Homework • Homework #5 Turing Machines – Returned today • Homework #7 (due Thursday) – Help after lecture And the languages they handle Before We Start Languages • Any questions? • Is that your final answer? – What is a language? – What is a class of languages? Now our picture looks like Turing Machine Context Free Languages • A Machine consists of: – A state machine Deterministic Context Free Languages – An input tape – A movable r/w tape head Regular Languages • A move of a Turning Machine Finite – Read the character on the tape at the current position of Languages the tape head – Change the character on the tape at the current position of the tape head – Move the tape head We’re going to start to look at – Change the state of the machine based on current state languages out here and character read 1 Turing Machine Turing Machine • Tape that holds • In the original Turing Machine Input tape (input/memory) ... character string – The read tape was infinite in both directions Read head • Movable tape head that – The text describes a semi-infinite TM where reads and writes • The tape is bounded on the left and infinite on the State character right Machine • It can be shown that the semi-infinite TM is • Machine that changes (program) equivalent to the basic TM. state based on what is – As not to confuse, I’ll follow the conventions in read in the book…(as much as that bothers me!) Turing Machines Turing Machines • About the input tape • About the machine states – Bounded to the left, infinite to the right – A Turing Machine does not have a set of – All cells on the tape are originally filled with a accepting states special “blank” character ∆ – Instead, each TM has a special halting state, h. – Tape head is read/write • Once in the halting state, the machine halts. – Tape head can not move to the left of the start – Unlike PDAs, the basic TM is deterministic! of the tape • If it tries, the machine “crashes” Turing Machines Turing Machines • Let’s formalize this • Transition function: – A Turing Machine M is a 5-tuple: – δ: Q x (Γ∪{ ∆ }) → (Q ∪ {h}) x (Γ∪{ ∆ }) x {R, L, Σ Γ δ S} – M = (Q, , , q0, ) where • Q = a finite set of states (assumed not to contain the halting state h) – Input: • Σ = input alphabet (strings to be used as input) • Current state • Γ = tape alphabet (chars that can be written onto the tape. • Tape symbol read at current position of tape head Includes symbols from Σ) – Output: • Both Σ and Γ are assumed not to contain the “blank” symbol • State in which to move the machine (can be the halting state) • δ = transition function • Tape symbol to write at the current position of the tape head (can be the “blank” symbol) • Direction in which to move the tape head (R = right, L = left, S= stationary) 2 Turing Machine Turing Machine • Accepting a string • Running a Turing Machine – A string x is accepted by a TM, if – The execution of a TM can result in 4 possible • Starting in the initial configuration cases: • With x on the input tape • The machine “halts” (ACCEPT) • The machine eventually ends up in the halting state. • The machine has nowhere to go (REJECT) –I.e. • The machine “crashes” (REJECT) ∆ * • The machine goes into an “infinite loop” (REJECT •(q0, x) a (h, xay) – for x,y ∈(Γ∪{∆})*, a ∈(Γ∪{∆}) but keeps us guessing!) Turing Machine Computation with Turing Machines • Language accepted by a TM • Computing functions with TMs – The language accepted by a TM is the set of all – The result of the function applied to an input input strings x on which the machine halts. string x, will be left on the tape when the machine halts. – Functions with multiple arguments can be placed on the input tape with arguments separated by blanks. • Questions? Computation with Turing Machines Computation with Turing Machines • Computing functions with TMs • Computing functions with TMs – Formally, – Formally (with multiple arguments) Σ Γ δ Σ Γ δ • Let T = (Q, , , q0, ) be a TM and let f be a • Let T = (Q, , , q0, ) be a TM and let f be a partial function on Σ*. We say T computes f if for partial function on (Σ*) k. We say T computes f if ∈Σ* every x where f is defined: for every (x1, x2. …, xk) where f is defined: ∆ * ∆ ∆ ∆ ∆ * ∆ –(q0, x) a (h, f(x)) –(q0, x1 x2… xk) a (h, f(x1, x2. …, xk)) • And for every other x, T fails to halt on input x. • And for every other k-tuple, T fails to halt on input x. 3 Computation with Turing Machines TMs and languages • Characteristic function of a set: • Two classes of languages that involve TMs: – For any language L, the characteristic function is defined as: – Languages, L accepted by TM •1 if x ∈ L • Halts on x ∈L, “rejects” on x ∉L • 0 otherwise – Languages, L recognized by a TM – If a TM computes the characteristic function of a • TM computes the characteristic function of L language, it will always halt with either a 1 or 0 left on • Halts on all x the tape • Give yes/no answer as to whether x ∈L • This is stronger than saying that x is accepted by the TM since if x ∉ L, for a TM that accepts L, there is no guarantee that the the machine will halt. TMs and languages TMs and languages • Two classes of languages that involve TMs: • First observation: – A language is recursively enumerable if there is – Every recursive language is also recursively a TM that accepts it enumerable – A language is recursive if there is a TM that • Modify the TM that recognizes the language so that recognizes it. it goes into a “nowhere state” just before placing the 0 on the tape indicating that a string is not in the language. TMs and languages TMs and languages • Game plan Is there something out 1. Show that there exists a language that is not Recursively here recursively enumerable Enumerable 2. Show that there exists a language that is recursively enumerable but not recursive. Recursive Is there something in here 4 A language that is not recursively enumerable Universal Turing Machine • Sorry, no pumping Lemma for TMs! • Universal Turing Machine –A TM Tu that takes as input, an encoded version of another TM M’ and an input to M’. • Instead we introduce the notion of a – M will simulate the processing of M’ Universal Turning Machine • If M’ halts on input x, Tu will halt • If M’ rejects on input x,Tu will reject. –Tu is a general purpose TM • M’ is a program run on the general purpose TM Universal Turing Machine Universal Turing Machine • Encoding scheme • Encoding scheme – We need a way to encode a TM so that it can be – Associate states, symbols, and directions with provided as input to T u binary numbers: – Define sets: • States: • = {q , q , … } = set of all possible states that may appear in Q 1 2 → any TM – s(h) 0 → i+1 – s(qi) 0 • S = {a1, a2, a3, …} = set of all possible tapes symbols that can be written on any TM. • Symbols –So for any TM –s(∆) → 0 ⊆ Γ⊆ → i+1 •Q Q and S – s(ai) 0 Universal Turing Machine Universal Turing Machine • Encoding scheme • Encoding scheme – Associate states, symbols, and directions with – Encoding transitions: binary numbers: • Encode each state, symbol, direction • Directions: • Separate by 1’a – s(S) → 0 – The transition δ (p,a) = (q, b, D) can be → – s(L) 00 encoded as: – s(R) → 000 • s(p)1s(a)1s(q)1s(b)1s(D) 5 Universal Turing Machine Universal Turing Machine • Encoding • The input to Tu will be – Encoding for a TM – an encoded TM and • Encode the start state – a string x to run on that TM • Encode each of it’s moves – We encode x by individually encoding the • Separate by 1’s characters of x (separated by 1’s) – e(T) = s(q)1e(m )1e(m )1…1e(m )1 1 2 k •x = z1z2…zn • e(x) = s(z1)1s(z2)1…1s(zn)1 Universal Turing Machine Universal Turing Machine • Encoding Scheme • Encoding example – Finally, the description of the TM and the – States •q– state 1 encoding of the string to run on the TM will be 0 • p – state 2 separated by 2 1’s • s – state 3 • Input to Tu – Symbols – e(M)11e(x) • a – symbol 1 • b – symbol 2 Universal Turing Machine Universal Turing Machine • Encoding example • Encoding scheme – States • s (h) = 0 – The actual scheme isn’t really important (in •s(q) = 00 0 fact, there are many) • s(p) = 000 • s(r) = 0000 – What is important is that we can encode a TM – Symbols and it’s input as a binary string. •s(∆) = 0 • s(a) = 00 • s(b) = 000 001 00101000101 000 – Directions • s(S) → 0 •s(L) → 00 • s(R) → 000 6 Universal Turing Machine Universal Turing Machine • For the mathematically inclined… • How it works – The set of all TMs is countably infinite –3 tape TM • We can define a one-to-one and onto function • Tape 1 – Input and output tape between the set of TMs and the set of natural • Tape 2 – Working tape numbers { 0, 1, 2, …} • Tape 3 – encoded form of the state that machine being simulated is in – Note that a 3 tape TM is equivalent to a “basic” TM. Universal Turing Machine Universal Turing Machine • How it works • On a single move, • How it works ... – The symbol on each – When we start ∆e(T)11e(x) is on tape 1 tape is read – Step 1: Move e(x) from tape 1 to tape 2 and ..