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Other Models of Computation

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Models of computation:

•Turing Machines•Recursive Functions•Post Systems•Rewriting Systems

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Church’s Thesis:

All models of computation are equivalent

Turing’s Thesis: A computation is mechanical if and only if it can be performed by a Turing Machine

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Church’s and Turing’s Thesis are similar:

Church-Turing Thesis

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Recursive Functions

An example function:

1)( 2 nnfDomain Range3 1010)3( f

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We need a way to define functions

We need a set of basic functions

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Zero function: 0)( xz

Successor function: 1)( xxs

Projection functions: 1211 ),( xxxp

2212 ),( xxxp

Basic Primitive Recursive Functions

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Building complicated functions:

Composition: )),(),,((),( 21 yxgyxghyxf

Primitive Recursion:

)),(),,(()1,( 2 yxfyxghyxf

)()0,( 1 xgxf

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Any function built fromthe basic primitive recursive functionsis called:

Primitive Recursive Function

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A Primitive Recursive Function: ),( yxadd

xxadd )0,( (projection)

)),(()1,( yxaddsyxadd

(successor function)

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5 )4(

))3(( )))0,3(((

))1,3(()2,3(

sssaddss

addsadd

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),( yxmult

0)0,( xmult

)),(,()1,( yxmultxaddyxmult

Another Primitive Recursive Function:

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Theorem: The set of primitive recursive functions is countable

Proof: Each primitive recursive function can be encoded as a string

Enumerate all strings in proper order

Check if a string is a functionCostas Busch - LSU

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Theorem there is a function that is not primitive recursive

Proof:Enumerate the primitive recursive functions

,,, 321 fff

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Define function 1)()( ifig i

g differs from every if

g is not primitive recursive

END OF PROOFCostas Busch - LSU

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A specific function that is notPrimitive Recursive:

Ackermann’s function:

)),(,1()1,()1,1()0,(

1),0(

yxAxAyxAxAxA

yyA

Grows very fast, faster than any primitive recursive function

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Recursive Functions

0),(such that smallest )),(( yxgyyxgy

Accerman’s function is a Recursive Function

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Primitive recursive functions

Recursive Functions

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Post Systems• Have Axioms

• Have Productions

Very similar with unrestricted grammars

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Example: Unary Addition

Axiom: 1111

Productions:

1111

321321

321321VVVVVVVVVVVV

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111111111111111111

11 321321 VVVVVV

11 321321 VVVVVV

A production:

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Post systems are good for proving mathematical statements from a set of Axioms

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Theorem: A language is recursively enumerable if and only if a Post system generates it

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Rewriting Systems

• Matrix Grammars

• Markov Algorithms

• Lindenmayer-Systems

They convert one string to another

Very similar to unrestricted grammarsCostas Busch - LSU

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Matrix GrammarsExample:

213

22112

211

,:,:

:

SSPcbSSaSSP

SSSP

A set of productions is applied simultaneously

Derivation:

aabbccccbbSaaScbSaSSSS 212121

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213

22112

211

,:,:

:

SSPcbSSaSSP

SSSP

}0:{ ncbaL nnn

Theorem: A language is recursively enumerable if and only if a Matrix grammar generates itCostas Busch - LSU

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Markov Algorithms

Grammars that produce

Example:

.

SSaSbSab

Derivation:

SaSbaaSbbaaabbb

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.

SSaSbSab

}0:{ nbaL nn

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In general: }:{*

wwL

Theorem:

A language is recursively enumerable if and only if a Markov algorithm generates it

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Lindenmayer-SystemsThey are parallel rewriting systems

Example: aaa

aaaaaaaaaaaaaaa Derivation:

}0:{ 2 naLn

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Lindenmayer-Systems are not generalAs recursively enumerable languages

Extended Lindenmayer-Systems: uyax ),,(

Theorem: A language is recursively enumerable if and only if an Extended Lindenmayer-System generates it

context

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