these will be \concise" versions of problems complete for classes at the polynomial level (like P, NP, PSPACE). We begin with a survey of some results of this type, and then work out one example in detail. 3 Variants of Bounded Halting To find some natural problems, we consider variations of the bounded halting problem: 1. M; x; 1t : M accepts x within t steps we know to be P-complete. 2. fhM; x; ti : M accepts x within t stepsg turns out to be EXP-complete. 3. M; 1t : 9x; M accepts x within t steps we know to be NP-complete. 4. fhM; ti : 9x; M accepts x within t stepsg turns out to be NEXP-complete. Thus by writing the time bound more concisely | in binary instead of unary | the complexity goes up an exponential. This phenomenon is like the converse of what we saw with translation/padding arguments.
    [Show full text]
  • Introduction to Interactive Proofs & the Sumcheck Protocol 1 Introduction
    CS294: Probabilistically Checkable and Interactive Proofs January 19, 2017 Introduction to Interactive Proofs & The Sumcheck Protocol Instructor: Alessandro Chiesa & Igor Shinkar Scribe: Pratyush Mishra 1 Introduction Traditional mathematical proofs are static objects: a prover P writes down a sequence of mathematical statements, and then at some later time a verifier V checks that these statements are consistent and correct. Over the years, computer science has changed the notion of a mathematical proof. The first such change was the observation that for all practical purposes, the verification procedure should be efficient; V should not have to expend large amounts of effort to verify the proof of a claim (at least much less than P expended to find the proof). This notion of “efficient verification” corresponds to the complexity class NP: Definition 1 A language L belongs to the class NP if and only if there exists an efficient algorithm V such that the following conditions hold. COMPLETENESS: For all x 2 L, there exists a proof π that makes V accept: V(x; π) = 1. SOUNDNESS: For all x 62 L, for all claimed proofs π∗, V rejects: V(x; π∗) = 0. Even though the verification procedure is now efficient, the proof is still a static object. Computer scientists in the 80’s and 90’s changed this view by introducing interaction and randomness into the mix: the prover and verifier were no longer required to be deterministic, and could now talk to each other. How did this change things? As we shall see, introducing interaction allows a computationally bounded (but randomized) verifier to check extraordinary claims efficiently.
    [Show full text]