Universal First-Order Logic Is Superfluous for Nl, P, Np and Co Np
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Infinitary Logic and Inductive Definability Over Finite Structures
University of Pennsylvania ScholarlyCommons Technical Reports (CIS) Department of Computer & Information Science November 1991 Infinitary Logic and Inductive Definability Over Finite Structures Anuj Dawar University of Pennsylvania Steven Lindell University of Pennsylvania Scott Weinstein University of Pennsylvania Follow this and additional works at: https://repository.upenn.edu/cis_reports Recommended Citation Anuj Dawar, Steven Lindell, and Scott Weinstein, "Infinitary Logic and Inductive Definability Over Finite Structures", . November 1991. University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-91-97. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_reports/365 For more information, please contact [email protected]. Infinitary Logic and Inductive Definability Over Finite Structures Abstract The extensions of first-order logic with a least fixed point operators (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [AV91b] investigated the relation of these two logics in the absence of an ordering, using a mchine model of generic computation. In particular, they showed that the two languages have equivalent expressive power if and only if P = PSPACE. These languages can also be seen as fragments of an infinitary logic where each ω formula has a bounded number of variables, L ∞ω (see, for instance, [KV90]). We present a treatment of the results in [AV91b] from this point of view. In particular, we show that we can write a formula of FO + LFP and P from ordered structures to classes of structures where every element is definable. -
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/. -
Canonical Models and the Complexity of Modal Team Logic
On the Complexity of Team Logic and its Two-Variable Fragment Martin Lück Leibniz Universität Hannover, Germany [email protected] Abstract. We study the logic FO(∼), the extension of first-order logic with team semantics by unrestricted Boolean negation. It was recently shown axiomatizable, but otherwise has not yet received much attention in questions of computational complexity. In this paper, we consider its two-variable fragment FO2(∼) and prove that its satisfiability problem is decidable, and in fact complete for the recently introduced non-elementary class TOWER(poly). Moreover, we classify the complexity of model checking of FO(∼) with respect to the number of variables and the quantifier rank, and prove a di- chotomy between PSPACE- and ATIME-ALT(exp, poly)-completeness. To achieve the lower bounds, we propose a translation from modal team logic MTL to FO2(∼) that extends the well-known standard translation from modal logic ML to FO2. For the upper bounds, we translate to a fragment of second-order logic. Keywords: team semantics, two-variable logic, complexity, satisfiability, model checking 2012 ACM Subject Classification: Theory of computation → Complexity theory and logic; Logic; 1. Introduction In the last decades, the work of logicians has unearthed a plethora of decidable fragments of first-order logic FO. Many of these cases are restricted quantifier prefixes, such as the BSR-fragment which contains only ∃∗∀∗-sentences [30]. Others include the guarded arXiv:1804.04968v1 [cs.LO] 13 Apr 2018 fragment GF [1], the recently introduced separated fragment SF [32, 34], or the two-variable fragment FO2 [13, 27, 31]. -
Sharp Lower Bounds for the Dimension of Linearizations of Matrix Polynomials∗
Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 17, pp. 518-531, November 2008 ELA SHARP LOWER BOUNDS FOR THE DIMENSION OF LINEARIZATIONS OF MATRIX POLYNOMIALS∗ FERNANDO DE TERAN´ † AND FROILAN´ M. DOPICO‡ Abstract. A standard way of dealing with matrixpolynomial eigenvalue problems is to use linearizations. Byers, Mehrmann and Xu have recently defined and studied linearizations of dimen- sions smaller than the classical ones. In this paper, lower bounds are provided for the dimensions of linearizations and strong linearizations of a given m × n matrixpolynomial, and particular lineariza- tions are constructed for which these bounds are attained. It is also proven that strong linearizations of an n × n regular matrixpolynomial of degree must have dimension n × n. Key words. Matrixpolynomials, Matrixpencils, Linearizations, Dimension. AMS subject classifications. 15A18, 15A21, 15A22, 65F15. 1. Introduction. We will say that a matrix polynomial of degree ≥ 1 −1 P (λ)=λ A + λ A−1 + ···+ λA1 + A0, (1.1) m×n where A0,A1,...,A ∈ C and A =0,is regular if m = n and det P (λ)isnot identically zero as a polynomial in λ. We will say that P (λ)issingular otherwise. A linearization of P (λ)isamatrix pencil L(λ)=λX + Y such that there exist unimod- ular matrix polynomials, i.e., matrix polynomials with constant nonzero determinant, of appropriate dimensions, E(λ)andF (λ), such that P (λ) 0 E(λ)L(λ)F (λ)= , (1.2) 0 Is where Is denotes the s × s identity matrix. Classically s =( − 1) min{m, n}, but recently linearizations of smaller dimension have been considered [3]. -
Complexity Theory
Complexity Theory Course Notes Sebastiaan A. Terwijn Radboud University Nijmegen Department of Mathematics P.O. Box 9010 6500 GL Nijmegen the Netherlands [email protected] Copyright c 2010 by Sebastiaan A. Terwijn Version: December 2017 ii Contents 1 Introduction 1 1.1 Complexity theory . .1 1.2 Preliminaries . .1 1.3 Turing machines . .2 1.4 Big O and small o .........................3 1.5 Logic . .3 1.6 Number theory . .4 1.7 Exercises . .5 2 Basics 6 2.1 Time and space bounds . .6 2.2 Inclusions between classes . .7 2.3 Hierarchy theorems . .8 2.4 Central complexity classes . 10 2.5 Problems from logic, algebra, and graph theory . 11 2.6 The Immerman-Szelepcs´enyi Theorem . 12 2.7 Exercises . 14 3 Reductions and completeness 16 3.1 Many-one reductions . 16 3.2 NP-complete problems . 18 3.3 More decision problems from logic . 19 3.4 Completeness of Hamilton path and TSP . 22 3.5 Exercises . 24 4 Relativized computation and the polynomial hierarchy 27 4.1 Relativized computation . 27 4.2 The Polynomial Hierarchy . 28 4.3 Relativization . 31 4.4 Exercises . 32 iii 5 Diagonalization 34 5.1 The Halting Problem . 34 5.2 Intermediate sets . 34 5.3 Oracle separations . 36 5.4 Many-one versus Turing reductions . 38 5.5 Sparse sets . 38 5.6 The Gap Theorem . 40 5.7 The Speed-Up Theorem . 41 5.8 Exercises . 43 6 Randomized computation 45 6.1 Probabilistic classes . 45 6.2 More about BPP . 48 6.3 The classes RP and ZPP . -
Delegating Computation: Interactive Proofs for Muggles∗
Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 108 (2017) Delegating Computation: Interactive Proofs for Muggles∗ Shafi Goldwasser Yael Tauman Kalai Guy N. Rothblum MIT and Weizmann Institute Microsoft Research Weizmann Institute [email protected] [email protected] [email protected] Abstract In this work we study interactive proofs for tractable languages. The (honest) prover should be efficient and run in polynomial time, or in other words a \muggle".1 The verifier should be super-efficient and run in nearly-linear time. These proof systems can be used for delegating computation: a server can run a computation for a client and interactively prove the correctness of the result. The client can verify the result's correctness in nearly-linear time (instead of running the entire computation itself). Previously, related questions were considered in the Holographic Proof setting by Babai, Fortnow, Levin and Szegedy, in the argument setting under computational assumptions by Kilian, and in the random oracle model by Micali. Our focus, however, is on the original inter- active proof model where no assumptions are made on the computational power or adaptiveness of dishonest provers. Our main technical theorem gives a public coin interactive proof for any language computable by a log-space uniform boolean circuit with depth d and input length n. The verifier runs in time n · poly(d; log(n)) and space O(log(n)), the communication complexity is poly(d; log(n)), and the prover runs in time poly(n). In particular, for languages computable by log-space uniform NC (circuits of polylog(n) depth), the prover is efficient, the verifier runs in time n · polylog(n) and space O(log(n)), and the communication complexity is polylog(n). -
5.2 Intractable Problems -- NP Problems
Algorithms for Data Processing Lecture VIII: Intractable Problems–NP Problems Alessandro Artale Free University of Bozen-Bolzano [email protected] of Computer Science http://www.inf.unibz.it/˜artale 2019/20 – First Semester MSc in Computational Data Science — UNIBZ Some material (text, figures) displayed in these slides is courtesy of: Alberto Montresor, Werner Nutt, Kevin Wayne, Jon Kleinberg, Eva Tardos. A. Artale Algorithms for Data Processing Definition. P = set of decision problems for which there exists a poly-time algorithm. Problems and Algorithms – Decision Problems The Complexity Theory considers so called Decision Problems. • s Decision• Problem. X Input encoded asX a finite“yes” binary string ; • Decision ProblemA : Is conceived asX a set of strings on whichs the answer to the decision problem is ; yes if s ∈ X Algorithm for a decision problemA s receives an input string , and no if s 6∈ X ( ) = A. Artale Algorithms for Data Processing Problems and Algorithms – Decision Problems The Complexity Theory considers so called Decision Problems. • s Decision• Problem. X Input encoded asX a finite“yes” binary string ; • Decision ProblemA : Is conceived asX a set of strings on whichs the answer to the decision problem is ; yes if s ∈ X Algorithm for a decision problemA s receives an input string , and no if s 6∈ X ( ) = Definition. P = set of decision problems for which there exists a poly-time algorithm. A. Artale Algorithms for Data Processing Towards NP — Efficient Verification • • The issue here is the3-SAT contrast between finding a solution Vs. checking a proposed solution.I ConsiderI for example : We do not know a polynomial-time algorithm to find solutions; but Checking a proposed solution can be easily done in polynomial time (just plug 0/1 and check if it is a solution). -
Lecture 10: Space Complexity III
Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Lecture 10: Space Complexity III Arijit Bishnu 27.03.2010 Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Outline 1 Space Complexity Classes: NL and L 2 Reductions 3 NL-completeness 4 The Relation between NL and coNL 5 A Relation Among the Complexity Classes Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Outline 1 Space Complexity Classes: NL and L 2 Reductions 3 NL-completeness 4 The Relation between NL and coNL 5 A Relation Among the Complexity Classes Definition for Recapitulation S c NPSPACE = c>0 NSPACE(n ). The class NPSPACE is an analog of the class NP. Definition L = SPACE(log n). Definition NL = NSPACE(log n). Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Space Complexity Classes Definition for Recapitulation S c PSPACE = c>0 SPACE(n ). The class PSPACE is an analog of the class P. Definition L = SPACE(log n). Definition NL = NSPACE(log n). Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Space Complexity Classes Definition for Recapitulation S c PSPACE = c>0 SPACE(n ). The class PSPACE is an analog of the class P. Definition for Recapitulation S c NPSPACE = c>0 NSPACE(n ). -
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). -
Lecture 4: Space Complexity II: NL=Conl, Savitch's Theorem
COM S 6810 Theory of Computing January 29, 2009 Lecture 4: Space Complexity II Instructor: Rafael Pass Scribe: Shuang Zhao 1 Recall from last lecture Definition 1 (SPACE, NSPACE) SPACE(S(n)) := Languages decidable by some TM using S(n) space; NSPACE(S(n)) := Languages decidable by some NTM using S(n) space. Definition 2 (PSPACE, NPSPACE) [ [ PSPACE := SPACE(nc), NPSPACE := NSPACE(nc). c>1 c>1 Definition 3 (L, NL) L := SPACE(log n), NL := NSPACE(log n). Theorem 1 SPACE(S(n)) ⊆ NSPACE(S(n)) ⊆ DTIME 2 O(S(n)) . This immediately gives that N L ⊆ P. Theorem 2 STCONN is NL-complete. 2 Today’s lecture Theorem 3 N L ⊆ SPACE(log2 n), namely STCONN can be solved in O(log2 n) space. Proof. Recall the STCONN problem: given digraph (V, E) and s, t ∈ V , determine if there exists a path from s to t. 4-1 Define boolean function Reach(u, v, k) with u, v ∈ V and k ∈ Z+ as follows: if there exists a path from u to v with length smaller than or equal to k, then Reach(u, v, k) = 1; otherwise Reach(u, v, k) = 0. It is easy to verify that there exists a path from s to t iff Reach(s, t, |V |) = 1. Next we show that Reach(s, t, |V |) can be recursively computed in O(log2 n) space. For all u, v ∈ V and k ∈ Z+: • k = 1: Reach(u, v, k) = 1 iff (u, v) ∈ E; • k > 1: Reach(u, v, k) = 1 iff there exists w ∈ V such that Reach(u, w, dk/2e) = 1 and Reach(w, v, bk/2c) = 1. -
Descriptive Complexity: a Logician's Approach to Computation
Descriptive Complexity a Logicians Approach to Computation Neil Immerman Computer Science Dept University of Massachusetts Amherst MA immermancsumassedu App eared in Notices of the American Mathematical Society A basic issue in computer science is the complexity of problems If one is doing a calculation once on a mediumsized input the simplest algorithm may b e the b est metho d to use even if it is not the fastest However when one has a subproblem that will havetobesolved millions of times optimization is imp ortant A fundamental issue in theoretical computer science is the computational complexity of problems Howmuch time and howmuch memory space is needed to solve a particular problem Here are a few examples of such problems Reachability Given a directed graph and two sp ecied p oints s t determine if there is a path from s to t A simple lineartime algorithm marks s and then continues to mark every vertex at the head of an edge whose tail is marked When no more vertices can b e marked t is reachable from s i it has b een marked Mintriangulation Given a p olygon in the plane and a length L determine if there is a triangulation of the p olygon of total length less than or equal to L Even though there are exp onentially many p ossible triangulations a dynamic programming algorithm can nd an optimal one in O n steps ThreeColorability Given an undirected graph determine whether its vertices can b e colored using three colors with no two adjacentvertices having the same color Again there are exp onentially many p ossibilities -
Descriptive Complexity Theories
Descriptive Complexity Theories Joerg FLUM ABSTRACT: In this article we review some of the main results of descriptive complexity theory in order to make the reader familiar with the nature of the investigations in this area. We start by presenting the characterization of automata recognizable languages by monadic second-order logic. Afterwards we explain the characteri- zation of various logics by fixed-point logics. We assume familiarity with logic but try to keep knowledge of complexity theory to a minimum. Keywords: Computational complexity theory, complexity classes, descriptive characterizations, monadic second-order logic, fixed-point logic, Turing machine. Complexity theory or more precisely, computational complexity theory (cf. Papadimitriou 1994), tries to classify problems according to the complexity of algorithms solving them. Of course, we can think of various ways of measuring the complexity of an al- gorithm, but the most important and relevant ones are time and space. That is, we think we have a type of machine capable, in principle, to carry out any algorithm, a so-called general purpose machine (or, general purpose computer), and we measure the com- plexity of an algorithm in terms of the time or the number of steps needed to carry out this algorithm. By space, we mean the amount of memory the algorithm uses. Time bounds yield (time) complexity classes consisting of all problems solvable by an algorithm keeping to the time bound. Similarly, space complexity classes are obtained. It turns out that these definitions are quite robust in the sense that, for reasonable time or space bounds, the corresponding complexity classes do not depend on the special type of machine model chosen.