Modeling algorithms

1. Modeling algorithms • Neither the class of DFAs nor the class of PDAs are sufficiently general to model the informal notion of an algorithm. • For example, no DFA or PDA can check membership in {0n1n2n | n > 0}. • There are several formalisms that attempt to model this informal notion. • The Turing machine is one of the simplest • It’s a fairly natural generalization of DFAs and PDAs.

2. Turing machines (informally) • A Turing machine (TM) is a DFA equipped with a tape, infinite in both directions. • The tape is divided into squares. • At any point in a TM computation, the TM is scanning exactly one of the squares. • At each move of a TM computation, the TM • may change state • may rewrite the current tape square, and • must move to the tape square to the left or right

3. Turing machine input and acceptance • The input to a TM appears at a designated position of the tape. • It remains on the tape unless it is rewritten. • When the TM enters an accepting state, it accepts the entire input string. • There's no notion of having to read the entire input.

4. Turing machine computations • TMs may compute forever on a given input • So one of three things may happen in a TM computation. It may • never halt, • halt in an accepting state • halt (crash) by being in a nonaccepting state with no legal move • Only in the second case does the TM accept its input.

5. Liberalized definitions • The definition of Turing machines can be liberalized in many ways • without changing the languages accepted • In each case, one shows this by showing that a TM of the more liberal type may be simulated by a TM of a less liberal type.

6. Sample liberalized definitions • For example, TMs may be allowed: • to have multiple tracks on their tape • to have multiple tapes • to check off symbols • to shift a string on the tape • subroutines

7. The power of Turing machines • By proceeding in this way, one may show that TMs have the power of any existing processor or high-level language. • More precisely, for any existing programming language (high or low level), TMs can accept any language that can be recognized by a program in the language.

8. Turing machines as computers of functions • A TM's tape allows it to have more general output than "yes" or "no". • So a TM can be used to compute functions • these functions needn’t be Boolean valued • When computing functions, a TM can compute no more and no fewer functions that existing programming languages can • if they were somehow given unbounded memory

9. Turing machines and algorithms • An algorithm corresponds to TM that halts on all inputs. • This notion of an algorithm is one version of what is known as Turing's thesis • or Church’s thesis, or the Church-Turing thesis.

10. Justifying Turing’s thesis • Turing's thesis cannot be proved, since it necessarily deals with an unformalized notion of an algorithm. • It would be fatally weakened, though, if someone were to find a language or machine that could compute more than a TM can. • So far no one has.

11. Nondeterministic TMs • Nondeterministic TMs can be shown to have no more power than ordinary deterministic TMs. • And quantum computing gives no more power than nondeterministic TMs have

12. Universal TMs • Real computers are universal • they aren't limited to a particular algorithm. • They can take and run different programs • representing different algorithms • A universal TM would be a TM that takes another TM M and a string x as input, and runs M on x. Such a TM can be defined • once we decide what its alphabet should be • In this sense, TMs model computers, and not just algorithms.

13. Unsolvable problems • Universal TMs make it easier to investigate the question of whether there are problems that can't be solved. • In the language of TMs, this would correspond to languages that can't be recognized.

14. Sample unsolvable problems • For example, there's no solution algorithm to the problem of whether a given TM M accepts a given input x. • Here the input to a solution algorithm would be an encoding of the pair (M,x). • The set of such pairs could not be recognized by a TM. • There's not even an algorithm to decide whether M halts on x.

15. Turing machines -- formally • A Turing machine (TM) has components • Q, S, q0, and F, as for DFAs and PDAs • a tape alphabetG containing S as a subset • a special blank symbol □, which is in G - S • and transition function d: Q x G → Q x G x {L, R}

16. Turing machines and computations • The input appears on the TM's tape • starting at a designated square 0 • one symbol per square • with blanks in all other squares • A TM computation begins in state q0 with its read/write head scanning the first symbol of the input. • d determines the new state and tape symbol, and whether to move left or right on the tape • after the tape symbol is updated

17. A sample TM • A TM to recognize L((a+b)*b) could have states q0, q1, and qf, with F = {qf} • Its transition function could be defined by • d(q0, a) = (q0,a,R) d(q0,b) = (q1,b,R) • d(q1, a) = (q1,a,R) d(q1,b) = (q1,b,R) • d(q1, □) = (qf, □,R) • Here q1 represents a state where b was last read, and q0 a state in which it wasn't

18. Diagrams for TMs • Transition diagrams may be used for TMs • one for our sample DFA is shown below • Each edge is labeled with • the old & new tape symbols for the current square • and the direction of movement on the tape

19. Instantaneous descriptions of TMs • An ID for TMs needs to specify at least • the state, • the (nonblank) tape contents, and • which tape square the is being scanned • This can be done by listing the tape contents, and inserting the state before the currently scanned tape symbol. We may assume that • Q and G are disjoint • only finitely many tape symbols are nonblank • The initial ID for an input of 012 is q0012

20. TM computations • Again we may define computation in terms of a ├ relation on IDs • If d(qi, c) = (qk,e,R), then aqicb ├ aeqkb • If d(qi, c) = (qk,e,L), then adqicb ├ aqkdeb

21. Acceptance by TMs • We use as before the symbol ├* for the transitive closure of the ├ relation • Then a TM M accepts x iff q0x |*- aqb • for some a,b in G* and q ε F • L(M) is again { x | M accepts x} • A language that is accepted by a TM is called recursively enumerable (r.e.).

22. A sample accepting computation • The accepting computation for ab for the TM of p. 232 of Linz is • q0ab ├ xq1b ├ q2xy ├ xq0y ├ xyq3 ├ xy□q4 • The intuition is that • q0 looks for a's and checks them off • q1 looks for b's and checks them off • q2 moves leftward looking for the leftmost remaining a • q3 checks that no b's remain if q0 finds no a

23. A sample nonaccepting computation • A TM rejects its input if the computation crashes • that is, halts because there is no legal move • A nonaccepting computation for the TM above is • q0abb ├ xq1bb ├ q2xyb ├ xq0yb ├ xyq3b • Here q3 is not expecting the symbol b

24. Another example • An accepting computation for our first sample TM (shown below) is • q0abb ├ aq0bb ├ abq1b ├ abbq1 ├ abb□q2

25. Halting by TMs • A given TM M may fail to halt on a given input string x • In this case, x is not in L(M) • It's not always possible to find an equivalent TM for L(M) that halts on all input. • A language that is accepted by a TM that halts on all input is called recursive. • The Church-Turing thesis says that a language is recursive iff there's an algorithm for deciding membership in the language.

26. Recursive and r.e. languages • The r.e. languages are those for which there's a procedure • that correctly answers "yes" whenever a string is in the language • but may compute forever if the answer is "no". • In particular, every recursive language is r.e. • It follows from a recent assertion that not every r.e. language is recursive