Automata and Formal Languages - CM0081 Introduction

Transcription

1 Automata and Formal Languages - CM0081 Introduction Andrés Sicard-Ramírez Universidad EAFIT Semester

2 Administrative Information Course Coordinator Andrés Sicard Ramírez Head of the Department of Mathematical Sciences Carlos Mario Vélez Sánchez Course web page Exams, programming labs, etc. See course web page. Convention The numbers assigned to examples, exercises or theorems correspond to the numbers in the textbook (Hopcroft, Motwani and Ullman 2007). Introduction 2/29

3 Computability (Decidability) Question What can a computer do at all? Introduction 3/29

4 Computability (Decidability) Question What can a computer do at all? Definition A computable (or decidable) problem is a problem than can be solved by a computer (informal definition). Introduction 4/29

5 Computability (Decidability) Example (The halting problem: An undecidable problem) Given an program P and an input I, to decide if the program will halt or will run forever. P Yes The halting algorithm I No Introduction 5/29

6 Algorithmic Complexity (Tractability) Comparison of several time complexity functions f(n) log n 2.3 sec 3.9 sec 4.6 sec n 10.0 sec 50.0 sec 1.7 min n min 41.7 min 2.8 h 2 n 17.1 min c c 3 n 16.4 h c c n! 42.0 d c c Introduction 6/29

7 Algorithmic Complexity (Tractability) Question What can a computer do efficiently? Introduction 7/29

8 Algorithmic Complexity (Tractability) Question What can a computer do efficiently? Definition A tractable problem is a problem than can be solved by a computer algorithm that run in polynomial time. Introduction 8/29

9 Algorithmic Complexity (Tractability) Definition A literal is an atomic formula (propositional variable) or the negation of an atomic formula. Definition A (propositional logic) formula F is said to be in conjunctive normal form, if and only if, F has the form F 1 F n, n 1, where each F 1,, F n is a disjunction of literals. Introduction 9/29

10 Algorithmic Complexity (Tractability) Example (3-SAT: An intractable problem) To determine the satisfiability of a propositional formula in conjunctive normal form where each disjunction of literals is limited to at most three literals. The problem was proposed in Karp s 21 NP-complete problems (Karp 1972). Introduction 10/29

11 Algorithmic Complexity (Tractability) Improvements on 3-SAT deterministic algorithmic complexity O( n ) Makino, Tamaki and Yamamoto (2011, 2013) O( n ) Moser and Scheder (2011) O(1.439 n ) Kutzkov and Scheder (2010) O(1.465 n ) Scheder (2008) O(1.473 n ) Brueggemann and Kern (2004) O(1.481 n ) Dantsin, Goerdt, Hirsch, Kannan, Kleinberg, Papadimitriou, Raghavan and Schöning (2002) O(1.497 n ) Schiermeyer (1996) O(1.505 n ) Kullmann (1999) O( n ) Monien and Speckenmeyer (1979, 1985) O(2 n ) Brute-force search Main sources: Hertli (2011, 2015). Last updated: July Introduction 11/29

12 Algorithmic Complexity (Tractability) Supercomputers Machines from: Petaflops (PFs): floating-point operations per second Date Machine PFs Summit Sunway TaihuLight Tianhe Blue Gene/Q K computer 8.16 Last updated: TOP500 List - June Introduction 12/29

13 Algorithmic Complexity (Tractability) Simulation: Running 3-SAT times on different supercomputers using the faster deterministic algorithm, i.e., O( n ) Machine PFs Summit sec 1.6 y c Sunway TaihuLight sec 2.1 y c Tianhe min 5.8 y c Blue Gene/Q min 12.0 y c K computer min 24.0 y c Introduction 13/29

14 Algorithmic Complexity (Tractability) Simulation: Running 3-SAT times for different deterministic algorithms using the faster supercomputer, i.e., PFs Complexity O( n ) 32.1 sec 1.6 y c O( n ) 45.5 sec 2.6 y c O(1.439 n ) 48.6 sec c c O(1.465 n ) 1.9 y c c O(2 n ) 3.7 y c c Introduction 14/29

15 Computability and Algorithmic Complexity Problem (TM-Computability) Computable (decidable) problem (Algorithm Complexity) Tractable problem Intractable problem Non-computable (undecidable) problem (Hypercomputation) Introduction 15/29

16 Course Outline Language Machine Other models Regular DFA Regular expressions NFA ε-nfa Context-free Pushdown automata Recursive Halting TMs λ-calculus Total recursive functions Recursively enumerable TMs λ-calculus Partial recursive functions DFA: NFA: ε-nfa: TM: Deterministic finite automata Non-deterministic finite automata Non-deterministic finite automata with ε-transitions Turing machine Introduction 16/29

17 Paradigms of Programming Some paradigms Imperative: Describe computation in terms of state-transforming operations such as assignment. Programming is done with statements. Logic: Predicate calculus as a programming language. Programming is done with sentences. Functional: Describe computation in terms of (mathematical) functions. Programming is done with expressions. Examples C Imperative C++ Java Logic CLP(R) Prolog Standard ML Functional Erlang Haskell Introduction 17/29

18 Pure Functional Programming Side effects A side effect introduces a dependency between the global state of the system and the behaviour of a function... Side effects are essentially invisible inputs to, or outputs from, functions. (O Sullivan, Goerzen and Stewart 2008, p. 27) Introduction 18/29

19 Pure Functional Programming Pure functions A function may be described as pure if both these statements about the function hold: 1. The function always evaluates the same result value given the same argument value(s). The function result value cannot depend on any... state that may change as program execution proceeds or between different executions of the program, nor can it depend on any external input from I/O devices. Wikipedia: Pure function (July 28, 2014). Introduction 19/29

20 Pure Functional Programming Pure functions A function may be described as pure if both these statements about the function hold: 1. The function always evaluates the same result value given the same argument value(s). The function result value cannot depend on any... state that may change as program execution proceeds or between different executions of the program, nor can it depend on any external input from I/O devices. 2. Evaluation of the result does not cause any semantically observable side effect or output, such as mutation of mutable objects or output to I/O devices. Wikipedia: Pure function (July 28, 2014). Introduction 20/29

21 Pure Functional Programming Pure functions...take all their input as explicit arguments, and produce all their output as explicit results. (Hutton 2016, 10.1) Introduction 21/29

22 Pure Functional Programming Referential transparency Equals can be replaced by equals Introduction 22/29

23 Pure Functional Programming Referential transparency Equals can be replaced by equals By definition, a function in Haskell defines a fixed relation between inputs and output: whenever a function f is applied to the argument value arg it will produce the same output no matter what the overall state of the computation is. Haskell, like any other pure functional language, is said to be referentially transparent or side-effect free. This property does not hold for imperative languages... (Grune, Bal, Jacobs and Langendoen 2003, pp ) Introduction 23/29

24 Pure Functional Programming Reasoning about (pure) functional programs Equational reasoning + induction + co-induction + Introduction 24/29

25 Reading Homework To read from the textbook the following sections: 1.1. Why Study Automata Theory? 1.2. Introduction to Formal Proofs 1.3. Additional Forms of Proofs Introduction 25/29

26 References Brueggemann, T. and Kern, W. (2004). An Improved Deterministic Local Search Algorithm for 3-SAT. In: Theoretical Computer Science , pp doi: /j.tcs (cit. on p. 11). Dantsin, E. et al. (2002). A Deterministic (2 2/(k + 1)) n Algorithm for k-sat Based on Local Search. In: Theoretical Computer Science 289.1, pp doi: /S (01) (cit. on p. 11). Grune, D., Bal, H. E., Jacobs, C. J. H. and Langendoen, K. G. (2003). Modern Compiler Desing. John Wiley & Sons (cit. on pp. 22, 23). Hertli, T. (2011). 3-SAT Faster and Simpler - Unique-SAT Bounds for PPSZ Hold in General. In: Proceedings of the 52nd Annual Symposium on Foundations of Computer Science (FOCS 2011). IEEE, pp doi: /FOCS (cit. on p. 11). (2015). Improved Exponential Algorithms for SAT and ClSP. PhD thesis. ETH Zurich. doi: /ethz-a (cit. on p. 11). Introduction 26/29

27 References Hopcroft, J. E., Motwani, R. and Ullman, J. D. (2007). Introduction to Automata theory, Languages, and Computation. 3rd ed. Pearson Education (cit. on p. 2). Hutton, G. (2016). Programming in Haskell. 2nd ed. Cambridge University Press (cit. on p. 21). Karp, R. M. (1972). Reducibility Among Combinatorial Problems. In: Complexity of Computer Computations. Ed. by Miller, R. E. and Thatcher, J. W. Plenum Press, pp doi: / _9 (cit. on p. 10). Kullmann, O. (1999). New Methods for 3-SAT Decision and Worst-Case Analysis. In: Theoretical Computer Science , pp doi: /S (98) (cit. on p. 11). Kutzkov, K. and Scheder, D. (2010). Using CSP to Improve Deterministic 3-SAT. CoRR abs/ url: (cit. on p. 11). Introduction 27/29

28 References Makino, K., Tamaki, S. and Yamamoto, M. (2011). Derandomizing HSSW Algorithm for 3-SAT. In: Computing and Combinatorics (COCOON 2011). Ed. by Fu, B. and Du, D.-Z. Vol Lecture Notes in Computer Science. Springer, pp doi: / _1 (cit. on p. 11). (2013). Derandomizing HSSW Algorithm for 3-SAT. In: Algorithmica 67.2, pp doi: /s (cit. on p. 11). Monien, B. and Speckenmeyer, E. (1979). 3-Satisfiability is Testable in O(1.62 r ) Steps. Tech. rep. 3/1979. Reihe Theoretische Informatik, Universität-Gesamthochschule-Paderborn (cit. on p. 11). (1985). Solving Satisfiability in less than 2 n Steps. In: Discrete Applied Mathematics 10.3, pp doi: / X(85) (cit. on p. 11). Moser, R. A. and Scheder, D. (2011). A Full Derandomization of Schöning s k-sat Algorithm. In: Proceedings of the Forty-third Annual ACM Symposium on Theory of Computing (STOC 2011), pp doi: / (cit. on p. 11). Introduction 28/29

29 References O Sullivan, B., Goerzen, J. and Stewart, D. (2008). Real World Haskell. O REILLY (cit. on p. 18). Scheder, D. (2008). Guided Search and a Faster Deterministic Algorithm for 3-SAT. In: Proc. of the 8th Latin American Symposium on Theoretical Informatic (LATIN 2008). Ed. by Laber, E. S., Bornstein, C., Nogueira, T. L. and Faria, L. Vol Lecture Notes in Computer Science. Springer, pp doi: / _6 (cit. on p. 11). Schiermeyer, I. (1996). Pure Literal Look Ahead: An O(1.497 n ) 3-Satisfability Algorithm (Extended Abstract). Workshop on the Satisfability Problem, Siena url: (cit. on p. 11). Introduction 29/29