BU CS 332 Theory of Computation Lecture 10: Reading: Turing - - PowerPoint PPT Presentation

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BU CS 332 Theory of Computation Lecture 10: Reading: Turing - - PowerPoint PPT Presentation

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BU CS 332 Theory of Computation Lecture 10: Reading: Turing Machines Sipser Ch 3.1 3.2 TM Variants Mark Bun February 26, 2020 Turing Machines Motivation So far in this class weve seen several limited models of computation

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BU CS 332 – Theory of Computation

Lecture 10:

  • Turing Machines
  • TM Variants

Reading: Sipser Ch 3.1‐3.2

Mark Bun February 26, 2020

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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 𝑏𝑐𝑑 𝑜 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

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David Hilbert 1862‐1943

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

  • perations whether the equation is

solvable in rational integers.

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1928 – The Entscheidungsproblem

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David Hilbert 1862‐1943

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

Wilhelm Ackermann 1896‐1962

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1936 – Solution to the Entscheidungsproblem

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Alonzo Church 1903‐1995 Alan Turing 1912‐1954

"An unsolvable problem of elementary number theory“ Model of computation: 𝜇‐calculus (CS 320) “On computable numbers, with an application to the Entscheidungsproblem” Model of computation: Turing Machine

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Turing Machines

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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

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0 → 0, 𝑆 ⊔ → ⊔, 𝑆

accept reject

0 → 0, 𝑆 ⊔ → ⊔, 𝑆

Example

𝑟0 𝑟1

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Example

0 → 0, 𝑆 ⊔ → ⊔, 𝑆

accept reject

0 → 0, 𝑆 ⊔ → ⊔, 𝑆 0 → 0, 𝑆 ⊔ → ⊔, 𝑀 𝑟0 𝑟1 𝑟3

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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 in , accept
  • If there is an odd number of s in

, reject

  • Delete half of the s in

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Example

Decide if

  • Implementation‐Level Description
  • 1. While moving the tape head left‐to‐right:

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

  • 2. Return the head to the left end of the tape
  • 3. Go back to step 1

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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
  • 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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  • …
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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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  • …

Example:

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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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