CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 13

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CD5560

FABER

Formal Languages, Automata and Models of Computation

Lecture 13

Mälardalen University

2010

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Content

Alan Turing and Hilbert Program Universal Turing Machine Chomsky Hierarchy DecidabilityReducibilityUncomputable FunctionsRice’s TheoremInteractive Computing, Persistent TM’s (Dina Goldin)

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http://www.turing.org.uk/turing/

Who was Alan Turing?Founder of computer science, mathematician, philosopher, codebreaker, visionary man before his time.

http://www.cs.usfca.edu/www.AlanTuring.net/turing_archive/index.html- Jack Copeland and Diane Proudfoot

http://www.turing.org.uk/turing/ The Alan Turing Home PageAndrew Hodges

http://www.turing.org.uk/turing/

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Alan Turing

1912 (23 June): Birth, London

1926-31: Sherborne School

1930: Death of friend Christopher Morcom

1931-34: Undergraduate at King's College, Cambridge University

1932-35: Quantum mechanics, probability, logic

1935: Elected fellow of King's College, Cambridge

1936: The Turing machine, computability, universal machine

1936-38: Princeton University. Ph.D. Logic, algebra, number theory

1938-39: Return to Cambridge. Introduced to German Enigma cipher machine

1939-40: The Bombe, machine for Enigma decryption

1939-42: Breaking of U-boat Enigma, saving battle of the Atlantic

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Alan Turing

1943-45: Chief Anglo-American crypto consultant. Electronic work.

1945: National Physical Laboratory, London

1946: Computer and software design leading the world.

1947-48: Programming, neural nets, and artificial intelligence

1948: Manchester University

1949: First serious mathematical use of a computer

1950: The Turing Test for machine intelligence

1951: Elected FRS. Non-linear theory of biological growth

1952: Arrested as a homosexual, loss of security clearance

1953-54: Unfinished work in biology and physics

1954 (7 June): Death (suicide) by cyanide poisoning, Wilmslow, Cheshire.

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Hilbert’s Program, 1900

Hilbert’s hope was that mathematics would be reducible to finding proofs (manipulating the strings of symbols) from a fixed system of axioms, axioms that everyone could agree were true.

Can all of mathematics be made algorithmic, or will there always be new problems that outstrip any given algorithm, and so require creative acts of mind to solve?

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TURING MACHINES

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Turing’s "Machines". These machines are humans who calculate. (Wittgenstein)

A man provided with paper, pencil, and rubber, and subject to strict discipline, is in effect a universal machine.

(Turing)

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............Tape

Read-Write headControl Unit

Standard Turing Machine

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............

Read-Write head

No boundaries -- infinite length

The head moves Left or Right

The Tape

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............

Read-Write head

1. Reads a symbol2. Writes a symbol3. Moves Left or Right

The head at each time step:

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ExampleTime 0

............ a a cb

Time 1............ a b k c

1. Reads a2. Writes k3. Moves Left

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Head starts at the leftmost positionof the input string

............

Blank symbol

head

a b ca

Input string

The Input String

#####

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1q 2qLba ,

Read Write Move Left

1q 2qRba ,

Move Right

States & Transitions

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............ # a b caTime 1

1q 2qRba ,

............ a b cbTime 2

1q

2q

# # # #

# # # # #

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Determinism

1q

2qRba ,

Allowed Not Allowed

3qLdb ,

1q

2qRba ,

3qLda ,

No lambda transitions allowed in standard TM!

Turing Machines are deterministic

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Formal Definitions for

Turing Machines

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Transition Function

1q 2qRba ,

),,(),( 21 Rbqaq

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Turing Machine

),#,,,,,( 0 FqQM

Transitionfunction

Initialstate

blank

Finalstates

States

Inputalphabet

Tapealphabet

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Universal Turing Machine

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A limitation of Turing Machines:

Better are reprogrammable machines.

Turing Machines are “hardwired”

they executeonly one program

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Solution: Universal Turing Machine

• Reprogrammable machine

• Simulates any other Turing Machine

Characteristics:

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simulates any other Turing Machine M

Input of Universal Turing Machine

• Description of transitions of M• Initial tape contents of M

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Description of Three tapes

MTape Contents of

Tape 2

State of M

Tape 3

M

Tape 1

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We describe Turing machine as a string of symbols:

We encode as a string of symbols

M

M

Description of M

Tape 1

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Alphabet Encoding

Symbols: a b c d

Encoding: 1 11 111 1111

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State Encoding

States: 1q 2q 3q 4q

Encoding: 1 11 111 1111

Head Move Encoding

Move:

Encoding:

L R

1 11

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Transition Encoding

Transition: ),,(),( 21 Lbqaq

Encoding: 10110110101

separator

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Machine Encoding

Transitions:

),,(),( 21 Lbqaq

Encoding:

10110110101

),,(),( 32 Rcqbq

110111011110101100

separator

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Tape 1 contents of Universal Turing Machine:

encoding of the simulated machine as a binary string of 0’s and 1’s

M

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A Turing Machine is described with a binary string of 0’s and 1’s.

The set of Turing machines forms a language:

Each string of the language isthe binary encoding of a Turing Machine.

Therefore:

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Language of Turing Machines

L = { 010100101,

00100100101111,

111010011110010101, …… }

(Turing Machine 1)

(Turing Machine 2)

……

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The Chomsky Hierarchy

34Non-recursively enumerable

Recursively-enumerable

Recursive

Context-sensitive

Context-free

Regular

The Chomsky Language Hierarchy

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Unrestricted Grammars

Productionsvu

String of variablesand terminals


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Example of unrestricted grammar

dAccAaB

aBcS

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A language is recursively enumerableif and only if it is generated by anunrestricted grammar.

L

Theorem

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Context-Sensitive Grammars

and |||| vu

Productionsvu



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The language }{ nnn cbais context-sensitive:

aaAaaaBBbbBBbccAcbAAb

aAbcabcS

|

|

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A language is context sensitiveif and only if

it is accepted by a Linear-Bounded automaton.

L

Theorem

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Linear Bounded Automata (LBAs)are the same as Turing Machineswith one difference:

The input string tape spaceis the only tape space allowed to use.

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[ ]a b c d e

Left-endmarker

Input string

Right-endmarker

Working space in tape

All computation is done between end markers.

Linear Bounded Automaton (LBA)

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There is a language which is context-sensitivebut not recursive.

Observation

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Decidability

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Consider problems with answer YES or NO.

Examples

• Does Machine have three states ?M• Is string a binary number? w• Does DFA accept any input? M

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A problem is decidable if some Turing machinesolves (decides) the problem.

Decidable problems:

• Does Machine have three states ?M• Is string a binary number? w• Does DFA accept any input? M

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Turing MachineInputprobleminstance

YES

NO

The Turing machine that solves a problemanswers YES or NO for each instance.

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The machine that decides a problem:

• If the answer is YES then halts in a yes state

• If the answer is NO then halts in a no state

These states may not be the final states.

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YES

NO

Turing Machine that decides a problem

YES and NO states are halting states

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Difference between Recursive Languages (“Acceptera”) and Decidable problems (“Avgöra”)

The YES states may not be final states.For decidable problems:

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Some problems are undecidable:

There is no Turing Machine thatsolves all instances of the problem.

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A famous undecidable problem:

The halting problem

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The Halting Problem

Input: • Turing Machine M• String w

Question: Does halt on ? M w

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Theorem

The halting problem is undecidable.

Proof

Assume to the contrary thatthe halting problem is decidable.

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There exists Turing Machinethat solves the halting problem

H

HM

w

YES M halts on w

M doesn’t halt on

wNO

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H

wwM 0qyq

nq

Input:initial tape contents

Encodingof M w

String

YES

NO

Construction of H

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Construct machine H

returns YES then loop forever. HIf

returns NO then halt.HIf

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H

wwM 0qyq

nq NO

aq bq

H

Loop forever

YES

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HConstruct machine

Input:

If M halts on input Mw

Then loop forever

Else halt

Mw (machine )M

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Mw MM wwcopy

Mw H

H

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HRun machine with input itself

Input:

If halts on input

Then loop forever

Else halt

Hw ˆ (machine )H

H Hw ˆ

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on input H Hw ˆ

If halts then loops forever.

If doesn’t halt then it halts.

:

H

H

CONTRADICTION !

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This means that


END OF PROOF

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Another proof of the same theorem

If the halting problem was decidable thenevery recursively enumerable languagewould be recursive.

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Theorem


Proof

Assume to the contrary thatthe halting problem is decidable.

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There exists Turing Machinethat solves the halting problem.

H

HM

w

YES M halts on w

M doesn’t halt on

wNO

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Let be a recursively enumerable language. L

Let be the Turing Machine that accepts .M L

We will prove that is also recursive: L

We will describe a Turing machine thataccepts and halts on any input.L

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M halts on ?wYES

NOM

w

Run with input

Mw

Hreject w

accept w

reject w

Turing Machine that acceptsand halts on any input

L

Halts on final state

Halts on non-final state

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Therefore L is recursive.

But there are recursively enumerablelanguages which are not recursive.

Contradiction!

Since is chosen arbitrarily, we have proven that every recursively enumerablelanguage is also recursive.

L

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Therefore, the halting problem is undecidable.

END OF PROOF

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A simple undecidable problem:

The Membership Problem

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The Membership Problem

Input: • Turing Machine M

• String w

Question: Does accept ? M w

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Theorem

The membership problem is undecidable.

Proof

Assume to the contrary thatthe membership problem is decidable.

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There exists a Turing Machinethat solves the membership problem

H

HM

w

YES M accepts w

NO M rejects w

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Let be a recursively enumerable language. L

Let be the Turing Machine that accepts .M L

We will prove that is also recursive: L

We will describe a Turing machine thataccepts and halts on any input.L

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M accepts ?wNO

YESM

w

Haccept w

Turing Machine that acceptsand halts on any input

L

reject w

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Therefore, L is recursive.

But there are recursively enumerablelanguages which are not recursive.

Contradiction!

Since is chosen arbitrarily, we have proven that every recursively enumerablelanguage is also recursive.

L

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Therefore, the membership problem is undecidable.

END OF PROOF

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Reducibility

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Problem is reduced to problemA B

If we can solve problem thenwe can solve problem .

BA

B

A

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If is decidable then is decidable.B A

If is undecidable then is undecidable.A B

Problem is reduced to problemA B

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Example

the halting problem

reduced to

the state-entry problem.

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The state-entry problem

Inputs:

Question:

M•Turing Machine

•State q

•String w

Does M enter state qon input ?w

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Theorem

The state-entry problem is undecidable.

ProofReduce the halting problem to

the state-entry problem.

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Suppose we have an algorithm (Turing Machine)

that solves the state-entry problem.

We will construct an algorithmthat solves the halting problem.

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Algorithm for state-entry problem

Mw

q

YES

NO

entersM q

doesn’t enter

M q

Assume we have the state-entry algorithm:

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Algorithm for Halting problem

M

w

YES

NO

halts onM w

doesn’t halt on

M w

We want to design the halting algorithm:

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Modify input machine M• Add new state q• From any halting state add transitions to q

M q

halting statesSinglehalt state

M

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M halts

M halts on state q

if and only if

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Algorithm for halting problem

Inputs: machine and stringM w

2. Run algorithm for state-entry problem with inputs: M wq, ,

1. Construct machine with state M q

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GenerateM M

w

M qw

State-entryalgorithm

Halting problem algorithm

YES

NO

YES

NO

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Since the halting problem is undecidable,it must be that the state-entry problemis also undecidable.

END OF PROOF

We reduced the halting problemto the state-entry problem.

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Another example

The halting problem

reduced to

the blank-tape halting problem.

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The blank-tape halting problem

Input: MTuring Machine

Question: Does M halt when started witha blank tape?

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ProofReduce the halting problem to the

blank-tape halting problem.

Theorem

The blank-tape halting problem is undecidable.

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Suppose we have an algorithmfor the blank-tape halting problem.

We will construct an algorithmfor the halting problem.

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Algorithm for blank-tape halting problem

M

YES

NO

halts onblank tapeM

doesn’t halton blank tape

M

Assume we have the blank-tape halting algorithm

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M

w

YES

NO

halts onM w

doesn’t halt on

M w

We want to design the halting algorithm:

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wMConstruct a new machine

• On blank tape writes w• Then continues execution like M

wM

Mthen write w

step 1 step2if blank tape execute

with input w

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M halts on input string

wM halts when started with blank tape.

if and only if

w

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1. Construct wM

2. Run algorithm for blank-tape halting problem with input wM

Inputs: machine and stringM w

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Generate

wMMw

Blank-tape halting algorithm

Halting problem algorithm

YES

NOwM

YES

NO

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Since the halting problem is undecidable,the blank-tape halting problem is also undecidable.

END OF PROOF

We reduced the halting problemto the blank-tape halting problem.

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Summary of Undecidable Problems

Halting Problem

Does machine halt on input ?M w

Membership problemDoes machine accept string ?M w

Is a string member of a

recursively enumerable language ?)Lw(In other words:

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Does machine halt when startingon blank tape?

Blank-tape halting problemM

State-entry Problem:

Does machine enter state on input ?

Mw

q

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Uncomputable Functions

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Uncomputable Functions

A function is uncomputable if it cannotbe computed for all of its domain.

Domain Rangef

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An uncomputable function:

)(nfmaximum number of moves untilany Turing machine with stateshalts when started with the blank tape.

n

Example

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TheoremFunction is uncomputable.)(nf

Then the blank-tape halting problem is decidable.

ProofAssume to the contrary that is computable.)(nf

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Algorithm for blank-tape halting problem

Input: machine M

1. Count states of : M m2. Compute )(mf3. Simulate for steps starting with empty tape

M )(mf

If halts then return YES otherwise return NO

M

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Therefore, the blank-tape haltingproblem must be decidable.

However, we know that the blank-tape halting problem is undecidable.

Contradiction!

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Therefore, function is uncomputable.)(nf

END OF PROOF

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Rice’s Theorem

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Non-trivial properties of recursively enumerable languages:

any property possessed by some (not all)recursively enumerable languages.

Definition

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Some non-trivial properties of recursively enumerable languages:

• is emptyL

L• is finiteL• contains two different strings

of the same length

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Rice’s Theorem

Any non-trivial property of a recursively enumerable languageis undecidable.

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We will prove some non-trivial propertieswithout using Rice’s theorem.

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TheoremFor any recursively enumerable language Lit is undecidable whether it is empty.

Proof

We will reduce the membership problemto the problem of deciding whether is empty.

L

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Membership problem:Does machine accept string ?wM

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Algorithm for empty languageproblem

M

YES

NO

Assume we have the empty language algorithm:

Let be the machine that accepts M L

)(ML

)(ML

empty

not empty

LML )(

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Algorithm for membershipproblem

M

w

YES

NO

acceptsM w

rejectsM w

We will design the membership algorithm:

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First construct machine : wM

When enters a final state, compare original input string with . w

M

Accept if original input is the same as .w

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Lw

)( wML is not empty

if and only if

}{)( wML w

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Algorithm for membership problem

Inputs: machine and string M w

1. Construct wM

2. Determine if is empty )( wML

YES: then )(MLw

NO: then )(MLw

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construct

wM

Check if)( wML

is empty

YES

NO

M

w

NO

YES

Membership algorithm

wM

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Since the membership problem is undecidable,the empty language problem is also undecidable.

END OF PROOF

We reduced the empty language problemto the membership problem.

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Decidability…continued…

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Theorem

For a recursively enumerable language Lit is undecidable to determine whether is finite. L

Proof

We will reduce the halting problemto the finite language problem.

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Assume we have the finite language algorithm:

Algorithm for finite languageproblem

M

YES

NO

)(ML

)(ML

finite

not finite

Let be the machine that accepts M LLML )(

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We will design the halting problem algorithm:


M

w

YES

NO

halts onM w

doesn’t halt on

M w

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First construct machine .wM

When enters a halt state, accept any input (infinite language).

M

Initially, simulates on input . M w

Otherwise accept nothing (finite language).

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M halts on

)( wML is not finite.

if and only if

w

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Algorithm for halting problem:Inputs: machine and string M w

1. Construct wM2. Determine if is finite )( wML

YES: then doesn’t halt on M wNO: then halts on M w

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construct

wM

Check if)( wML

is finite

YES

NO

M

w

NO

YES

Machine for halting problem

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Since the halting problem is undecidable,the finite language problem is also undecidable.

END OF PROOF

We reduced the finite language problemto the halting problem.

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TheoremFor a recursively enumerable language Lit is undecidable whether contains two different strings of same length.

L

ProofWe will reduce the halting problemto the two strings of equal length- problem.

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Assume we have the two-strings algorithm:

Let be the machine that accepts M LLML )(

Algorithm for two-stringsproblem

M

YES

NO

)(ML

)(ML

contains

doesn’t contain

two equal length strings

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We will design the halting problem algorithm:


M

w

YES

NO

halts onM w

doesn’thalt on

M w

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First construct machine . wM

When enters a halt state, accept symbols or .

M

Initially, simulates on input . M w

a b(two equal length strings)

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M halts on

wM

if and only if

w

accepts and a b

(two equal length strings)

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Inputs: machine and string M w

1. Construct wM2. Determine if accepts two strings of equal length

wM

YES: then halts on M wNO: then doesn’t halt on M w

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construct

wM

Check if)( wML

has twoequal lengthstrings

YES

NO

M

w

YES

NO

Machine for halting problem

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Since the halting problem is undecidable,the two strings of equal length problem is also undecidable.

END OF PROOF

We reduced the two strings of equal length - problem to the halting problem.

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Rices sats

Om är en mängd av Turing-accepterbara språk som innehåller något men inte alla sådana språk, så kan ingen TM avgöra för ett godtyckligt Turing-accepterbart språk L om L tillhör eller ej.

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Exempel

Givet en Turingmaskin M, kan man avgöra om alla strängar som accepteras av M börjar och slutar på samma tecken?

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Oavgörbart

Problemet handlar om en icke-trivial språkegenskap. Det finns TM:er vars accepterade strängar har egenskapen i fråga, och det finns TM:er vars accepterade strängar inte har egenskapen.

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Formellt:

= { L | TM accepterbara språkvars strängar börjar och slutar på samma tecken. }

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Interaction: Conjectures, Results, and Myths

Dina GoldinUniv. of Connecticut, Brown University

http://www.cse.uconn.edu/~dqg

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Fundamental Questions Underlying Theory of

Computation

What is computation?

How do we model it?

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Shared Wisdom(from our undergraduate Theory of Computation courses)

computation: finite transformation of input to outputinput: finite size (e.g. string or number)closed system: all input available at start, all output generated at end

behavior: functions, transformation of input data to output dataChurch-Turing thesis: Turing Machines capture this (algorithmic) notion of computation

Mathematical worldview: All computable problems

are function-based.

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“The theory of computability and non-computability [is] usually referred to as the theory of recursive functions... the notion of TM has been made central in the development."

Martin Davis, Computability & Unsolvability, 1958

“Of all undergraduate CS subjects, theoretical computer science has changed the least over the decades.”

SIGACT News, March 2004

“A TM can do anything that a computer can do.”Michael Sipser, Introduction to the Theory of Computation, 1997

The Mathematical Worldview

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The Operating System Conundrum

Real programs, such as operating systems and word processors, often receive an unbounded amount of input over time, and never "finish" their task. Turing machines do not model such ongoing computation well…

[TM entry, Wikipedia]

If a computation does not terminate,

it’s “useless” – but aren’t OS’s

useful??

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Rethinking Shared Wisdom:(what do computers do?)

computation: finite transformation of input to outputinput: finite-size (string or number)

closed system: all input available at start, all output generated at end

behavior: functions, algorithmic transformation of input data to output dataChurch-Turing thesis: Turing Machines capture this (algorithmic) notion of computation

computation: ongoing process which performs a task or delivers a service

dynamically generated stream of input tokens (requests, percepts, messages)

open system: later inputs depend on earlier outputs and vice versa (I/O entanglement, history dependence)

behavior: processes, components, control devices, reactive systems, intelligent agents

Wegner’s conjecture: Interaction is more powerful than algorithms

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Example: Driving home from work

Algorithmic input: a description of the world (a static “map”)

Output: a sequence of pairs of #s (time-series data)- for turning the wheel- for pressing gas/break

Similar to classic AI search/planning problems.

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But… in a real-world environment, the output depends on every grain of sand in the road (chaotic behavior).

Can we possibly have a map that’s detailed enough?

Worse yet… the domain is dynamic. The output depends on weather conditions, and on other drivers and pedestrians.

We can’t possibly be expected to predict that in advance!

Nevertheless the problem is solvable!

Google “autonomous vehicle research”

Driving home from work (cont.)

?

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Driving home from work (cont.)

The problem is solvable interactively.

Interactive input: stream of video camera images, gathered as we are driving

Output: the desired time-series data, generated as we are driving

similar to control systems, or online computation

A paradigm shift in the conceptualization of computational problem solving.

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• Rethinking the mathematical worldview • Persistent Turing Machines (PTMs)• PTM expressiveness• Sequential Interaction

– Sequential Interaction Thesis• The Myth of the Church-Turing Thesis

– the origins of the myth• Conclusions and future work

Outline

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Sequential Interaction

• Sequential interactive computation:

system continuously interacts with its environment by alternately accepting an input string

and computing a corresponding output string.

• Examples:- method invocations of an object instance

in an OO language- a C function with static variables- queries/updates to single-user databases- recurrent neural networks

- control systems- online computation- transducers- dynamic algorithms- embedded systems

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Sequential Interaction Thesis

• Universal PTM: simulates any other PTM– Need additional input describing the PTM (only once)

• Example: simulating Answering Machine(simulate AM, will-do), (record hello, ok), (erase, done), (record John, ok),(record Hopkins, ok), (playback, John Hopkins), …

Simulation of other sequential interactive systems is analogous.

Whenever there is an effective method for performing sequential interactive computation, this computation

can be performed by a Persistent Turing Machine

160

Church-Turing Thesis Revisited• Church-Turing Thesis:

Whenever there is an effective method for obtaining the values of a mathematical function, the function can be computed by a Turing Machine

• Common Reinterpretation (Strong Church-Turing Thesis) A TM can do (compute) anything that a computer can do

• The equivalence of the two is a myth – the function-based behavior of algorithms does not capture all

forms of computation– this myth has been dogmatically accepted by the CS community

• Turing himself would have denied it– in the same paper where he introduced what we now call Turing

Machines, he also introduced choice machines, as a distinct model of computation

– choice machines extend Turing Machines to interaction by allowing a human operator to make choices during the computation.

161

Origins of the Church-Turing Thesis Myth

A TM can do anything that a computer can do.

Based on several claims:1. A problem is solvable if there exists a Turing Machine

for computing it.2. A problem is solvable if it can be specified by an algorithm.3. Algorithms are what computers do.

Each claim is correct in isolationprovided we understand the underlying assumptions

Together, they induce an incorrect conclusionTMs = solvable problems = algorithms = computation

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Deconstructing the Turing Thesis Myth (1)

TMs = solvable problems

• Assumes:All computable problems are function-based.

• Reasons: – Theory of Computation started as a field of mathematics; mathematical

principles were adopted for the fundamental notions of computation, identifying computability with the computation of functions, as well as with Turing Machines.

– The batch-based modus operandi of original computers did not lend itself to other conceptualizations of computation.

163


solvable problems = algorithms

Assumes:- Algorithmic computation is also function based;

i.e., the computational role of an algorithm is to transform input data to output data.

• Reasons: – Original (mathematical) meaning of “algorithms”

E.g. Euclid’s greatest common divisor algorithm

– Original (Knuthian) meaning of “algorithms” “An algorithm has zero or more inputs, i.e., quantities which are given to it initially before the algorithm begins.“ [Knuth’68]

164


algorithms = computation

• Reasons: – The ACM Curriculum (1968): Adopted algorithms as the central

concept of CS without explicit agreement on the meaning of this term.

– Textbooks: When defining algorithms, the assumption of their closed function-based nature was often left implicit, if not forgotten

“An algorithm is a recipe, a set of instructions or the specifications of a process for doing something. That something is usually solving a problem of some sort.” [Rice&Rice’69]

“An algorithm is a collection of simple instructions for carrying out some task. Commonplace in everyday life, algorithms sometimes are called procedures or recipes...” [Sipser’97]

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• Rethinking the mathematical worldview • Persistent Turing Machines (PTMs)• PTM expressiveness• Sequential Interaction • The Myth of the Church-Turing Thesis• Conclusions and future work

Outline

166

The Shift to Interaction in CS

Computation = transforming input to output

Computation = carrying out a task over time

Logic and search in AI Intelligent agents, partially observable environments, learning

Procedure-oriented programming

Object-oriented programming

Closed systems Open systemsCompositional behavior Emergent behaviorRule-based reasoning Simulation, control, semi-Markov

processes

Algorithmic Interactive

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The Interactive Turing Test

• From answering questions to holding discussions.• Learning from -- and adapting to -- the questioner.• “Book intelligence” vs. “street smarts”.

“It is hard to draw the line at what is intelligence and what is environmental interaction. In a sense, it does not really matter which is which, as all intelligent systems must be situated in some world or other if they are to be useful entities.” [Brooks]

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• Many other interactive models– Reactive [MP] and embedded systems– Dataflow, I/O automata [Lynch], synchronous languages, finite/pushdown

automata over infinite words– Interaction games [Abramsky], online algorithms [Albers]– TM extensions: on-line Turing machines [Fischer], interactive Turing machines

[Goldreich]...

• Concurrency Theory– Focuses on communication (between concurrent agents/processes) rather than

computation [Milner]– Orthogonal to the theory of computation and TMs.

• What makes PTMs unique?– Provably more expressive than TMs.– Bridging the gap between concurrency theory (labeled transition systems) and

traditional TOC.

Modeling Interactive Computation: PTMs in Perspective

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• Theory of Sequential Interactionconjecture: notions analogous to computational complexity, logic, and recursive functions can be developed for sequential interaction computation

• Multi-stream interaction– From hidden variables to hidden interfaces conjecture: multi-stream interaction is more

powerful than sequential interaction [Wegner’97]• Formalizing indirect interaction

– Interaction via persistent, observable changes to the common environment

– In contrast to direct interaction (via message passing) conjecture: direct interaction does not capture all

forms of multi-agent behaviors

Future Work: 3 conjectures

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Referenceshttp://www.cse.uconn.edu/~dqg/papers/

[Wegner’97] Peter WegnerWhy Interaction is more Powerful than AlgorithmsCommunications of the ACM, May 1997

[EGW’04] Eugene Eberbach, Dina Goldin, Peter Wegner Turing's Ideas and Models of Computationbook chapter, in Alan Turing: Life and Legacy of a Great Thinker, Springer 2004

[I&C’04] Dina Goldin, Scott Smolka, Paul Attie, Elaine SondereggerTuring Machines, Transition Systems, and InteractionInformation & Computation Journal, 2004

[GW’04] Dina Goldin, Peter WegnerThe Church-Turing Thesis: Breaking the Mythpresented at CiE 2005, Amsterdam, June 2005 to be published in LNCS

CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 13

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