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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