BU CS 332 Theory of Computation Lecture 11: Reading: TM Variants - - PowerPoint PPT Presentation

BU CS 332 – Theory of Computation

Lecture 11:

• TM Variants
• Closure Properties

Reading: Sipser Ch 3.2

Mark Bun March 1, 2020

The Basic Turing Machine (TM)

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Tape 𝑏 𝑐 𝑏 𝑏 Finite control …

• Input is written on an infinitely long tape
• Head can both read and write, and move in both

directions

• Computation halts when control reaches

“accept” or “reject” state

Input

0 → 0, 𝑆 ⊔ → ⊔, 𝑆

accept reject

0 → 0, 𝑆 ⊔ → ⊔, 𝑆

Example

𝑟0 𝑟1

Formal Definition of a TM

A TM is a 7‐tuple

• is a finite set of states
• is the input alphabet (does not include )
• is the tape alphabet (contains

and )

• is the transition function

…more on this later

• is the start state
• is the accept state
• is the reject state (

)

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TM Transition Function

means “move left” and means “move right” means:

• Replace 𝑏 with 𝑐 in current cell
• Transition from state 𝑞 to state 𝑟
• Move tape head right

means:

• Replace 𝑏 with 𝑐 in current cell
• Transition from state 𝑞 to state 𝑟
• Move tape head left UNLESS we are at left end of tape, in

which case don’t move

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Configuration of a TM

A string with captures the state of a TM together with the contents of the tape

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• …

Configuration of a TM: Formally

A configuration is a string where and

∗

• Tape contents =

(followed by blanks )

• Current state =
• Tape head on first symbol of

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• …

How a TM Computes

Start configuration: One step of computation:

• yields

if

• yields

if

• yields

if Accepting configuration: = Rejecting configuration: =

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How a TM Computes

accepts input if there is a sequence of configurations

• such that:
• =
• yields for every
• is an accepting configuration

the set of all strings which accepts is Turing‐recognizable if for some TM :

• halts on

in state

• halts on

in state OR runs forever

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Recognizers vs. Deciders

the set of all strings which accepts is Turing‐recognizable if for some TM :

• halts on

in state

• halts on

in state OR runs forever is (Turing‐)decidable if for some TM which halts on every input

• halts on

in state

• halts on

in state

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Back to Hilbert’s Tenth Problem

Computational Problem: Given a Diophantine equation, does it have a solution over the integers?

• is Turing‐recognizable
• is not decidable (1949‐70)

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TM Variants

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How Robust is the TM Model?

Does changing the model result in different languages being recognizable / decidable? So far we’ve seen… ‐ We can require that FAs/PDAs have a single accept state ‐ (CFGs can always be put in Chomsky Normal Form) ‐ Adding nondeterminism does not change the languages recognized by finite automata Turing machines have an astonishing level of robustness

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Extensions that do not increase the power of the TM model

• TMs that are allowed to “stay put” instead of moving

left or right Proof that TMs with “stay put” are no more powerful: Simulation: Convert any TM with “stay put” into an equivalent TM without Replace every “stay put” instruction in with a move right instruction, followed by a move left instruction in ’

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Extensions that do not increase the power of the TM model

• TMs with a 2‐way infinite tape, unbounded left to right

Proof that TMs with 2‐way infinite tapes are no more powerful: Simulation: Convert any TM with 2‐way infinite tape into a 1‐way infinite TM with a “two‐track tape”

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Tape 𝑏 𝑐 𝑏 … Input …

Formalizing the Simulation

• New tape alphabet:

New state set:

• means “ , working on upper track”

means “ , working on lower track” New transitions:

If 𝜀 𝑞, 𝑏 𝑟, 𝑐, 𝑀, let 𝜀′ 𝑞, , 𝑏, 𝑏 𝑟, , 𝑐, 𝑏 , 𝑆 Also need new transitions for moving right, lower track, hitting $, initializing input into 2‐track format

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TMs are equivalent to…

• TMs with “stay put”
• TMs with 2‐way infinite tapes
• Multi‐tape TMs
• Nondeterministic TMs
• Random access TMs
• Enumerators
• Finite automata with access to an unbounded queue = 2‐

stack PDAs

• Primitive recursive functions
• Cellular automata

…

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Church‐Turing Thesis

The equivalence of these models is a mathematical theorem Church‐Turing Thesis: Each of these models captures our intuitive notion of algorithms The Church‐Turing Thesis is not a mathematical statement!

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Multi‐Tape TMs

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𝑐 𝑐 𝑏 𝑏 𝑏 Finite control 𝑏 𝑐 ⊔ 𝑏 𝑏 ⊔ 𝑐 𝑏 𝑏 𝑑

Fixed number of tapes (can’t change during computation) Transition function

Multi‐Tape TMs are Equivalent to Single‐Tape TMs

Theorem: Every ‐tape TM with can be simulated by an equivalent single‐tape TM

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𝑐 𝑐 𝑏 𝑏 Finite control 𝑏 𝑐 ⊔ 𝑏 ⊔ 𝑐 𝑏 𝑏 ⊔ 𝑐 𝑏 𝑏 𝑑 # 𝑏 𝑐 ⊔ 𝑏 # 𝑐 𝑐 𝑏 𝑏 # Finite control