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Home , Alonzo Church, Decidability (logic), Turing machine

Lecture 14: Menu Church -Turing Thesis • Finish computing model for TM • The Most Bogus Sentence • Robustness of TM Model (Church-Turing Thesis)

Alonzo Church (1903-1995) (1912-1954) cs302: Theory of Reminder: PS4 is due Tuesday University of Virginia David Evans Science http://www.cs.virginia.edu/evans Lecture 14: Church-Turing Thesis 2

Turing Formal Description

...... FSM 7-: (Q, Σ, Γ, δ, q , q , q ) FSM 0 accept reject Infinite tape: Γ* Q: finite set of states Tape head: read current square on tape, Σ: input alphabet (cannot include blank symbol, _) write into current square, Γ: tape alphabet, includes Σ and _ move one square left or right δ: transition : Q × Γ → Q × Γ × { , } ∈ FSM: like PDA, except: q0: start state, q0 Q ∈ transitions also include direction ( left /right ) qaccept : accepting state, qaccept Q ∈ final accepting and rejecting states qreject : rejecting state, qreject Q (Sipser’s notation)

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Turing Machine Computing Model TM Computing Model

x∈Σ Initial configuration: δ*: Γ* × Q × Γ* → Γ* × Q × Γ*

x x x x x x x ____ . . . The qaccept and qreject states are final:

blanks δ*( L, qaccept , R) → (L, qaccept , R) input FSM δ*( L, qreject , R) → (L, qreject , R) q0 TM Configuration: Γ* × Q × Γ*

tape contents current tape contents left of head FSM state head and right

Lecture 14: Church-Turing Thesis 5 Lecture 14: Church-Turing Thesis 6 TM Computing Model TM Computing Model δ*: Γ* × Q × Γ* → Γ* × Q × Γ* δ*: Γ* × Q × Γ* → Γ* × Q × Γ* u, v ∈ Γ*, a, b ∈ Γ u, v ∈ Γ*, a, b ∈ Γ

a b . . . a b . . .

u FSM v u FSM v q q δ*( ua , q, bv ) = δ*( uac , q , v) if δ(q, b) = ( q , c, R) δ*( ua , q, bv ) = δ*( uac , q , v) if δ(q, b) = ( q , c, R) r r r r δ*( ua , q, bv ) = δ*( u, q , acv ) if δ(q, b) = ( q , c, L) δ*( ua , q, bv ) = δ*( u, q , acv ) if δ(q, b) = ( q , c, L) r r r r δ L δ*( ε, q, bv ) = δ*( ε, qr, cv ) if (q, b) = ( qr, c, ) Also: need a rule to cover what happens at left edge of tape Do we need a rule for the right edge of the tape?

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TM Computing Model Termination

δ*: Γ* × Q × Γ* → Γ* × Q × Γ* • DFAs, DPDAs: A string w is in the language of Turing – Consume one input symbol each step Machine T if – Must terminate • NFAs: δ*(ε, q0, w) = ( α, qaccept ,β) – Equivalent to DFA: must terminate A string w is not in the language of Turing Machine T if • Turing Machine: – Can move left and right: no “progress” guarantee δ*(ε, q0, w) = ( α, qreject ,β) Does this cover all possibilities?

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Possible Outcomes Recognizing vs. Deciding

1. Running TM M on input w eventually leads • Turing-recognizable : A language L is “Turing- recognizable” if there exists a TM M such that for all to qaccept . strings w:

2. Running TM M on input w eventually leads – If w ∈ L eventually M enters qaccept – If w ∉ L either M enters q to q . reject reject or M never terminates 3. Running TM M on input w runs forever • Turing-decidable : A language L is “decidable” if (never terminates). there exists a TM M such that for all strings w:

– If w ∈ L, M enters qaccept .

– If w ∉ L, M enters qreject .

Lecture 14: Church-Turing Thesis 11 Lecture 14: Church-Turing Thesis 12 Decidable and Recognizable Languages Decider vs. Recognizer? Recognizable Decidable Deciders always terminate.

Recognizers can run forever without

deciding. Do we know this picture is right yet?

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A bogus sentence (but not the one I The Most Bogus Sentence Guesses had in mind) • “Intuitive notion of equals Turing machine algorithms.” • “Some of these models are very much like Turing , but others are quite different.” • “To show that two models are equivalent • “Think of these as 'virtual' tapes and heads,” on page 149. The we simply need to show that we can quotation marks around virtual imply that the tapes and heads are not virtual, which is false. simulate one by the other.” • “If you feel the need to review nondeterminism, turn to Section Winner: David Horres 1.2 (page 47).” (By this point, one should have a firm grasp of nondeterminism.) A B • “Proving an doesn't exist requires having a clear definition of algorithm.” For set equivalence, need to show A ⊆ B and B ⊆ A. • “For mathematicians of that period to come to this conclusion [(Hilbert’s 10th Problem’s accepted solution)] with their intuitive For machine equivalence, need to show concept of algorithm would have been virtually impossible.” A can simulate B and B can simulate A. I don’t find any of these statement bogus.

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The Most Bogus Sentence Things Real Can Do

“A Turning machine can do everything a real computer can do.”

On the first page! Generate Heat

Provide an adequate habitat for fish Winners: Erin Carson, Emily Lam, Ruixin Yang, Stop a Door

Lecture 14: Church-Turing Thesis 17 Lecture 14: Church-Turing Thesis 18 Computational Thing Most Real Computers Can Do (that Turing Machines can’t) Church-Turing Thesis

Generate randomness

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Alonzo Church’s “Less Successful” Alan Turing (1912-1954) PhD Students • Published On Computable , with an Application to the (1936) – Introduced the – Formal Michael Rabin (now known as “Turing Machine”) Hartley Rogers • Codebreaker at Bletchley Park – Involved in breaking Enigma Cipher • After the war: convicted of homosexuality (then a crime in Britain), Stephen Kleene committed suicide eating cyanide apple John Kemeny Martin Davis See http://www.genealogy.ams.org/id.php?id=8011 for full list

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Church-Turing Thesis

• As stated by Kleene: Every effectively calculable function (effectively decidable predicate) is general recursive . “Since a precise mathematical definition of the term effectively calculable (effectively decidable) has been wanting, we can take this thesis ... as a definition of it...”

Yes, this is circular: everything calculable can be computed by a TM, and we define what is calculable as what can be computed by a TM.

Lecture 14: Church-Turing Thesis 23 Lecture 14: Church-Turing Thesis 24 Church-Turing Thesis Examples

• Any mechanical computation can be performed by a Turing • [Last class and PS4] Equivalence of TM and 2-stack Machine deterministic PDA + ε-transitions • There is a TM-n corresponding to every computable problem • [PS4] Making the tape infinite in both directions adds no power • We can model any mechanical computer with a TM • [Soon] Adding a second tape adds no power • The set of languages that can be decided by a TM is identical to • [Church] is equivalent to TM the set of languages that can be decided by any mechanical • [Chomsky] Unrestricted replacement grammars are equivalent computing machine to TM • If there is no TM that decides problem P, there is no algorithm • [Takahara and Yokomori] DNA is at least as powerful as a TM that solves problem P. • [Hotly Debated] Is the human brain equivalent to a TM?

All of these statements are implied by the Church-Turing thesis “Some of these models are very much like Turing machines, but others are quite different.” (not such a bogus sentence)

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Charge

• Next week: what languages cannot be recognized by a TM? • Read Chapter 4: – I don’t think it has any extremely bogus sentences, but if you find one send it to me…

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