Some Hardness Escalation Results in Computational Complexity Theory Pritish Kamath
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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. -
Automating Cutting Planes Is NP-Hard
Automating Cutting Planes is NP-Hard Mika G¨o¨os† Sajin Koroth Ian Mertz Toniann Pitassi Stanford University Simon Fraser University Uni. of Toronto Uni. of Toronto & IAS April 20, 2020 Abstract We show that Cutting Planes (CP) proofs are hard to find: Given an unsatisfiable formula F , (1) it is NP-hard to find a CP refutation of F in time polynomial in the length of the shortest such refutation; and (2) unless Gap-Hitting-Set admits a nontrivial algorithm, one cannot find a tree-like CP refutation of F in time polynomial in the length of the shortest such refutation. The first result extends the recent breakthrough of Atserias and M¨uller (FOCS 2019) that established an analogous result for Resolution. Our proofs rely on two new lifting theorems: (1) Dag-like lifting for gadgets with many output bits. (2) Tree-like lifting that simulates an r-round protocol with gadgets of query complexity O(log r) independent of input length. Contents 1 Introduction1 4.1 Subcubes from simplices . .9 1.1 Cutting Planes . .1 4.2 Simplified proof . 11 1.2 Dag-like result . .2 4.3 Accounting for error . 12 1.3 Tree-like result . .2 5 Tree-like definitions 13 2 Overview of proofs3 5.1 Tree-like dags/proofs . 13 2.1 Dag-like case . .3 5.2 Real protocols . 14 arXiv:2004.08037v1 [cs.CC] 17 Apr 2020 2.2 Tree-like case . .4 6 Tree-like lifting 14 3 Dag-like definitions6 6.1 Proof of real lifting . 14 3.1 Standard models . -
Arxiv:Quant-Ph/0202111V1 20 Feb 2002
Quantum statistical zero-knowledge John Watrous Department of Computer Science University of Calgary Calgary, Alberta, Canada [email protected] February 19, 2002 Abstract In this paper we propose a definition for (honest verifier) quantum statistical zero-knowledge interactive proof systems and study the resulting complexity class, which we denote QSZK. We prove several facts regarding this class: The following natural problem is a complete promise problem for QSZK: given instructions • for preparing two mixed quantum states, are the states close together or far apart in the trace norm metric? By instructions for preparing a mixed quantum state we mean the description of a quantum circuit that produces the mixed state on some specified subset of its qubits, assuming all qubits are initially in the 0 state. This problem is a quantum generalization of the complete promise problem of Sahai| i and Vadhan [33] for (classical) statistical zero-knowledge. QSZK is closed under complement. • QSZK PSPACE. (At present it is not known if arbitrary quantum interactive proof • systems⊆ can be simulated in PSPACE, even for one-round proof systems.) Any honest verifier quantum statistical zero-knowledge proof system can be parallelized to • a two-message (i.e., one-round) honest verifier quantum statistical zero-knowledge proof arXiv:quant-ph/0202111v1 20 Feb 2002 system. (For arbitrary quantum interactive proof systems it is known how to parallelize to three messages, but not two.) Moreover, the one-round proof system can be taken to be such that the prover sends only one qubit to the verifier in order to achieve completeness and soundness error exponentially close to 0 and 1/2, respectively. -
Four Results of Jon Kleinberg a Talk for St.Petersburg Mathematical Society
Four Results of Jon Kleinberg A Talk for St.Petersburg Mathematical Society Yury Lifshits Steklov Institute of Mathematics at St.Petersburg May 2007 1 / 43 2 Hubs and Authorities 3 Nearest Neighbors: Faster Than Brute Force 4 Navigation in a Small World 5 Bursty Structure in Streams Outline 1 Nevanlinna Prize for Jon Kleinberg History of Nevanlinna Prize Who is Jon Kleinberg 2 / 43 3 Nearest Neighbors: Faster Than Brute Force 4 Navigation in a Small World 5 Bursty Structure in Streams Outline 1 Nevanlinna Prize for Jon Kleinberg History of Nevanlinna Prize Who is Jon Kleinberg 2 Hubs and Authorities 2 / 43 4 Navigation in a Small World 5 Bursty Structure in Streams Outline 1 Nevanlinna Prize for Jon Kleinberg History of Nevanlinna Prize Who is Jon Kleinberg 2 Hubs and Authorities 3 Nearest Neighbors: Faster Than Brute Force 2 / 43 5 Bursty Structure in Streams Outline 1 Nevanlinna Prize for Jon Kleinberg History of Nevanlinna Prize Who is Jon Kleinberg 2 Hubs and Authorities 3 Nearest Neighbors: Faster Than Brute Force 4 Navigation in a Small World 2 / 43 Outline 1 Nevanlinna Prize for Jon Kleinberg History of Nevanlinna Prize Who is Jon Kleinberg 2 Hubs and Authorities 3 Nearest Neighbors: Faster Than Brute Force 4 Navigation in a Small World 5 Bursty Structure in Streams 2 / 43 Part I History of Nevanlinna Prize Career of Jon Kleinberg 3 / 43 Nevanlinna Prize The Rolf Nevanlinna Prize is awarded once every 4 years at the International Congress of Mathematicians, for outstanding contributions in Mathematical Aspects of Information Sciences including: 1 All mathematical aspects of computer science, including complexity theory, logic of programming languages, analysis of algorithms, cryptography, computer vision, pattern recognition, information processing and modelling of intelligence. -
Prahladh Harsha
Prahladh Harsha Toyota Technological Institute, Chicago phone : +1-773-834-2549 University Press Building fax : +1-773-834-9881 1427 East 60th Street, Second Floor Chicago, IL 60637 email : [email protected] http://ttic.uchicago.edu/∼prahladh Research Interests Computational Complexity, Probabilistically Checkable Proofs (PCPs), Information Theory, Prop- erty Testing, Proof Complexity, Communication Complexity. Education • Doctor of Philosophy (PhD) Computer Science, Massachusetts Institute of Technology, 2004 Research Advisor : Professor Madhu Sudan PhD Thesis: Robust PCPs of Proximity and Shorter PCPs • Master of Science (SM) Computer Science, Massachusetts Institute of Technology, 2000 • Bachelor of Technology (BTech) Computer Science and Engineering, Indian Institute of Technology, Madras, 1998 Work Experience • Toyota Technological Institute, Chicago September 2004 – Present Research Assistant Professor • Technion, Israel Institute of Technology, Haifa February 2007 – May 2007 Visiting Scientist • Microsoft Research, Silicon Valley January 2005 – September 2005 Postdoctoral Researcher Honours and Awards • Summer Research Fellow 1997, Jawaharlal Nehru Center for Advanced Scientific Research, Ban- galore. • Rajiv Gandhi Science Talent Research Fellow 1997, Jawaharlal Nehru Center for Advanced Scien- tific Research, Bangalore. • Award Winner in the Indian National Mathematical Olympiad (INMO) 1993, National Board of Higher Mathematics (NBHM). • National Board of Higher Mathematics (NBHM) Nurture Program award 1995-1998. The Nurture program 1995-1998, coordinated by Prof. Alladi Sitaram, Indian Statistical Institute, Bangalore involves various topics in higher mathematics. 1 • Ranked 7th in the All India Joint Entrance Examination (JEE) for admission into the Indian Institutes of Technology (among the 100,000 candidates who appeared for the examination). • Papers invited to special issues – “Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding” (with Eli Ben- Sasson, Oded Goldreich, Madhu Sudan, and Salil Vadhan). -
Spring 2020 (Provide Proofs for All Answers) (120 Points Total)
Complexity Qual Spring 2020 (provide proofs for all answers) (120 Points Total) 1 Problem 1: Formal Languages (30 Points Total): For each of the following languages, state and prove whether: (i) the lan- guage is regular, (ii) the language is context-free but not regular, or (iii) the language is non-context-free. n n Problem 1a (10 Points): La = f0 1 jn ≥ 0g (that is, the set of all binary strings consisting of 0's and then 1's where the number of 0's and 1's are equal). ∗ Problem 1b (10 Points): Lb = fw 2 f0; 1g jw has an even number of 0's and at least two 1'sg (that is, the set of all binary strings with an even number, including zero, of 0's and at least two 1's). n 2n 3n Problem 1c (10 Points): Lc = f0 #0 #0 jn ≥ 0g (that is, the set of all strings of the form 0 repeated n times, followed by the # symbol, followed by 0 repeated 2n times, followed by the # symbol, followed by 0 repeated 3n times). Problem 2: NP-Hardness (30 Points Total): There are a set of n people who are the vertices of an undirected graph G. There's an undirected edge between two people if they are enemies. Problem 2a (15 Points): The people must be assigned to the seats around a single large circular table. There are exactly as many seats as people (both n). We would like to make sure that nobody ever sits next to one of their enemies. -
Computational Complexity Theory
Computational Complexity Theory Markus Bl¨aser Universit¨atdes Saarlandes Draft|February 5, 2012 and forever 2 1 Simple lower bounds and gaps Lower bounds The hierarchy theorems of the previous chapter assure that there is, e.g., a language L 2 DTime(n6) that is not in DTime(n3). But this language is not natural.a But, for instance, we do not know how to show that 3SAT 2= DTime(n3). (Even worse, we do not know whether this is true.) The best we can show is that 3SAT cannot be decided by a O(n1:81) time bounded and simultaneously no(1) space bounded deterministic Turing machine. aThis, of course, depends on your interpretation of \natural" . In this chapter, we prove some simple lower bounds. The bounds in this section will be shown for natural problems. Furthermore, these bounds are unconditional. While showing the NP-hardness of some problem can be viewed as a lower bound, this bound relies on the assumption that P 6= NP. However, the bounds in this chapter will be rather weak. 1.1 A logarithmic space bound n n Let LEN = fa b j n 2 Ng. LEN is the language of all words that consists of a sequence of as followed by a sequence of b of equal length. This language is one of the examples for a context-free language that is not regular. We will show that LEN can be decided with logarithmic space and that this amount of space is also necessary. The first part is easy. Exercise 1.1 Prove: LEN 2 DSpace(log). -
IAN MERTZ Contact Research Theory Group [email protected] Department of Computer Science University of Toronto
IAN MERTZ Contact Research www.cs.toronto.edu/∼mertz Theory Group [email protected] Department of Computer Science University of Toronto EDUCATION University of Toronto, Toronto, ON ................................................................... Winter 2018- Ph.D. in Computer Science Advisor: Toniann Pitassi University of Toronto, Toronto, ON .......................................................... Fall 2016-Winter 2018 Masters in Science, Computer Science Advisor: Toniann Pitassi, Stephen Cook Rutgers University, Piscataway, NJ ........................................................... Fall 2012-Spring 2016 B.S. in Computer Science, B.A. in Mathematics, B.A. in East Asian Language Studies (concentration in Japanese) Summa cum laude, honors program, Dean's List PUBLICATIONS Lifting is as easy as 1,2,3 Ian Mertz, Toniann Pitassi In submission, STOC '21. Automating Cutting Planes is NP-hard Mika G¨o¨os,Sajin Koroth, Ian Mertz, Toniann Pitassi In Proc. 52nd ACM Symposium on Theory of Computing (STOC '20), Association for Computing Machinery (ACM), 84 (132) pp. 68-77, 2020. Catalytic Approaches to the Tree Evaluation Problem James Cook, Ian Mertz In Proc. 52nd ACM Symposium on Theory of Computing (STOC '20), Association for Computing Machinery (ACM), 84 (132) pp. 752-760, 2020. Short Proofs Are Hard to Find Ian Mertz, Toniann Pitassi, Yuanhao Wei In Proc. 46th International Colloquium on Automata, Languages and Programming (ICALP '19), Leibniz International Proceedings in Informatics (LIPIcs), 84 (132) pp. 1-16, 2019. Dual VP Classes Eric Allender, Anna G´al,Ian Mertz computational complexity, 25 pp. 1-43, 2016. An earlier version appeared in Proc. 40th International Symposium on Mathematical Foundations of Computer Science (MFCS '15), Lecture Notes in Computer Science, 9235 pp. 14-25, 2015. -
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. -
DTIME, NTIME, DSPACE, Measured on Multi-Tape Turing Machines. Th
CMPSCI 601: Recall From Last Time Lecture 9 Def: DTIME, NTIME, DSPACE, measured on Multi-tape Turing Machines. ¡¤£¥§ © £¡¤£¥§§© Th: DTIME ¢¡¤£¦¥¨§ © RAM-TIME DTIME ¥© L DSPACE + ¥! #"%$'&(© ¥ © ) * P DTIME +-, DTIME $ + #"%$'& ¥ © ¥ © * ) NP NTIME +-, NTIME $ + ¥! ."/$ &0© ¢¥ © ) * PSPACE DSPACE +-, DSPACE $ ¥43657£¦¥§81 ¥ Th: For ¡¤£¥§21 , ¢¡¤£¦¥§:© ¡9£¦¥§ © DTIME ¡9£¦¥§ © NTIME DSPACE #">"@?9&%& =< © DSPACE ;57£¦¥§:© DTIME Cor: L P NP PSPACE 1 CMPSCI 601: NTIME and NP Lecture 9 ¢¡¤£¦¥¨§ © £¦¡¤£¦¥§§ NTIME probs. accepted by NTMs in time + ."/$ & ¢¥ © ¥ © ) * NP NTIME +-, NTIME $ Theorem 9.1 For any function ¡¤£¦¥§ , ¢¡¤£¦¥§:© ¡9£¦¥§ © DTIME ¡9£¦¥§ © NTIME DSPACE Proof: The first inclusion is obvious. For the second, £¡¤£¥§§ note that in space we can simulate all computa- £¡¤£¦¥¨§§ tions of length , so we will find the shortest ac- cepting one if it exists. ¡ =< #"£¢ " ?9&%&(© Recall: DSPACE ¡¤£¥§ © DTIME Corollary 9.2 L P NP PSPACE 2 So we can simulate NTM’s by DTM’s, at the cost of an exponential increase in the running time. It may be pos- sible to improve this simulation, though no essentially better one is known. If the cost could be reduced to poly- nomial, we would have that P NP. There is probably such a quantitative difference between the power of NTM’s and DTM’s. But note that qualita- tively there is no difference. If ¡ is the language of some NTM ¢ , it must also be r.e. because there is a DTM that £ searches through all computations of ¢ on , first of one ¡ ¤ ¥ step, then of two steps, and so on. If £ , will eventually find an accepting computation. If not, it will search forever. What about an NTM-based definition of “recursive” or “Turing-decidable” sets? This is less clear because NTM’s don’t decide – they just have a range of possible actions. -
Query-To-Communication Lifting for PNP∗†
Query-to-Communication Lifting for PNP∗† Mika Göös1, Pritish Kamath2, Toniann Pitassi3, and Thomas Watson4 1 Harvard University, Cambridge, MA, USA [email protected] 2 Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 3 University of Toronto, Toronto, ON, Canada [email protected] 4 University of Memphis, Memphis, TN, USA [email protected] Abstract We prove that the PNP-type query complexity (alternatively, decision list width) of any boolean function f is quadratically related to the PNP-type communication complexity of a lifted version of f. As an application, we show that a certain “product” lower bound method of Impagliazzo and Williams (CCC 2010) fails to capture PNP communication complexity up to polynomial factors, which answers a question of Papakonstantinou, Scheder, and Song (CCC 2014). 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes Keywords and phrases Communication Complexity, Query Complexity, Lifting Theorem, PNP Digital Object Identifier 10.4230/LIPIcs.CCC.2017.12 1 Introduction Broadly speaking, a query-to-communication lifting theorem (a.k.a. communication- to-query simulation theorem) translates, in a black-box fashion, lower bounds on some type of query complexity (a.k.a. decision tree complexity) [38, 6, 19] of a boolean function f : {0, 1}n → {0, 1} into lower bounds on a corresponding type of communication complexity [23, 19, 27] of a two-party version of f. Table 1 lists several known results in this vein. In this work, we provide a lifting theorem for PNP-type query/communication complexity. PNP decision trees. Recall that a deterministic (i.e., P-type) decision tree computes an n-bit boolean function f by repeatedly querying, at unit cost, individual bits xi ∈ {0, 1} of the input x until the value f(x) is output at a leaf of the tree.