Probabilistic Proof Systems: a Primer
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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”. -
Interactive Proof Systems and Alternating Time-Space Complexity
Theoretical Computer Science 113 (1993) 55-73 55 Elsevier Interactive proof systems and alternating time-space complexity Lance Fortnow” and Carsten Lund** Department of Computer Science, Unicersity of Chicago. 1100 E. 58th Street, Chicago, IL 40637, USA Abstract Fortnow, L. and C. Lund, Interactive proof systems and alternating time-space complexity, Theoretical Computer Science 113 (1993) 55-73. We show a rough equivalence between alternating time-space complexity and a public-coin interactive proof system with the verifier having a polynomial-related time-space complexity. Special cases include the following: . All of NC has interactive proofs, with a log-space polynomial-time public-coin verifier vastly improving the best previous lower bound of LOGCFL for this model (Fortnow and Sipser, 1988). All languages in P have interactive proofs with a polynomial-time public-coin verifier using o(log’ n) space. l All exponential-time languages have interactive proof systems with public-coin polynomial-space exponential-time verifiers. To achieve better bounds, we show how to reduce a k-tape alternating Turing machine to a l-tape alternating Turing machine with only a constant factor increase in time and space. 1. Introduction In 1981, Chandra et al. [4] introduced alternating Turing machines, an extension of nondeterministic computation where the Turing machine can make both existential and universal moves. In 1985, Goldwasser et al. [lo] and Babai [l] introduced interactive proof systems, an extension of nondeterministic computation consisting of two players, an infinitely powerful prover and a probabilistic polynomial-time verifier. The prover will try to convince the verifier of the validity of some statement. -
If Np Languages Are Hard on the Worst-Case, Then It Is Easy to Find Their Hard Instances
IF NP LANGUAGES ARE HARD ON THE WORST-CASE, THEN IT IS EASY TO FIND THEIR HARD INSTANCES Dan Gutfreund, Ronen Shaltiel, and Amnon Ta-Shma Abstract. We prove that if NP 6⊆ BPP, i.e., if SAT is worst-case hard, then for every probabilistic polynomial-time algorithm trying to decide SAT, there exists some polynomially samplable distribution that is hard for it. That is, the algorithm often errs on inputs from this distribution. This is the ¯rst worst-case to average-case reduction for NP of any kind. We stress however, that this does not mean that there exists one ¯xed samplable distribution that is hard for all probabilistic polynomial-time algorithms, which is a pre-requisite assumption needed for one-way func- tions and cryptography (even if not a su±cient assumption). Neverthe- less, we do show that there is a ¯xed distribution on instances of NP- complete languages, that is samplable in quasi-polynomial time and is hard for all probabilistic polynomial-time algorithms (unless NP is easy in the worst case). Our results are based on the following lemma that may be of independent interest: Given the description of an e±cient (probabilistic) algorithm that fails to solve SAT in the worst case, we can e±ciently generate at most three Boolean formulae (of increasing lengths) such that the algorithm errs on at least one of them. Keywords. Average-case complexity, Worst-case to average-case re- ductions, Foundations of cryptography, Pseudo classes Subject classi¯cation. 68Q10 (Modes of computation (nondetermin- istic, parallel, interactive, probabilistic, etc.) 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 68Q17 Compu- tational di±culty of problems (lower bounds, completeness, di±culty of approximation, etc.) 94A60 Cryptography 2 Gutfreund, Shaltiel & Ta-Shma 1. -
On the Randomness Complexity of Interactive Proofs and Statistical Zero-Knowledge Proofs*
On the Randomness Complexity of Interactive Proofs and Statistical Zero-Knowledge Proofs* Benny Applebaum† Eyal Golombek* Abstract We study the randomness complexity of interactive proofs and zero-knowledge proofs. In particular, we ask whether it is possible to reduce the randomness complexity, R, of the verifier to be comparable with the number of bits, CV , that the verifier sends during the interaction. We show that such randomness sparsification is possible in several settings. Specifically, unconditional sparsification can be obtained in the non-uniform setting (where the verifier is modelled as a circuit), and in the uniform setting where the parties have access to a (reusable) common-random-string (CRS). We further show that constant-round uniform protocols can be sparsified without a CRS under a plausible worst-case complexity-theoretic assumption that was used previously in the context of derandomization. All the above sparsification results preserve statistical-zero knowledge provided that this property holds against a cheating verifier. We further show that randomness sparsification can be applied to honest-verifier statistical zero-knowledge (HVSZK) proofs at the expense of increasing the communica- tion from the prover by R−F bits, or, in the case of honest-verifier perfect zero-knowledge (HVPZK) by slowing down the simulation by a factor of 2R−F . Here F is a new measure of accessible bit complexity of an HVZK proof system that ranges from 0 to R, where a maximal grade of R is achieved when zero- knowledge holds against a “semi-malicious” verifier that maliciously selects its random tape and then plays honestly. -
Ab Pcpchap.Pdf
i Computational Complexity: A Modern Approach Sanjeev Arora and Boaz Barak Princeton University http://www.cs.princeton.edu/theory/complexity/ [email protected] Not to be reproduced or distributed without the authors’ permission ii Chapter 11 PCP Theorem and Hardness of Approximation: An introduction “...most problem reductions do not create or preserve such gaps...To create such a gap in the generic reduction (cf. Cook)...also seems doubtful. The in- tuitive reason is that computation is an inherently unstable, non-robust math- ematical object, in the the sense that it can be turned from non-accepting to accepting by changes that would be insignificant in any reasonable metric.” Papadimitriou and Yannakakis [PY88] This chapter describes the PCP Theorem, a surprising discovery of complexity theory, with many implications to algorithm design. Since the discovery of NP-completeness in 1972 researchers had mulled over the issue of whether we can efficiently compute approxi- mate solutions to NP-hard optimization problems. They failed to design such approxima- tion algorithms for most problems (see Section 1.1 for an introduction to approximation algorithms). They then tried to show that computing approximate solutions is also hard, but apart from a few isolated successes this effort also stalled. Researchers slowly began to realize that the Cook-Levin-Karp style reductions do not suffice to prove any limits on ap- proximation algorithms (see the above quote from an influential Papadimitriou-Yannakakis paper that appeared a few years before the discoveries described in this chapter). The PCP theorem, discovered in 1992, gave a new definition of NP and provided a new starting point for reductions. -
The Complexity of Space Bounded Interactive Proof Systems
The Complexity of Space Bounded Interactive Proof Systems ANNE CONDON Computer Science Department, University of Wisconsin-Madison 1 INTRODUCTION Some of the most exciting developments in complexity theory in recent years concern the complexity of interactive proof systems, defined by Goldwasser, Micali and Rackoff (1985) and independently by Babai (1985). In this paper, we survey results on the complexity of space bounded interactive proof systems and their applications. An early motivation for the study of interactive proof systems was to extend the notion of NP as the class of problems with efficient \proofs of membership". Informally, a prover can convince a verifier in polynomial time that a string is in an NP language, by presenting a witness of that fact to the verifier. Suppose that the power of the verifier is extended so that it can flip coins and can interact with the prover during the course of a proof. In this way, a verifier can gather statistical evidence that an input is in a language. As we will see, the interactive proof system model precisely captures this in- teraction between a prover P and a verifier V . In the model, the computation of V is probabilistic, but is typically restricted in time or space. A language is accepted by the interactive proof system if, for all inputs in the language, V accepts with high probability, based on the communication with the \honest" prover P . However, on inputs not in the language, V rejects with high prob- ability, even when communicating with a \dishonest" prover. In the general model, V can keep its coin flips secret from the prover. -
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. -
Computational Complexity: a Modern Approach
i Computational Complexity: A Modern Approach Draft of a book: Dated January 2007 Comments welcome! Sanjeev Arora and Boaz Barak Princeton University [email protected] Not to be reproduced or distributed without the authors’ permission This is an Internet draft. Some chapters are more finished than others. References and attributions are very preliminary and we apologize in advance for any omissions (but hope you will nevertheless point them out to us). Please send us bugs, typos, missing references or general comments to [email protected] — Thank You!! DRAFT ii DRAFT Chapter 9 Complexity of counting “It is an empirical fact that for many combinatorial problems the detection of the existence of a solution is easy, yet no computationally efficient method is known for counting their number.... for a variety of problems this phenomenon can be explained.” L. Valiant 1979 The class NP captures the difficulty of finding certificates. However, in many contexts, one is interested not just in a single certificate, but actually counting the number of certificates. This chapter studies #P, (pronounced “sharp p”), a complexity class that captures this notion. Counting problems arise in diverse fields, often in situations having to do with estimations of probability. Examples include statistical estimation, statistical physics, network design, and more. Counting problems are also studied in a field of mathematics called enumerative combinatorics, which tries to obtain closed-form mathematical expressions for counting problems. To give an example, in the 19th century Kirchoff showed how to count the number of spanning trees in a graph using a simple determinant computation. Results in this chapter will show that for many natural counting problems, such efficiently computable expressions are unlikely to exist. -
Lecture 14 (Feb 28): Probabilistically Checkable Proofs (PCP) 14.1 Motivation: Some Problems and Their Approximation Algorithms
CMPUT 675: Computational Complexity Theory Winter 2019 Lecture 14 (Feb 28): Probabilistically Checkable Proofs (PCP) Lecturer: Zachary Friggstad Scribe: Haozhou Pang 14.1 Motivation: Some problems and their approximation algorithms Many optimization problems are NP-hard, and therefore it is unlikely to efficiently find optimal solutions. However, in some situations, finding a provably good-enough solution (approximation of the optimal) is also acceptable. We say an algorithm is an α-approximation algorithm for an optimization problem iff for every instance of the problem it can find a solution whose cost is a factor of α of the optimum solution cost. Here are some selected problems and their approximation algorithms: Example: Min Vertex Cover is the problem that given a graph G = (V; E), the goal is to find a vertex cover C ⊆ V of minimum size. The following algorithm gives a 2-approximation for Min Vertex Cover: Algorithm 1 Min Vertex Cover Algorithm Input: Graph G = (V; E) Output: a vertex cover C of G. C ; while some edge (u; v) has u; v2 = C do C C [ fu; vg end while return C Claim 1 Let C∗ be an optimal solution, the returned solution C satisfies jCj ≤ 2jC∗j. Proof. Let M be the edges considered by the algorithm in the loop, we have jCj = 2jMj. Also, jC∗j covers M and no two edges in M share an endpoint (M is a matching), so jC∗j ≥ jMj. Therefore, jCj = 2jMj ≤ 2jC∗j. Example: Max SAT is the problem that given a CNF formula, the goal is to find a assignment that satisfies as many clauses as possible. -
The Complexity Zoo
The Complexity Zoo Scott Aaronson www.ScottAaronson.com LATEX Translation by Chris Bourke [email protected] 417 classes and counting 1 Contents 1 About This Document 3 2 Introductory Essay 4 2.1 Recommended Further Reading ......................... 4 2.2 Other Theory Compendia ............................ 5 2.3 Errors? ....................................... 5 3 Pronunciation Guide 6 4 Complexity Classes 10 5 Special Zoo Exhibit: Classes of Quantum States and Probability Distribu- tions 110 6 Acknowledgements 116 7 Bibliography 117 2 1 About This Document What is this? Well its a PDF version of the website www.ComplexityZoo.com typeset in LATEX using the complexity package. Well, what’s that? The original Complexity Zoo is a website created by Scott Aaronson which contains a (more or less) comprehensive list of Complexity Classes studied in the area of theoretical computer science known as Computa- tional Complexity. I took on the (mostly painless, thank god for regular expressions) task of translating the Zoo’s HTML code to LATEX for two reasons. First, as a regular Zoo patron, I thought, “what better way to honor such an endeavor than to spruce up the cages a bit and typeset them all in beautiful LATEX.” Second, I thought it would be a perfect project to develop complexity, a LATEX pack- age I’ve created that defines commands to typeset (almost) all of the complexity classes you’ll find here (along with some handy options that allow you to conveniently change the fonts with a single option parameters). To get the package, visit my own home page at http://www.cse.unl.edu/~cbourke/. -
Lecture 9 1 Interactive Proof Systems/Protocols
CS 282A/MATH 209A: Foundations of Cryptography Prof. Rafail Ostrovsky Lecture 9 Lecture date: March 7-9, 2005 Scribe: S. Bhattacharyya, R. Deak, P. Mirzadeh 1 Interactive Proof Systems/Protocols 1.1 Introduction The traditional mathematical notion of a proof is a simple passive protocol in which a prover P outputs a complete proof to a verifier V who decides on its validity. The interaction in this traditional sense is minimal and one-way, prover → verifier. The observation has been made that allowing the verifier to interact with the prover can have advantages, for example proving the assertion faster or proving more expressive languages. This extension allows for the idea of interactive proof systems (protocols). The general framework of the interactive proof system (protocol) involves a prover P with an exponential amount of time (computationally unbounded) and a verifier V with a polyno- mial amount of time. Both P and V exchange multiple messages (challenges and responses), usually dependent upon outcomes of fair coin tosses which they may or may not share. It is easy to see that since V is a poly-time machine (PPT), only a polynomial number of messages may be exchanged between the two. P ’s objective is to convince (prove to) the verifier the truth of an assertion, e.g., claimed knowledge of a proof that x ∈ L. V either accepts or rejects the interaction with the P . 1.2 Definition of Interactive Proof Systems An interactive proof system for a language L is a protocol PV for communication between a computationally unbounded (exponential time) machine P and a probabilistic poly-time (PPT) machine V such that the protocol satisfies the properties of completeness and sound- ness. -
Lecture 19,20 (Nov 15&17, 2011): Hardness of Approximation, PCP
,20 CMPUT 675: Approximation Algorithms Fall 2011 Lecture 19,20 (Nov 15&17, 2011): Hardness of Approximation, PCP theorem Lecturer: Mohammad R. Salavatipour Scribe: based on older notes 19.1 Hardness of Approximation So far we have been mostly talking about designing approximation algorithms and proving upper bounds. From no until the end of the course we will be talking about proving lower bounds (i.e. hardness of approximation). We are familiar with the theory of NP-completeness. When we prove that a problem is NP-hard it implies that, assuming P=NP there is no polynomail time algorithm that solves the problem (exactly). For example, for SAT, deciding between Yes/No is hard (again assuming P=NP). We would like to show that even deciding between those instances that are (almost) satisfiable and those that are far from being satisfiable is also hard. In other words, create a gap between Yes instances and No instances. These kinds of gaps imply hardness of approximation for optimization version of NP-hard problems. In fact the PCP theorem (that we will see next lecture) is equivalent to the statement that MAX-SAT is NP- hard to approximate within a factor of (1 + ǫ), for some fixed ǫ> 0. Most of hardness of approximation results rely on PCP theorem. For proving a hardness, for example for vertex cover, PCP shows that the existence of following reduction: Given a formula ϕ for SAT, we build a graph G(V, E) in polytime such that: 2 • if ϕ is a yes-instance, then G has a vertex cover of size ≤ 3 |V |; 2 • if ϕ is a no-instance, then every vertex cover of G has a size > α 3 |V | for some fixed α> 1.