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Turing Machines

Turing Machines. (At last!). Designing Universal Computational Devices Was Not The Only Contribution from Alan Turing…. Enter the year 1940:. The world is at war Nazi Germany has succeeded in conquering most of west Europe Britain is under siege British supply lines are threaten by German

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Turing Machines

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  1. Turing Machines (At last!)

  2. Designing Universal Computational Devices Was Not The Only Contribution from Alan Turing… Enter the year 1940: • The world is at war • Nazi Germany has succeeded in conquering most of west Europe • Britain is under siege • British supply lines are threaten by German • Germany used the Enigma Code, considered unbreakable • Alan Turing led a group of scientist that broke the enigma code Enter the year 2006: The National Security Agency (NSA) needs people! (www.nsa.gov)

  3. Pushdown Automata accepts context-free languages only For example, {anbncn : w  *} is not context-free Background • Automata accepts regular languages only For example, {anbn : n = 0, 1, …} is not regular • We can easily write an algorithm (in C) recognizing if a sequence of characters have the form anbncn or not

  4. The Key Question • Is it possible to design a formal model of a computational device that capture the capabilities of any algorithm? Alan Turing, 1940’s: Yes!

  5. Basic Idea Turing decomposed operations in the device as follows: • A pencil will be “observing” symbols from a paper • The pencil can move left and right • Computational steps erase a symbol observed by the pencil and write a new one in its place • The decision about which symbol should be written and which will be observed next depend on: • The current symbol being observed • The “state of mind” of the device This device is called a Turing Machine

  6. Turing Machine Looks Similar to Finite Automata • Common: • A “state of mind” sounds like the collection of states • A “pencil” sounds like the pointer for the next word Main differences: • Turing machines can write on the “paper” • Turing machines can backup and read symbols again This turns out to be a major difference!

  7. In Turing’s Own Words ... which can be made to do the work of any special-purpose machine, that is to say to carry out any piece of computing, if a tape bearing suitable "instructions" is inserted into it.

  8. Expressivity of Turing Machines (TMs) • TMs can simulate any data structure • TMs can simulate major components of imperative languages: sequence, branching and loop • TMs can control branching and loops • Several extensions have been attempted: adding multiple heads (pencils), tapes, non-determinism, etc. But we can show that TMs can simulate each of these extensions

  9. Church-Turing Thesis Every computer algorithm can be implemented as a Turing machine Therefore, C, C++, Prolog, Lisp, Small talk, and Java programs can be simulated in Turing machines Definition: a programming language is Turing-complete if it is equivalent to a Turing machine.

  10. Control head Tape cell Informal Description … a1 a2 • The head: • Reads the symbol from the cell it is pointing to, and • Either: • Writes a new symbol in the cell, or • Moves one cell to the left or right

  11. Informal Description (II) • New cells can be added to the right of the tape as needed (similar to RAM memory) • These new cells contain the blank symbol,  • The tape is bounded to the left

  12. Transitions can be described by (Case I): ((s,a),(q,b,)) If the machine is in state s and the current cell has an a then jump to state q and write b in the current cell and moves head to right Preliminaries to Formal Description •  denotes the alphabet of characters in language as usual. •  denotes the set of symbols that can be written on tape • It contains  and all symbols in 

  13. Transitions can be described by (Case II): ((s,a),(q,b,)) If the machine is in state s and the current cell has an a then jump to state q and write b in the current cell and moves head to left Preliminaries to Formal Description (II)

  14. Formal Definition Definition. A Turing machine is a 7-tuple (Q, , , , q0, qaccept, qreject), where: • Q is a set of states •  is a set of symbols (the alphabet) •  is a set of symbols that can be written in tape,    and    • q0 Q is the initial state • qaccept is the accepting state • qreject is the rejecting state, qreject  qaccept

  15. Formal Definition (II) •  is a collection of transitions defined by the function: • : (Q  {qaccept, qreject})    Q    {, }

  16. My First Turing Machine • Construct a Turing machine such that: • input: a substring over the alphabet  ={a,b}, the substring contains at least 1 character b • the header is pointing to the first cell in tape • When the Turing machine halts, the header must be pointing to the first occurrence of b in the input a b  a  a b a

  17. Discussion: Are you sure that such a simple model can simulate my C++ program for computing the Minimum Spanning Tree?

  18. The “Eraser” TM Construct a Turing machine that receives as input a substring of a’s and replace each a for a blank space,  a a  a    

  19. (q,h,LS,RS) • The current state, q • The symbol pointed by the head, h • The string left of the head, LS • The string right of the head, RS Configuration for Turing Machines Configuration:

  20. Configuration for Turing Machines (2) Instead of writing: (q,h,LS,RS), we write:LSqhRS If q = qaccept, then LSqhRS is called an accepting configuration If q = qreject, then LSqhRS is called a rejecting configuration Examples: - Initial configuration with 3 a’s for the eraser TM: eq0aaa - Accepting configuration starting with eq0aaa: qaccept

  21. C yields C’, written C * C’: There is a sequence of configurations in one step:     C C0 C1 … C’ Computation in Turing Machines • A configuration C1 = LS1q1h1RS1yields a configuration C2 = LS2h2q2RS2 in one step, written C1 C2, if:  There is a transition: (q1, h1) = (q2,h2,) • The other case is analogous for (q1, h1) = (q2,h2,) • Example: eq0aaa yieldseq0

  22. Two Forms of Computability • Definition. A word w is accepted by a Turing machine M if M yields an accepting configuration starting from the configuration eq0w • Definition. The language L(M) recognized by a Turing machine M: • L(M) = {w | w accepted by M} • Definition. A language L is Turing-recognizable if there is a Turing machine that recognizes L • Definition. A language L is decidable if there is a Turing machine M such that: • L(M) = L • For any word w L, M yields a rejecting configuration starting from the configuration eq0w

  23. Example (programming): Given two nn matrixes A and B to compute ABT we: • Construct a procedure to compute BT, transverse(B,C) • Construct a procedure to multiply two matrices, multiply(B,C,D) • Call transverse(B,C) and then multiply(A,C,D). D contains the answer Constructing Complex Turing Machines Idea: Use a modular approach similar to the one use to construct complex programs

  24. Convention. Let A and B be 2 TMs, and a be a character in  then: a A B Means that: • If A halts in a configuration of the form w1qacceptaw2 wherew1or w2 are any words • Then B starts with the configuration w1q0aw2 Graphical Conventions We will use a convention similar to a flow chart to indicate how to put together Turing machines (TMs). A; if a then B

  25. A B M Means that: • If M halts in a configuration of the form w1qacceptaw2, then A starts with the configuration w1q0aw2 • If M halts in a configuration of the form w1 qacceptw2 with   a, then B starts with the configuration w1q0w2 Graphical Conventions (2) a a M; If a then A else B

  26. A Means that: • If the initial configuration has the form w1q0w2, then A starts with this configuration if   a, where s is the initial state in A • If A halts in a configuration of the form w1qacceptw2 with   a, then A starts with the configuration w1q0w2 Graphical Conventions (3) a while (not a) do A a

  27. Denotes “write a on the current cell” (a is a character in ) Denotes “move to the right of the current cell” (the book uses “R” instead of “”) Denotes “move to the left of the current cell” (the book uses “L” instead of “”) a   Basic Turing Machines

  28. b a Other Simple Turing Machines What it does Machine a “if current cell has an a,then replace it with a b” “if current cell has a character a different than , then move once to the right and write that a”

  29. The  TuringMachine –  –  • What it does? Move to the right until if finds 

  30. –  –   The  TuringMachine Draw diagram for 

  31.  and  – –  – Move to the left until the first non-blank space cell is found   – Move to the right until the first non-blank space cell is found 

  32. R R “move to the right until you find “ “move to the right until you find a symbol other than “ The Conventions • 2 simple rules: • Rule # 1: • Rule # 2: Note: first check. Then move (think of a “while”) “Never forget Rule # 1” (Mr Miogi, The Karate Kid)

  33. Example: Deciding the language (ab)* “yes” a b  a b a b  a a “no”

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