# CS代考计算机代写 algorithm BU CS 332 – Theory of Computation

BU CS 332 – Theory of Computation

Lecture 10:

• Turing Machines

Reading:

Sipser Ch 3.1‐3.2

• TM Variants

Mark Bun February 26, 2020

Turing Machines – Motivation

So far in this class we’ve seen several limited models of computation

Finite Automata / Regular Expressions

• Can do simple pattern matching (e.g., substrings), check parity, addition

• Can’t recognize palindromes

Pushdown Automata / Context‐Free Grammars

• Can count and compare, parse math expressions • Can’t recognize 𝑎𝑏𝑐 𝑛 0

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Turing Machines – Motivation

Goal:

Define a model of computation that is

1) General purpose. Captures all algorithms that can be implemented in any programming language.

2) Mathematically simple. We can hope to prove that things are not computable in this model.

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A Brief History

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1900 – Hilbert’s Tenth Problem

Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.

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David Hilbert

1862‐1943

1928 – The Entscheidungsproblem The “Decision Problem”

Wilhelm Ackermann 1896‐1962

David Hilbert

1862‐1943

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Is there an algorithm which takes as input a formula (in first‐order logic) and decides whether it’s logically valid?

1936 – Solution to the Entscheidungsproblem “An unsolvable problem of elementary

Alonzo Church 1903‐1995

Alan Turing 1912‐1954

Model of computation: Turing Machine CS332 ‐ Theory of Computation 7

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number theory“

Model of computation: 𝜆‐calculus (CS 320)

“On computable numbers, with an application to the Entscheidungsproblem”

Turing Machines

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The Basic Turing Machine (TM)

Tape 𝑎𝑏𝑎𝑎 Finite

…

Input

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

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Example

𝑞0

⊔ → ⊔, 𝑅

accept

0 → 0,𝑅

⊔ → ⊔, 𝑅 𝑞1

0 → 0,𝑅 reject

Example

reject

𝑞0

⊔ → ⊔, 𝑅

𝑞1

0 → 0,𝑅

accept

0 → 0,𝑅

⊔ → ⊔, 𝑅

𝑞3

0 → 0,𝑅 ⊔ → ⊔, 𝐿

TMs vs. Finite / Pushdown Automata

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Three Levels of Abstraction

High‐Level Description

An algorithm (like CS 330)

Implementation‐Level Description

Describe (in English) the instructions for a TM • How to move the head

• What to write on the tape

Low‐Level Description

State diagram or formal specification

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Example

Decide if

High‐Level Description

Repeat the following:

• If there is exactly one

• If there is an odd number of • Delete half of the s in

, accept s in

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in

, reject

Example

Decide if

Implementation‐Level Description

1.

While moving the tape head left‐to‐right:

2. 3.

Return the head to the left end of the tape Go back to step 1

a) Cross off every other 0

b) If there is exactly one 0 when we reach the right end of the tape, accept

c) If there is an odd number of 0s when we reach the right end of the tape, reject

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Example

Decide if

Low‐Level Description

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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

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is the accept state

is the reject state ( )

TM Transition Function

means “move left” and means “move right”

means:

• Replace 𝑎 with 𝑏 in current cell

• Transition from state 𝑝 to state 𝑞

• Move tape head right

means:

• Replace 𝑎 with 𝑏 in current cell

• Transition from state 𝑝 to state 𝑞

• 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

• Tape contents = (followed by blanks • Current state =

• Tape head on first symbol of

and )

∗

Example:

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…

How a TM Computes

Start configuration:

One step of computation:

• • •

yields

if

yields yields

if if

Accepting configuration: Rejecting configuration:

= =

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