Lecture 7: Nonuniformity 1 Time-Space Lower Bounds For
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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. -
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. -
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/. -
Algorithms and NP
Algorithms Complexity P vs NP Algorithms and NP G. Carl Evans University of Illinois Summer 2019 Algorithms and NP Algorithms Complexity P vs NP Review Algorithms Model algorithm complexity in terms of how much the cost increases as the input/parameter size increases In characterizing algorithm computational complexity, we care about Large inputs Dominant terms p 1 log n n n n log n n2 n3 2n 3n n! Algorithms and NP Algorithms Complexity P vs NP Closest Pair 01 closestpair(p1;:::; pn) : array of 2D points) 02 best1 = p1 03 best2 = p2 04 bestdist = dist(p1,p2) 05 for i = 1 to n 06 for j = 1 to n 07 newdist = dist(pi ,pj ) 08 if (i 6= j and newdist < bestdist) 09 best1 = pi 10 best2 = pj 11 bestdist = newdist 12 return (best1, best2) Algorithms and NP Algorithms Complexity P vs NP Mergesort 01 merge(L1,L2: sorted lists of real numbers) 02 if (L1 is empty and L2 is empty) 03 return emptylist 04 else if (L2 is empty or head(L1) <= head(L2)) 05 return cons(head(L1),merge(rest(L1),L2)) 06 else 07 return cons(head(L2),merge(L1,rest(L2))) 01 mergesort(L = a1; a2;:::; an: list of real numbers) 02 if (n = 1) then return L 03 else 04 m = bn=2c 05 L1 = (a1; a2;:::; am) 06 L2 = (am+1; am+2;:::; an) 07 return merge(mergesort(L1),mergesort(L2)) Algorithms and NP Algorithms Complexity P vs NP Find End 01 findend(A: array of numbers) 02 mylen = length(A) 03 if (mylen < 2) error 04 else if (A[0] = 0) error 05 else if (A[mylen-1] 6= 0) return mylen-1 06 else return findendrec(A,0,mylen-1) 11 findendrec(A, bottom, top: positive integers) 12 if (top -
CS 579: Computational Complexity. Lecture 2 Space Complexity
CS 579: Computational Complexity. Lecture 2 Space complexity. Alexandra Kolla Today Space Complexity, L,NL Configuration Graphs Log- Space Reductions NL Completeness, STCONN Savitch’s theorem SL Turing machines, briefly (3-tape) Turing machine M described by tuple (Γ,Q,δ), where ◦ Γ is “alphabet” . Contains start and blank symbol, 0,1,among others (constant size). ◦ Q is set of states, including designated starting state and halt state (constant size). ◦ Transition function δ:푄 × Γ3 → 푄 × Γ2 × {퐿, 푆, 푅}3 describing the rules M uses to move. Turing machines, briefly (3-tape) NON-DETERMINISTIC Turing machine M described by tuple (Γ,Q,δ0,δ1), where ◦ Γ is “alphabet” . Contains start and blank symbol, 0,1,among others (constant size). ◦ Q is set of states, including designated starting state and halt state (constant size). ◦ Two transition functions δ0, δ1 :푄 × Γ3 → 푄 × Γ2 × {퐿, 푆, 푅}3. At every step, TM makes non- deterministic choice which one to Space bounded turing machines Space-bounded turing machines used to study memory requirements of computational tasks. Definition. Let 푠: ℕ → ℕ and 퐿 ⊆ {0,1}∗. We say that L∈ SPACE(s(n)) if there is a constant c and a TM M deciding L s.t. at most c∙s(n) locations on M’s work tapes (excluding the input tape) are ever visited by M’s head during its computation on every input of length n. We will assume a single work tape and no output tape for simplicity. Similarly for NSPACE(s(n)), TM can only use c∙s(n) nonblank tape locations, regardless of its nondeterministic choices Space bounded turing machines Read-only “input” tape. -
Lecture 11 — October 16, 2012 1 Overview 2
6.841: Advanced Complexity Theory Fall 2012 Lecture 11 | October 16, 2012 Prof. Dana Moshkovitz Scribe: Hyuk Jun Kweon 1 Overview In the previous lecture, there was a question about whether or not true randomness exists in the universe. This question is still seriously debated by many physicists and philosophers. However, even if true randomness does not exist, it is possible to use pseudorandomness for practical purposes. In any case, randomization is a powerful tool in algorithms design, sometimes leading to more elegant or faster ways of solving problems than known deterministic methods. This naturally leads one to wonder whether randomization can solve problems outside of P, de- terministic polynomial time. One way of viewing randomized algorithms is that for each possible setting of the random bits used, a different strategy is pursued by the algorithm. A property of randomized algorithms that succeed with bounded error is that the vast majority of strategies suc- ceed. With error reduction, all but exponentially many strategies will succeed. A major theme of theoretical computer science over the last 20 years has been to turn algorithms with this property into algorithms that deterministically find a winning strategy. From the outset, this seems like a daunting task { without randomness, how can one efficiently search for such strategies? As we'll see in later lectures, this is indeed possible, modulo some hardness assumptions. There are two main objectives in this lecture. The first objective is to introduce some probabilis- tic complexity classes, by bounding time or space. These classes are probabilistic analogues of deterministic complexity classes P and L. -
User's Guide for Complexity: a LATEX Package, Version 0.80
User’s Guide for complexity: a LATEX package, Version 0.80 Chris Bourke April 12, 2007 Contents 1 Introduction 2 1.1 What is complexity? ......................... 2 1.2 Why a complexity package? ..................... 2 2 Installation 2 3 Package Options 3 3.1 Mode Options .............................. 3 3.2 Font Options .............................. 4 3.2.1 The small Option ....................... 4 4 Using the Package 6 4.1 Overridden Commands ......................... 6 4.2 Special Commands ........................... 6 4.3 Function Commands .......................... 6 4.4 Language Commands .......................... 7 4.5 Complete List of Class Commands .................. 8 5 Customization 15 5.1 Class Commands ............................ 15 1 5.2 Language Commands .......................... 16 5.3 Function Commands .......................... 17 6 Extended Example 17 7 Feedback 18 7.1 Acknowledgements ........................... 19 1 Introduction 1.1 What is complexity? complexity is a LATEX package that typesets computational complexity classes such as P (deterministic polynomial time) and NP (nondeterministic polynomial time) as well as sets (languages) such as SAT (satisfiability). In all, over 350 commands are defined for helping you to typeset Computational Complexity con- structs. 1.2 Why a complexity package? A better question is why not? Complexity theory is a more recent, though mature area of Theoretical Computer Science. Each researcher seems to have his or her own preferences as to how to typeset Complexity Classes and has built up their own personal LATEX commands file. This can be frustrating, to say the least, when it comes to collaborations or when one has to go through an entire series of files changing commands for compatibility or to get exactly the look they want (or what may be required). -
A Study of the NEXP Vs. P/Poly Problem and Its Variants by Barıs
A Study of the NEXP vs. P/poly Problem and Its Variants by Barı¸sAydınlıoglu˘ A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Computer Sciences) at the UNIVERSITY OF WISCONSIN–MADISON 2017 Date of final oral examination: August 15, 2017 This dissertation is approved by the following members of the Final Oral Committee: Eric Bach, Professor, Computer Sciences Jin-Yi Cai, Professor, Computer Sciences Shuchi Chawla, Associate Professor, Computer Sciences Loris D’Antoni, Asssistant Professor, Computer Sciences Joseph S. Miller, Professor, Mathematics © Copyright by Barı¸sAydınlıoglu˘ 2017 All Rights Reserved i To Azadeh ii acknowledgments I am grateful to my advisor Eric Bach, for taking me on as his student, for being a constant source of inspiration and guidance, for his patience, time, and for our collaboration in [9]. I have a story to tell about that last one, the paper [9]. It was a late Monday night, 9:46 PM to be exact, when I e-mailed Eric this: Subject: question Eric, I am attaching two lemmas. They seem simple enough. Do they seem plausible to you? Do you see a proof/counterexample? Five minutes past midnight, Eric responded, Subject: one down, one to go. I think the first result is just linear algebra. and proceeded to give a proof from The Book. I was ecstatic, though only for fifteen minutes because then he sent a counterexample refuting the other lemma. But a third lemma, inspired by his counterexample, tied everything together. All within three hours. On a Monday midnight. I only wish that I had asked to work with him sooner. -
A Short History of Computational Complexity
The Computational Complexity Column by Lance FORTNOW NEC Laboratories America 4 Independence Way, Princeton, NJ 08540, USA [email protected] http://www.neci.nj.nec.com/homepages/fortnow/beatcs Every third year the Conference on Computational Complexity is held in Europe and this summer the University of Aarhus (Denmark) will host the meeting July 7-10. More details at the conference web page http://www.computationalcomplexity.org This month we present a historical view of computational complexity written by Steve Homer and myself. This is a preliminary version of a chapter to be included in an upcoming North-Holland Handbook of the History of Mathematical Logic edited by Dirk van Dalen, John Dawson and Aki Kanamori. A Short History of Computational Complexity Lance Fortnow1 Steve Homer2 NEC Research Institute Computer Science Department 4 Independence Way Boston University Princeton, NJ 08540 111 Cummington Street Boston, MA 02215 1 Introduction It all started with a machine. In 1936, Turing developed his theoretical com- putational model. He based his model on how he perceived mathematicians think. As digital computers were developed in the 40's and 50's, the Turing machine proved itself as the right theoretical model for computation. Quickly though we discovered that the basic Turing machine model fails to account for the amount of time or memory needed by a computer, a critical issue today but even more so in those early days of computing. The key idea to measure time and space as a function of the length of the input came in the early 1960's by Hartmanis and Stearns. -
NP-Hardness of Circuit Minimization for Multi-Output Functions
Electronic Colloquium on Computational Complexity, Report No. 21 (2020) NP-Hardness of Circuit Minimization for Multi-Output Functions Rahul Ilango∗ Bruno Loffy Igor C. Oliveiraz MIT University of Porto University of Warwick February 21, 2020 Abstract Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP- complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive. In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : f0; 1gn ! f0; 1gm is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators. Complementing these results, we investigate the computational hardness of minimizing com- munication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless P = NP, no polynomial-time computable function can approx- imate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions. -
Randomized Computation Eugene Santos Looked at Computability for Probabilistic TM
Randomized Computation Eugene Santos looked at computability for Probabilistic TM. John Gill studied complexity classes defined by Probabilistic TM. 1. Eugene Santos. Probabilistic Turing Machines and Computability. Proc. American Mathematical Society, 22: 704-710, 1969. 2. Eugene Santos. Computability by Probabilistic Turing Machines. Trans. American Mathematical Society, 159: 165-184, 1971. 3. John Gill. Computational Complexity of Probabilistic Turing Machines. STOC, 91-95, 1974. 4. John Gill. Computational Complexity of Probabilistic Turing Machines. SIAM Journal Computing 6(4): 675-695, 1977. Computational Complexity, by Fu Yuxi Randomized Computation 1 / 108 Synopsis 1. Tail Distribution 2. Probabilistic Turing Machine 3. PP 4. BPP 5. ZPP 6. Random Walk and RL Computational Complexity, by Fu Yuxi Randomized Computation 2 / 108 Tail Distribution Computational Complexity, by Fu Yuxi Randomized Computation 3 / 108 Markov's Inequality For all k > 0, 1 Pr[X ≥ kE[X ]] ≤ ; k or equivalently E[X ] Pr[X ≥ v] ≤ : v I Observe that d · Pr[X ≥ d] ≤ E[X ]. I We are done by letting d = kE[X ]. Computational Complexity, by Fu Yuxi Randomized Computation 4 / 108 Moment and Variance Information about a random variable is often expressed in terms of moments. k I The k-th moment of a random variable X is E[X ]. The variance of a random variable X is Var(X ) = E[(X − E[X ])2] = E[X 2] − E[X ]2: The standard deviation of X is σ(X ) = pVar(X ): Fact. If X1;:::; Xn are pairwise independent, then n n X X Var( Xi ) = Var(Xi ): i=1 i=1 Computational Complexity, by Fu Yuxi Randomized Computation 5 / 108 Chebyshev Inequality For all k > 0, 1 Pr[jX − E[X ]j ≥ kσ] ≤ ; k2 or equivalently σ2 Pr[jX − E[X ]j ≥ k] ≤ : k2 Apply Markov's Inequality to the random variable (X − E[X ])2. -
A Perspective for Logically Safe, Extensible, Powerful and Interactive Formal Method Tools
Plugins for the Isabelle Platform: A Perspective for Logically Safe, Extensible, Powerful and Interactive Formal Method Tools Burkhart Wolff Université Paris-Sud (Technical Advice by: M akarius Wenzel, Université Paris-Sud) What I am not Talking About What I am not Talking About Isabelle as: “Proof - Assistent” or “Theorem Prover” What I will Talk About Isabelle as: Formal Methods Tool Framework What I will Talk About Isabelle as: Formal Methods Tool Framework “The ECLIPSE of FM - Tools” Overview ● Three Histories Overview ● Three Histories ● Evolution of the ITP Programme and Evolution of the Isabelle - Architecture ● Evolution of Isabelle - LCF - Kernels ● Evolution of Tools built upon Isabelle The ITP Research Programme and The Evolution of the Isabelle/Architecture The “Interactive Proof” Research Programme ● 1969 : Automath ● 1975 : Stanford LCF LISP based Goal-Stack, orientation vs. functional Programming, orientation vs. embedding in interactive sh. Invention: Parametric Polymorphism, Shallow Types The “Interactive Proof” Research Programme ● 1980 : Edinburgh LCF ML implementation and toplevel shell, tactic programming via conversions over goal stack, adm meta-program, Invention: thm, goal-stack transforms thms Invention: Basic Embedding Techniques The “Interactive Proof” Research Programme ● 1985 : HOL4, Isabelle, Coq Further search to more foundational and logically safe systems lead to abandon of LCF; HOL became replacement. Invention: Coq: Dependent types, executable “proofs”, proofobjects Invention: HOL: recursion embeddable,