A Quantum Query Complexity Trichotomy for Regular Languages
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Automata, Semigroups and Duality
Automata, semigroups and duality Mai Gehrke1 Serge Grigorieff2 Jean-Eric´ Pin2 1Radboud Universiteit 2LIAFA, CNRS and University Paris Diderot TANCL’07, August 2007, Oxford LIAFA, CNRS and University Paris Diderot Outline (1) Four ways of defining languages (2) The profinite world (3) Duality (4) Back to the future LIAFA, CNRS and University Paris Diderot Part I Four ways of defining languages LIAFA, CNRS and University Paris Diderot Words and languages Words over the alphabet A = {a, b, c}: a, babb, cac, the empty word 1, etc. The set of all words A∗ is the free monoid on A. A language is a set of words. Recognizable (or regular) languages can be defined in various ways: ⊲ by (extended) regular expressions ⊲ by finite automata ⊲ in terms of logic ⊲ by finite monoids LIAFA, CNRS and University Paris Diderot Basic operations on languages • Boolean operations: union, intersection, complement. • Product: L1L2 = {u1u2 | u1 ∈ L1,u2 ∈ L2} Example: {ab, a}{a, ba} = {aa, aba, abba}. • Star: L∗ is the submonoid generated by L ∗ L = {u1u2 · · · un | n > 0 and u1,...,un ∈ L} {a, ba}∗ = {1, a, aa, ba, aaa, aba, . .}. LIAFA, CNRS and University Paris Diderot Various types of expressions • Regular expressions: union, product, star: (ab)∗ ∪ (ab)∗a • Extended regular expressions (union, intersection, complement, product and star): A∗ \ (bA∗ ∪ A∗aaA∗ ∪ A∗bbA∗) • Star-free expressions (union, intersection, complement, product but no star): ∅c \ (b∅c ∪∅caa∅c ∪∅cbb∅c) LIAFA, CNRS and University Paris Diderot Finite automata a 1 2 The set of states is {1, 2, 3}. b The initial state is 1. b a 3 The final states are 1 and 2. -
Introduction to the Theory of Computation, Michael Sipser
Introduction to the Theory of Computation, Michael Sipser Chapter 0: Introduction Automata, Computability and Complexity: · They are linked by the question: o “What are the fundamental capabilities and limitations of computers?” · The theories of computability and complexity are closely related. In complexity theory, the objective is to classify problems as easy ones and hard ones, whereas in computability theory he classification of problems is by those that are solvable and those that are not. Computability theory introduces several of the concepts used in complexity theory. · Automata theory deals with the definitions and properties of mathematical models of computation. · One model, called the finite automaton, is used in text processing, compilers, and hardware design. Another model, called the context – free grammar, is used in programming languages and artificial intelligence. Strings and Languages: · The string of the length zero is called the empty string and is written as e. · A language is a set of strings. Definitions, Theorems and Proofs: · Definitions describe the objects and notions that we use. · A proof is a convincing logical argument that a statement is true. · A theorem is a mathematical statement proved true. · Occasionally we prove statements that are interesting only because they assist in the proof of another, more significant statement. Such statements are called lemmas. · Occasionally a theorem or its proof may allow us to conclude easily that other, related statements are true. These statements are called corollaries of the theorem. Chapter 1: Regular Languages Introduction: · An idealized computer is called a “computational model” which allows us to set up a manageable mathematical theory of it directly. -
CS 273 Introduction to the Theory of Computation Fall 2006
Welcome to CS 373 Theory of Computation Spring 2010 Madhusudan Parthasarathy ( Madhu ) [email protected] What is computable? • Examples: – check if a number n is prime – compute the product of two numbers – sort a list of numbers – find the maximum number from a list • Hard but computable: – Given a set of linear inequalities, maximize a linear function Eg. maximize 5x+2y 3x+2y < 53 x < 32 5x – 9y > 22 Theory of Computation Primary aim of the course: What is “computation”? • Can we define computation without referring to a modern c computer? • Can we define, mathematically, a computer? (yes, Turing machines) • Is computation definable independent of present-day engineering limitations, understanding of physics, etc.? • Can a computer solve any problem, given enough time and disk-space? Or are they fundamental limits to computation? In short, understand the mathematics of computation Theory of Computation - What can be computed? Computability - Can a computer solve any problem, given enough time and disk-space? - How fast can we solve a problem? Complexity - How little disk-space can we use to solve a problem -What problems can we solve given Automata really very little space? (constant space) Theory of Computation What problems can a computer solve? Not all problems!!! Computability Eg. Given a C-program, we cannot check if it will not crash! Verification of correctness of programs Complexity is hence impossible! (The woe of Microsoft!) Automata Theory of Computation What problems can a computer solve? Even checking whether a C-program will Computability halt/terminate is not possible! input n; assume n>1; No one knows Complexity while (n !=1) { whether this if (n is even) terminates on n := n/2; on all inputs! else n := 3*n+1; Automata } 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. -
Complexity Theory Lectures 1–6
Complexity Theory 1 Complexity Theory Lectures 1–6 Lecturer: Dr. Timothy G. Griffin Slides by Anuj Dawar Computer Laboratory University of Cambridge Easter Term 2009 http://www.cl.cam.ac.uk/teaching/0809/Complexity/ Cambridge Easter 2009 Complexity Theory 2 Texts The main texts for the course are: Computational Complexity. Christos H. Papadimitriou. Introduction to the Theory of Computation. Michael Sipser. Other useful references include: Computers and Intractability: A guide to the theory of NP-completeness. Michael R. Garey and David S. Johnson. Structural Complexity. Vols I and II. J.L. Balc´azar, J. D´ıaz and J. Gabarr´o. Computability and Complexity from a Programming Perspective. Neil Jones. Cambridge Easter 2009 Complexity Theory 3 Outline A rough lecture-by-lecture guide, with relevant sections from the text by Papadimitriou (or Sipser, where marked with an S). Algorithms and problems. 1.1–1.3. • Time and space. 2.1–2.5, 2.7. • Time Complexity classes. 7.1, S7.2. • Nondeterminism. 2.7, 9.1, S7.3. • NP-completeness. 8.1–8.2, 9.2. • Graph-theoretic problems. 9.3 • Cambridge Easter 2009 Complexity Theory 4 Outline - contd. Sets, numbers and scheduling. 9.4 • coNP. 10.1–10.2. • Cryptographic complexity. 12.1–12.2. • Space Complexity 7.1, 7.3, S8.1. • Hierarchy 7.2, S9.1. • Descriptive Complexity 5.6, 5.7. • Cambridge Easter 2009 Complexity Theory 5 Complexity Theory Complexity Theory seeks to understand what makes certain problems algorithmically difficult to solve. In Data Structures and Algorithms, we saw how to measure the complexity of specific algorithms, by asymptotic measures of number of steps. -
A Categorical Approach to Syntactic Monoids 11
Logical Methods in Computer Science Vol. 14(2:9)2018, pp. 1–34 Submitted Jan. 15, 2016 https://lmcs.episciences.org/ Published May 15, 2018 A CATEGORICAL APPROACH TO SYNTACTIC MONOIDS JIRˇ´I ADAMEK´ a, STEFAN MILIUS b, AND HENNING URBAT c a Institut f¨urTheoretische Informatik, Technische Universit¨atBraunschweig, Germany e-mail address: [email protected] b;c Lehrstuhl f¨urTheoretische Informatik, Friedrich-Alexander-Universit¨atErlangen-N¨urnberg, Ger- many e-mail address: [email protected], [email protected] Abstract. The syntactic monoid of a language is generalized to the level of a symmetric monoidal closed category D. This allows for a uniform treatment of several notions of syntactic algebras known in the literature, including the syntactic monoids of Rabin and Scott (D “ sets), the syntactic ordered monoids of Pin (D “ posets), the syntactic semirings of Pol´ak(D “ semilattices), and the syntactic associative algebras of Reutenauer (D = vector spaces). Assuming that D is a commutative variety of algebras or ordered algebras, we prove that the syntactic D-monoid of a language L can be constructed as a quotient of a free D-monoid modulo the syntactic congruence of L, and that it is isomorphic to the transition D-monoid of the minimal automaton for L in D. Furthermore, in the case where the variety D is locally finite, we characterize the regular languages as precisely the languages with finite syntactic D-monoids. 1. Introduction One of the successes of the theory of coalgebras is that ideas from automata theory can be developed at a level of abstraction where they apply uniformly to many different types of systems. -
Syntactic Semigroups
Syntactic semigroups Jean-Eric Pin LITP/IBP, CNRS, Universit´e Paris VI, 4 Place Jussieu, 75252 Paris Cedex 05, FRANCE 1 Introduction This chapter gives an overview on what is often called the algebraic theory of finite automata. It deals with languages, automata and semigroups, and has connections with model theory in logic, boolean circuits, symbolic dynamics and topology. Kleene's theorem [70] is usually considered as the foundation of this theory. It shows that the class of recognizable languages (i.e. recognized by finite automata), coincides with the class of rational languages, which are given by rational expressions. The definition of the syntactic mono- id, a monoid canonically attached to each language, was first given in an early paper of Rabin and Scott [128], where the notion is credited to Myhill. It was shown in particular that a language is recognizable if and only if its syntactic monoid is finite. However, the first classification results were rather related to automata [89]. A break-through was realized by Schutzen-¨ berger in the mid sixties [144]. Schutzenb¨ erger made a non trivial use of the syntactic monoid to characterize an important subclass of the rational languages, the star-free languages. Schutzenb¨ erger's theorem states that a language is star-free if and only if its syntactic monoid is finite and aperiodic. This theorem had a considerable influence on the theory. Two other im- portant \syntactic" characterizations were obtained in the early seventies: Simon [152] proved that a language is piecewise testable if and only if its syntactic monoid is finite and J -trivial and Brzozowski-Simon [41] and in- dependently, McNaughton [86] characterized the locally testable languages. -
Natural Proofs Barrier
CS 388T: Advanced Complexity Theory Spring 2020 Natural Proofs Prof. Dana Moshkovitz Scribe: Maruth Goyal 1 Overview Complexity theorists have long been working on proving relations between various complexity classes. Their proofs would often treat Turing Machines as black boxes. However, they seemed to be unable to prove equality or separation for NP 6⊂ P. A reason for this was found with Baker, Gill, and Solovay's Relativization barrier [BGS75], which should any proof for such a separation must be non-relativizing. i.e., it must not hold with respect to arbitrary oracles. Hence, researchers turned to circuits as a way of analyzing various classes by considering more explicit constructions. This begs the question, is their a similar barrier for separating NP from P=poly? To separate a function f from some circuit complexity class C, mathematicians often consider some larger property of f, and then separate that property from C. For instance, in the case of PARITY [FSS84], we observed it cannot be simplified even under arbitrary random restrictions, while AC0 functions can be. While this seemed like a powerful proof technique, in 1997 Razborov and Rudich [RR] showed that to separate NP from P=poly, there is a large classification of properties which cannot be the basis for such a proof. They called these properties \natural properties". 2 Natural Properties Definition 1. A property P is said to be natural if: 1. Useful: 8g 2 C:g2 = P but f 2 P 2. Constructivity: Given g : f0; 1gn ! f0; 1g can check if g 2 P in time 2O(n). -
Typed Monoids – an Eilenberg-Like Theorem for Non Regular Languages
Electronic Colloquium on Computational Complexity, Report No. 35 (2011) Typed Monoids – An Eilenberg-like Theorem for non regular Languages Christoph Behle Andreas Krebs Stephanie Reifferscheid∗ Wilhelm-Schickard-Institut, Universität Tübingen, fbehlec,krebs,reiff[email protected] Abstract Based on different concepts to obtain a finer notion of language recognition via finite monoids we develop an algebraic structure called typed monoid. This leads to an algebraic description of regular and non regular languages. We obtain for each language a unique minimal recognizing typed monoid, the typed syntactic monoid. We prove an Eilenberg-like theorem for varieties of typed monoids as well as a similar correspondence for classes of languages with weaker closure properties. 1 Introduction We present an algebraic viewpoint on the study of formal languages and introduce a new algebraic structure to describe formal languages. We extend the known approach to use language recognition via monoids and morphisms by equipping the monoids with additional information to obtain a finer notion of recognition. In the concept of language recognition by monoids, a language L ⊆ Σ∗ is recognized by a monoid M if there exists a morphism h : Σ∗ ! M and a subset A of M such that L = h−1(A). The syntactic monoid, which is the smallest monoid recognizing L, has emerged as an important tool to study classes of regular languages. Besides problems applying this tool to non regular languages, even for regular languages this instrument lacks precision. One problem is, that not only the syntactic monoid but also the syntactic mor- phism, i.e. the morphism recognizing L via the syntactic monoid, plays an important role: Consider the two languages Lparity, Leven over the alphabet Σ = fa; bg where the first one consists of all words with an even number of a’s while the latter consists of all words of even length. -
Improved Learning of AC0 Functions
0 Improved Learning of AC Functions Merrick L. Furst Jeffrey C. Jackson Sean W. Smith School of Computer Science School of Computer Science School of Computer Science Carnegie Mellon University Carnegie Mellon University Carnegie Mellon University Pittsburgh, PA 15213 Pittsburgh, PA 15213 Pittsburgh, PA 15213 [email protected] [email protected] [email protected] Abstract A variant of one of our algorithms gives a more general learning method which we conjecture produces reasonable 0 Two extensions of the Linial, Mansour, Nisan approximations of AC functions for a broader class of in- 0 put distributions. A brief outline of the general algorithm AC learning algorithm are presented. The LMN method works when input examples are drawn is presented along with a conjecture about a class of distri- uniformly. The new algorithms improve on theirs butions for which it might perform well. by performing well when given inputs drawn Our two learning methods differ in several ways. The from unknown, mutually independent distribu- direct algorithm is very similar to the LMN algorithm and tions. A variant of the one of the algorithms is depends on a substantial generalization of their theory. This conjectured to work in an even broader setting. algorithm is straightforward, but our bound on its running time is very sensitive to the probability distribution on the inputs. The indirect algorithm, discovered independently 1 INTRODUCTION by Umesh Vazirani [Vaz], is more complicated but also relatively simple to analyze. Its time bound is only mildly Linial, Mansour, and Nisan [LMN89] introduced the use affected by changes in distribution, but for distributions not too far from uniform it is likely greater than the direct of the Fourier transform to accomplish Boolean function 0 bound. -
Theory of Computer Science May 22, 2017 — E2
Theory of Computer Science May 22, 2017 | E2. P, NP and Polynomial Reductions Theory of Computer Science E2.1 P and NP E2. P, NP and Polynomial Reductions E2.2 Polynomial Reductions Malte Helmert University of Basel E2.3 NP-Hardness and NP-Completeness May 22, 2017 E2.4 Summary Malte Helmert (University of Basel) Theory of Computer Science May 22, 2017 1 / 30 Malte Helmert (University of Basel) Theory of Computer Science May 22, 2017 2 / 30 Overview: Course Overview: Complexity Theory contents of this course: logic I X Complexity Theory . How can knowledge be represented? . How can reasoning be automated? E1. Motivation and Introduction E2. P, NP and Polynomial Reductions I automata theory and formal languages X . What is a computation? E3. Cook-Levin Theorem I computability theory X E4. Some NP-Complete Problems, Part I . What can be computed at all? E5. Some NP-Complete Problems, Part II I complexity theory . What can be computed efficiently? Malte Helmert (University of Basel) Theory of Computer Science May 22, 2017 3 / 30 Malte Helmert (University of Basel) Theory of Computer Science May 22, 2017 4 / 30 Further Reading (German) Further Reading (English) Literature for this Chapter (German) Literature for this Chapter (English) Theoretische Informatik { kurz gefasst Introduction to the Theory of Computation by Uwe Sch¨oning(5th edition) by Michael Sipser (3rd edition) I Chapters 3.1 and 3.2 I Chapter 7.1{7.4 Malte Helmert (University of Basel) Theory of Computer Science May 22, 2017 5 / 30 Malte Helmert (University of Basel) Theory of Computer Science May 22, 2017 6 / 30 E2. -
Theory of Computation Michael Sipser 18.404/6.840 Fall 2021 Course Information
Theory of Computation Michael Sipser 18.404/6.840 Fall 2021 Course Information Instructor: Michael Sipser, 2{438, 617{253{4992, [email protected], office hours: Tu 4:15{5, zoom hours Mo 1-2pm (see website for zoom link) Homepage: math.mit.edu/18.404 TAs: Fadi Atieh, is [email protected] Kerri Lu, [email protected] Zhezheng Luo, [email protected] Rene Reyes, [email protected] George Stefanakis, [email protected] Sam Tenka, [email protected] Nathan Weckwerth, [email protected] Course Outline I Automata and Language Theory (2 weeks). Finite automata, regular expressions, push-down automata, context free grammars, pumping lemmas. II Computability Theory (3 weeks). Turing machines, Church{Turing thesis, decidability, halting problem, reducibility, recursion theorem. { Midterm Exam: Thursday, October 21, 2021. III Complexity Theory (7 weeks). Time and space measures of complexity, complexity classes P, NP, L, NL, PSPACE, BPP and IP, complete problems, P versus NP conjecture, quantifiers and games, hierarchy theorems, provably hard problems, relativized computation and oracles, probabilistic computation, interactive proof systems. { Final Exam: 3 hours, emphasizing second half of the course. Prerequisites To succeed in this course you require experience and skill with mathematical concepts, theorems, and proofs. If you did reasonably well in 6.042, 18.200, or any other proof-oriented mathematics course, you should be fine. The course moves quickly, covering about 90% of the textbook. The homework assignments generally require proving some statement, and creativity in finding proofs will be necessary. Note that 6.045 has a significant overlap with 18.404, depending on who is teaching 6.045, and it often uses the same book though it generally covers less, with less depth, and with easier problem sets. -
A Fixed-Depth Size-Hierarchy Theorem for AC0 for Any S(N) Exp(No(1))
A Fixed-Depth Size-Hierarchy Theorem for AC0[ ] via the Coin ⊕ Problem Nutan Limaye ∗ Karteek Sreenivasaiah † Srikanth Srinivasan ‡ Utkarsh Tripathi § S. Venkitesh ¶ February 21, 2019 Abstract In this work we prove the first Fixed-depth Size-Hierarchy Theorem for uniform AC0[ ]. In ⊕ 0 particular, we show that for any fixed d, the class d,k of functions that have uniform AC [ ] formulas of depth d and size nk form an infinite hierarchy.C We show this by exibiting the first⊕ class of explicit functions where we have nearly (up to a polynomial factor) matching upper and lower bounds for the class of AC0[ ] formulas. The explicit functions are derived⊕ from the δ-Coin Problem, which is the computational problem of distinguishing between coins that are heads with probability (1+ δ)/2 or (1 δ)/2, where δ is a parameter that is going to 0. We study the complexity of this problem and− make progress on both upper bound and lower bound fronts. Upper bounds. For any constant d 2, we show that there are explicit monotone • AC0 formulas (i.e. made up of AND and≥ OR gates only) solving the δ-coin problem that have depth d, size exp(O(d(1/δ)1/(d−1))), and sample complexity (i.e. number of inputs) poly(1/δ). This matches previous upper bounds of O’Donnell and Wimmer (ICALP 2007) and Amano (ICALP 2009) in terms of size (which is optimal) and improves the sample complexity from exp(O(d(1/δ)1/(d−1))) to poly(1/δ). Lower bounds.