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Automata, Grammars and Languages

This article provides an overview of Turing Machines (TM), their basic features, and their applications in computer science. It explains the components of a TM, its acceptance and recognition capabilities, and its relationship with other models of computation such as RAM. The article also discusses the concept of TM extensions and Church's Thesis.

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Automata, Grammars and Languages

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  1. Automata, Grammars and Languages Discourse 05 Turing Machines C SC 473 Automata, Grammars & Languages

  2. b The TM Model current state 0 1 2 3  Unbounded blank tape    • Model of ``effectively computable procedure” (including all algorithms) • Primitive as possible (math. simplicity & constructions) • Arbitrary definition choice, but standard • Easy comparison with other models (RAM, etc.) • Finitely describable (just a formatted string) • 1 machine = 1 procedure (no stored program -- different FSA for each TM) • Computes in discrete steps (moves); each step physically trivial • Simple complexity measures (steps=time, cells=space) ⊔ ⊔ Fixed # bits R/W head C SC 473 Automata, Grammars & Languages

  3. b {⊔} The TM Model 0 1 2 3     • Control unit: FSA with state set Q • Input: character in cell under R/W head • Output: new state & overstrike character & head moves L or R • Initial state, accept state, reject state: • Tape unit • Finite length string of characters with blanks ⊔⊔⊔⊔⋯ to right • ⊔ marks R end of input • Fixed alphabet  ( ⊔  ); input alphabet • Tape bounded on left & unbounded on right  Finite- state control C SC 473 Automata, Grammars & Languages

  4. TM Examples input and R/W head left adjusted at start • TM to scan R over non-blanks to halt over first blank • TM to accept odd parity # 1’s 0 0, R ⊔ ⊔, L  1 1, R 0 0, R 1 1, R 0 0, R 1 1, R ⊔ ⊔, L ⊔ ⊔, L C SC 473 Automata, Grammars & Languages

  5. TM Definition (1-tape deterministic TM) • A 7-tuple • Q = finite set of states •  = finite tape alphabet; blank ⊔;input alphabet • Transition function • initial state ; halt (accept/reject) states: • Configuration • (1-step) yields relation between configurations    ⊔  Note: Below says you can move left from leftmost cell, although you don’t really move left! This convention makes accept, reject the only halt states.   C SC 473 Automata, Grammars & Languages

  6. Acceptance • Move relation (multi-step): • w* is accepted by M  • Language recognized by M: • TM-recognizable language – a set such that • Configuration I is halting: • Nowhere to move: from definition, only2 ways to halt: • A rejecting or accepting configuration: • A configuration for which no move is defined • Initial configuration: • Any halting configuration without is non-accepting • Can always “force” non-accepting, halting configs to be rejecting • For undefined just add transitions to * * *  C SC 473 Automata, Grammars & Languages

  7. yes yes Recognizer for L Decider for L w w no Turing-decidable vs. Turing-recognizable • A language L is Turing-recognizableiff for some TM M . • M need not, but might, halt rejecting on some strings  L • Such a TM is an acceptor or recognizer for L • A language L is Turing-decidable iff for some TM M that halts on every input string (accepts or rejects). Such a TM is called a decider or algorithm. “procedure” “algorithm” Note: ``yes’’ means enters and halts ``no’’ means enters and halts C SC 473 Automata, Grammars & Languages

  8. Example BB, L CC, L bb, R CC, R • Acceptor for aa, R BB, R a A,R cC, L bB, R B B, R AA, L aa, L bb, L cc, L ⊔ ⊔,L A A, R B B, R C C, R C C, R All transitions not shown go to (See dotted example) So halts rejecting unless accepting statereached ⊔ ⊔,L C SC 473 Automata, Grammars & Languages

  9.    Acceptance (Cont.) • Sequence of configurations is a computation of length (time) t • Computation is in a tight loop iff (a config. repeats) (1) How does a TM accept w? (2) How does a TM reject w? • in a tight loop • a halting & rejecting config • computation never reaches a halting config., and no configuration repeats. Call this a divergent computation. NOTE: Can make (1) coincide with halting. Can eliminate (a) in favor of (b). (How?) But there is no algorithm to detect (c ) and eliminate (as we shall see)    + * *   *   C SC 473 Automata, Grammars & Languages

  10. Acceptance (Cont.) • Recognizers and Deciders – summary • M recognizes • M decides • recognizes • rejects (& halts) • S is a Turing-recognizable language if there is some TM that recognizes it • S is a Decidable language iff there is some TM that decides it • NOTE: every TM recognizes some language, but only “special” TMs—ones that halt for every input—decide the language they recognize M is called a decider or algotithm C SC 473 Automata, Grammars & Languages

  11. A Look Ahead sets Decidable  C Turing- Recognizable Context-free  B Regular  A  D    E C SC 473 Automata, Grammars & Languages

  12. TM Extensions: Church’s Thesis • Evidence for Church’s Thesis • Church’s Thesis (Church-Turing Thesis): the effectively computable functions are those characterized by one the “standard formalisms” such as the TM. • Convenience of having alternate models • Two models of computation are equivalent  • whenever a language L is recognized by a machine in one model, there is an algorithm to construct a machine recognizing L in the second, and vice versa • whenever a function f is computed by a machine in one model, there is an algorithm to construct a machine computing f in the second, and vice versa • I.e., Simulation both directions by a “compiling” algorithm C SC 473 Automata, Grammars & Languages

  13. a b c d e f B . . . A B C D E F G . . . a A TM Extensions: k-tape TM • Tapes simulated by one tape with k tracks & software heads • a in cell iof tape 1  a on track 1 of cell i • a in cell iof tape 1 & head 1 scanning cell  on track 1 of cell i • Tape alphabet is head sweeps L to R until  store k scanned chars. In states. Sweep R to L & mark changes. Sweep L to R then R to L to move head marks. End cycle scanning cell 0. B B 0 1 2 3 … 0 1 2 3 … b d e f B . . . B C D E G . . . C SC 473 Automata, Grammars & Languages

  14. Nondeterministic TM • A 7-tuple • Transition function • Transition function • For each (q,a),  (q,a) provides a set of choices • Acceptance means: sequence of configurations leads to a configuration with state . • No use for reject state (rejection is not by entering a state) • Rejection means no possible chain of configurations leads to one with . • No next move is possible: can have C SC 473 Automata, Grammars & Languages

  15. Nondeterministic TM • Acceptance: • note the existential quantifier: w is accepted if there is some sequence of “guesses” that drive the initial tape configuration to an accepting configuration • A word w is not accepted iff every possible computation starting with fails to enter an accepting configuration    C SC 473 Automata, Grammars & Languages

  16. Choice Numbers • Choices for a given state and symbol • For each q and a assign each choice in (q,a) a number • For each machine, there is some largest number of choices for some transition–call it b • Strings over can be interpreted as deterministic directions for which choice to make from each configuration C SC 473 Automata, Grammars & Languages

  17. Computation Tree Choice sequence    C SC 473 Automata, Grammars & Languages

  18. NTM Equivalent to TM • Theorem 3.16. If L is accepted by an NTM N , there is a DTM D, constructable by algorithm, that accepts L. Proof: Simulate all possible computations on N for all possible choice strings and halt if a sequence of choices is found that leads to an accepting configuration. D has 3 tapes: read-only input, worktape simulating that of N, and an enumeration tape. On the latter, enumerate all choice sequences in lexicographic order Main cycle: generate the next choice seq. c on the enum. Tape. Use this sequence to drive computation of N for a total of |c| steps. If an accepting configuration is reached, D accepts and halts. If the computation halts rejecting, move to the next main cycle after clearing the worktape. Otherwise move to the next main cycle after |c| steps. Iterate the main cycle while an accepting configuration has not been found. If an accepting c exists, D will eventually find it. If none exists, the input will not be accepted. So L(D) = L(N). QED C SC 473 Automata, Grammars & Languages

  19. The Universal TM • Any (hardware) TM M can be encoded as a formatted string (software) • Encoding details below • The UTM U readsand simulates the action of M on w • The UTM U is one, fixed, finite machine, capable of simulating any TM (an interpreter) • For example, U reads and gives the same result as for input • We shall see that, whenever universality exists, unsolvability is an inevitable consequence C SC 473 Automata, Grammars & Languages

  20. Canonical Encoding of TM M • Let • Encode over 9 symbol alphabet object in Mencoding in C SC 473 Automata, Grammars & Languages

  21. Canonical Encoding of TM M (cont.) • is a word over • Final encoding is ASCII (binary) of that string • Notational convention: an encoding of a TM M ``is” (the binary representation of) a natural number • number is called the Gödel number of M • Conversely if e is a natural number, is the TM with that Gödel number • If e is a syntactically invalid code, is by definition a TM that halts and prints 0 on every input C SC 473 Automata, Grammars & Languages

  22. UTM Construction • Use 4 tapes: input program tape, state tape w & worktape of M being simulated • Parse input, copying to program tape. If invalid, put [0] on worktape and halt. Else copy w to worktape & put [0] on state tape simulating state • While the state tape [1] do { • If the state tape contains count over totuples for that state in program Use the character a under scan on the worktape ([0],[1],…) to scan to the correct 5-tuple • Overwrite the scanned character c(a) on the worktape by c(b) and move the head direction D on the worktape. • Copy the new state string from the program tape to the state tape (after erasing the previous state string) } C SC 473 Automata, Grammars & Languages

  23. UTM Construction • 4 tapes: input to U, tape to hold , tape to hold state of and worktape of . . . read-only input read-only program [ 0 ] B B B . . . [ 0 ] [ 1 ] . . . state of M worktape of M C SC 473 Automata, Grammars & Languages

  24. 2-State, 3-Tape Symbol UTM • http://www.wolframscience.com/prizes/tm23/ C SC 473 Automata, Grammars & Languages

  25. TM Extensions: Enumerators • TM that prints out a list of strings, separated by blanks • Starts with an empty tape • Defines a set of strings (language) • Strings need not be unique (repeated strings allowed) • Strings can be enumerated in no particular order • Usually does not halt; runs on forever printing strings • Finite or not, an enumerator E defines a set of strings Theorem 3.21: A language is Turing-recognizable iff it is enumerable by some enumerator. Pf: Must show 2 directions: a language defined by an enumerator has a Turing recognizer, and any TM-recognizable language has an enumerator. We will show there are algorithms for going in both directions. C SC 473 Automata, Grammars & Languages

  26. TM Extensions: Enumerators (cont’d) • Enumerator  Recognizer. Let A = L(E) where E is an enumerator. Using the code for E construct a recognizer M as follows. • M = “On input w: • Continue running E and output the next string s. Pause E. Compare s to w. • If w = s, halt and accept. Otherwise, continue at step 1.” • Machine M recognizes just those strings A that E enumerates. C SC 473 Automata, Grammars & Languages

  27. TM Extensions: Enumerators (cont’d) • Recognizer  Enumerator. Let A = L(M) where M is a TM recognizer. Using the code for M construct an enumerator E as follows. Let be a list of all strings in *. (It is easy to build a routine that generates strings in lexical order: ) • E = “Ignore the input. • for i = 1 to  do { • Run M for i steps on each of the inputs • If any computation sequence accepts an input print it out. } “ • E(eventually) enumerates every string M accepts • Technique: “interleaving computations” C SC 473 Automata, Grammars & Languages

  28. What is ``Effectively Computable’’? • Early 1900s: logicians sought a universal algorithm that would enable “Automatic Theorem-Proving” • Worked to find “deciders” to tell: • whether formulas like are logically true via a “mechanical” method (Decision Problem for First Order Logic) • whether multi-variable polynomial equations like have integer solutions (Hilbert’s 10th Problem) • Want to decide any such questions by ‘computation’ that is ‘effectively performable’ with a finite number of computation steps • Had no idea there were “undecidable” problems C SC 473 Automata, Grammars & Languages

  29. ``Effectively Computable’’ (cont’d) • But what did “effectively computable” mean? • An intuitive notion: ‘you know it when you see it’ • There were no programming languages or computers • Each author had to pick a “mechanically performable” method to illustrate what was computable • 1936: Alan Turing proposed Turing Machines • 1936: Alonzo Church proposed -calculus • 2 remarkable outcomes: • Both definition methods proved to be equivalent (agreed on what was a computable set or function) • Once a method was picked, it was discovered that there must be undecidable sets; • Eventually many more methods proved equivalent: C SC 473 Automata, Grammars & Languages

  30. Computability: The Evidence for Church-Turing’s Thesis Gödel & Herbrand (1934) Kleene (1936) General Recursive Functions Turing (1936) TM Church(1936)-Calculus Markov (1954) Markov Algorithms TM-computable Functions / TM-recognizable Sets Phrase-structure (Type 0) Grammar(1959) Shepherdson & Sturgis (1963) Register Machines(RAM) Any programming language to date Post (1943) Canonical Systems Elgot & Robinson (1964) RASPs For each pair of representations, there is a uniform algorithm (“compiler”) for translating one to the other C SC 473 Automata, Grammars & Languages

  31. Intuitive concept Turing-machine of algorithm algorithm Intuitive concept RAM of algorithm algorithm Intuitive concept Markov of algorithm algorithm Church-Turing Thesis • All methods also agree on what is a “decidable set” and what is a “recognizable set” Thesis [OED] A proposition laid down or stated, esp. as a theme to be discussed and proved, or to be maintained against attack ; a statement, assertion, tenet. C SC 473 Automata, Grammars & Languages

  32. ``Effectively Computable’’ (cont’d) • Both decision problems (Hilberts 10th, 1st Order Predicate Calculus) eventually shown to be undecidable • In our terminology, a decisionproblem is translated into a language. The problem is decidable iff the language has a Turing decider. • None of the following languages have a decider; they are undecidable languages. • This last is called the Halting or Membership Problem. C SC 473 Automata, Grammars & Languages

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