Unit 10. Turing machine Decidability Decision problems in FA

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1 Unit 10 Turing machine Decidability Decision problems in FA 1

2 Turing Machines Proposed by Alan Turing, Unlimited and unrestricted memory. A model of a general purpose computer. Can do anything that a real computer can do. Cannot solve all problems!! 2

3 The Turing Machine An infinite tape - unlimited memory. A tape head - can move left and right. Can read from the tape and write on the tape. head b b a c b b d a c 3

4 Input/Output Initial Configuration Input q 0 * w w1w 2... wn... * 0 1 w0 w_ 1w_ 2... w_ n Final Configuration Output q accept _... 4

5 Initial Configuration Input: Appears on the leftmost n slots of the tape. The rest of the tape is blank ( _ ). Head starts on the left-most square. Initial state: q 0 head q 0... * b b wa _ w1w_ 2... w_ n 5

6 Final Configuration During the computation the head can read and write to the tape and move right or left until it reach reject or accept states. Final state: q {q accept, q reject } The tape contains uw. The string u appears to the left of the head. The string w appears under the head and its right. The rest of the tape is blank. The output is the string u. q accept * 0 0 w1 0w 1w w_ n u w 6

7 Possible Outcomes of a TM The possible outcomes of a TM on a word w can be: stop with reject state. stop with accept state. loop forever. 7

8 Differences Between Finite Automata and TM A TM can both read and write. The head can move left and right. The tape is infinite. The special cases of reject and accept take an immediate action. A TM can produce an output. 8

9 TM: Definition Formally: A TM is defined by (Q,,,, q 0, q accept, q reject ) Q is the set of states. is the input alphabet _. is the tape alphabet, _ and. : Qx Qx x{r,l} the transition function. q 0, q accept, q reject the start, accept and reject states. q accept q reject 9

10 The Transition Function The transition function : Qx Qx x{r,l} The transition (p,a) (q,b,r) is depicted as: p a b,r q p a,c b,r a b,r c b,r q 10

11 Example: L={w#w w {0,1} * } # _ x # _ x # x _... accept x x x x # x x x x _ 11

12 Example: L={w#w w {0,1} * } Process: 1. Scan and check that the input contains a single #. If not reject. 2. Zig-zag to check whether corresponding positions on either sides of # contain the same symbol. If not, reject. Cross off the checked symbols. 3. When all symbols to the left of # are crossed, check for remaining symbols to the right of #. If any symbol, reject. Otherwise, accept. 12

13 TM s state diagram recognizing: B={w#w w {0,1} * } 13

14 The Turing-Church Conjecture 1936 If an algorithm exists for solving a problem then there is an equivalent Turing machine solving that algorithm. This gives a formal model for the intuitive notion of algorithms. This means that a Turing machine can do what any computer can do, and there isn t a more powerful computational model than TM. 14

15 Decision Problems Problems for which the answer is yes/no. Any problem can be reduced to a decision problem. 15

16 Arithmetic as a decision problem A membership problem determines whether a word w is in a language L. Any problem can be reduced to the membership problem of a word in a language. Example: An arithmetical problem of finding the answer to: 3*7=? This problem can be formulated as a question: which of the following words belong to the language of correct arithmetic statements: 3*7=0, 3*7=1,, 3*7=21, 3*7=22, 16

17 Informally: Decidability Decidability of problem X whether the problem has an algorithm which solves it (decides it), i.e. whether there is an TM which: 1. always halts 2. with the correct answer A decidable problem has such an algorithm/tm/grammar. An undecidable problem does not have such an algorithm/tm/grammar. 17

18 DFA Membership Input: DFA A and a string w Output: yes if w L(A); no otherwise. Theorem: DFA membership is decidable. 18

19 Proof of decidability The algorithm to decide the problem is: Simulate A with a TM: start A on w from q 0 if (q 0,w) F return yes otherwise return no The algorithm will always halt after w steps. 19

20 NFA Membership Input: NFA B and a string w Output: yes if w L(B); no otherwise. Theorem: NFA membership is decidable. 20

21 Proof of decidability The algorithm to decide the problem is: Convert NFA B to an equivalent DFA C Apply DFA membership TM to C and w If TM accepts w, accept, otherwise reject. The algorithm will always halt after w steps. 21

22 CFG Membership Input: CFG G and a string w Output: yes if w L(G); no otherwise. Theorem: CFG membership is decidable. 22

23 Proof of decidability Option 1: List all derivations ={u S * u} Check whether w. Will it work? Option2: Convert G into equivalent Chomsky NF. List all derivations ={u S * u} with 2 w -1 steps Check whether w 23

24 TM Membership Input: TM M and a string w Output: yes if M accepts w; no otherwise. Theorem: TM membership is undecidable. 24

25 Proof of un-decidability In next course! 25

26 Classes of Languages Turing -recognizable decidable context-free regular 26

27 Recap 27

28 Recap Regular Languages: Finite Automata (Q,,,q 0,F) Regular languages Closures of RL (union, intersection, complements, minus) Non deterministic automaton: : Qx 2 Q Equivalence DFA NFA Closures of RL (concatenation, Kleene star). 28

29 Recap Regular Expressions Equivalence DFA RE Minimal DFA and equivalent DFAs Non regular languages: The Pumping Lemma 29

30 Recap Context-Free Languages: Context-Free Grammar G=(V,, S, R) Regular grammar (A w, A wb) Equivalence NFA regular-grammar Closure of CFL (Union, Concatenation, Kleene star) Chomsky Normal Form (A BC, A ) Chomsky Hierarchy of grammars 30

31 Recap Pushdown automata (Q,,,, q 0, F) Equivalence PDA CFG Non context-free languages: The pumping lemma Turing Machines Decidability 31

32 Aims Practical techniques and tools - useful in compilation, translation, data compression etc. Formalization of the concept of language - useful in formal linguistics etc. Computational models - useful in answering basic questions about computations. Understanding of the computer abilities and its limits In a problem Solvable? At what resources? 32

33 THE END 33

Automata and Computability. Solutions to Exercises

Automata and Computability. Solutions to Exercises Automata and Computability Solutions to Exercises Fall 25 Alexis Maciel Department of Computer Science Clarkson University Copyright c 25 Alexis Maciel ii Contents Preface vii Introduction 2 Finite Automata