Domain Theory I

Domain Theory I

Wolfgang Schreiner

Research Institute for Symb olic Computation RISCLinz

Johannes Kepler University A Linz Austria

WolfgangSchreinerriscunilinzacat

httpwwwriscunilinzacatp eopleschreine

Wolfgang Schreiner RISCLinz

Domain Theory I

Semantics of Lo ops

B Bo oleanexpression

C Command

C j while B do C j

Cwhile B do C

s BBs Cwhile B do CCCs s

Problem meaning of a syntax phrase may b e

dened only in terms of its proper subparts

Cwhile B do C w

where w Store Store

? ?

sBBs w CCs s w =

Recursion in syntax exchanged for recursion

in function notation

Wolfgang Schreiner

Domain Theory I

Recursive Function Denitions

Function Denition

q = nn equals zero one [] q (n plus one)

Possible Function Graphs

fzero oneg

= fzero one one two g

fzero one one six two six g

fzero one one k two k g

Several functions satisfy sp ecication which

one shall we cho ose

Wolfgang Schreiner

Domain Theory I

Least Fixed Point Semantics

Theory that establishes meaning of recursive

sp ecications

Guarantees that every sp ecication has a

function satisfying it

Provides means for cho osing the b est

function out of the set of p ossibilities

Ensures that the selected function cor

resp onds to the conventional op erational

treatment of recursion

Argument is mapp ed to dened answer i simplication

of the sp ecication yields a result in a nite numb er of

recursive invo cations

Wolfgang Schreiner

Domain Theory I

The Factorial Function

facn n equals zero one

n times facn minus one

Only one function satises sp ecication

graphfactorial

fzero one one one

two two three six

i i! g

Wolfgang Schreiner

Domain Theory I

Simplication

facthree

three equals zero

one three times facthree minus one

three times facthree minus one

three times factwo

three times two equals zero

one two times factwo minus one

three times two times facone

three times two times one equals zero

one one times facone minus one

three times two times one times faczero

three times two times one times zero equals zero

one zero times faczero minus one

three times two times one times one

six

Wolfgang Schreiner

Domain Theory I

Partial Functions

Answer is produced in a nite numb er of

unfolding steps

Idea place limit on numb er of unfoldings

and investigate resulting graphs

zero fg

one fzero oneg

two fzero one one oneg

i + 1 fzero one one one i i!g

Graph at stage i denes function fac

i

Consistency with each other

graph(fac ) graph(fac )

i i+1

Consistency with ultimate solution

graph(fac ) graph(factorial )

i

Consequently

S

1

graph(fac ) graph(factorial )

i

i=0

Wolfgang Schreiner

Domain Theory I

Partial Functions

Any result is computed in a nite numb er

of unfoldings

(a b) graph(factorial )

(a b) graph(fac ) for some i

i

Consequently

S

1

graph(factorial ) graph(fac )

i

i=0

Thus

S

1

graph(factorial ) = graph(fac )

i

i=0

Factorial function can b e totally understo o d

in terms of the nite subfunctions fac

i

Wolfgang Schreiner

Domain Theory I

Partial Functions

Representations of subfunctions

fac n

0

fac n n equals zero one

i+1

n times fac n minus one

i

Each denition is nonrecursive

Recursive sp ecication can b e understo o d in terms of

a family of nonrecursive ones

Common format can b e extracted

Functional F

F Nat Nat Nat Nat

? ?

F f n n equals zero one

n times f n minus one

Each subfunction is an instance of the func

tional

Wolfgang Schreiner

Domain Theory I

Functional and Fixed Point

Partial functions

i

fac Ffac F

i+1 i

:= (n)

Function graph

S

1 i

graph(factorial ) = graph(F ())

i=0

Fixed p oint property

graphFfactorial graphfactorial

Ffactorial factorial

The function factorial is a xed p oint of the

functional F

Wolfgang Schreiner

Domain Theory I

q Function

Q = q nn equals zero

one [] q (n plus one)

0

Q (n)

0

graphQ fg

1

Q nn equals zero

one [] (n)(n plus one)

nn equals zero one []

1

graphQ fzero oneg

2 1

Q QQ nn equals zero

one [] ((n plus one) equals zero one [] )

2

graphQ fzero oneg

Wolfgang Schreiner

Domain Theory I

q Function

Convergence has o ccured

i

graphQ fzero oneg i 1

Resulting graph

S

i 1

graph(Q ()) = f(zero one)g

i=0

Fix p oint property

Q(qlimit) = qlimit

Still many solutions p ossible

graphq

k

f zero one one k i k g

Least xed p oint property

graphqlimit graphq

k

The function qlimit is the least xed p oint of

the functional Q

Wolfgang Schreiner

Domain Theory I

Recursive Sp ecications

The meaning of a recursive sp ecication f = F (f ) is

taken to b e xF the least xed p oint of the functional

denoted by F

S

1 i

graph(x F ) = graph(F ())

i=0

The domain D of F must b e a p ointed

cp o

partial ordering on D

every chain in D has a least upp er b ound in D

D has a least element

F must b e continuous

Preserves limits of chains

Semantic domains are cp os and their op er

ations are continuous

Pointed cp os are created from primitive domains and

union domains by lifting

Wolfgang Schreiner

Domain Theory I

Factorial Function

F f n n equals zero one

n times fn minus one

Simplication rule

x F Fx F

x Fthree

F x Fthree

f n n equals zero one

n times fn minus onex Fthree

n n equals zero one

n times x Fn minus onethree

three equals zero one

three times x Fthree minus one

three times x Ftwo

three times F x Ftwo

three times two times x Ftwo

Fixed p oint property justies rec unfolding

Wolfgang Schreiner

Domain Theory I

Double Recursion

g n n equals zero one

gn minus one plus

gn minus one minus one

0

graphF fg

1

graphF fzero oneg

1

graphF fzero one one oneg

2

graphF fzero one one one two oneg

i+1

graphF fzero one i oneg

x F n one

Stepwise construction of graph yields insight

Wolfgang Schreiner

Domain Theory I

Simultaneous Denitions

f g Nat Nat

?

f x x equals zero gzero

fgx minus one plus two

g y y equals zero zero

y times fy minus one

T Nat Nat

?

F T T T T

F fg

0

F fg fg

1

F fg fzero zerog

2

F fzerozerog fzero zerog

5

F fzerozero one two two twog

fzero zero one zero

two four three sixg

i 5

F F i 5

xF fg

Wolfgang Schreiner

Domain Theory I

While Lo ops

Cwhile B do C

xf s BBs f CCs s

Function Store Store

? ?

Example

Cwhile A do AA BB

x F where

F f s test s f adjust s s

test BA

adjust CAA BB

Partial function graphs

Each pair in graph shows store prior to lo op

entry and after lo op exit

i+1

Each graph F () contains those pairs

whose input stores nish processing in at

most i iterations

Wolfgang Schreiner

Domain Theory I

Example

0

graphF fg

1

graphF f

f Azero Bzero g

f Azero Bzero g

f Azero Bfour g

f Azero Bfour g g

2

graphF f

f Azero Bzero g

f Azero Bzero g

f Azero Bfour g

f Azero Bfour g

f Aone Bzero g

f Azero Bone g

f Aone Bfour g

f Azero Bve g g

Wolfgang Schreiner

Domain Theory I

While Lo ops

Representation by nite subfunctions

F

Cwhile B do C f

s

sBBs s

sBBs BBCCs CCs

s

sBBs BBCCs

BBCCCCs

CCCCs CCs s g

F

f

Cdiverge

Cif B then diverge else skip

Cif B then C if B then diverge else skip

else skip

Cif B then C if B then

C if B then diverge else skip

else skip else skip g

Lo op iteration can b e understo o d by sequence

of noniterating programs

Wolfgang Schreiner

Domain Theory I

Reasoning ab out Least Fixed Points

Fixed Point Induction Principle

To prove P (x F ) it suces to prove

P ()

P (d) P (F (d)) for arbitrary d D

for p ointed cp o D continuous functional F : D D and

inclusive predicate P : D B

Inclusiveness of predicates

If predicate holds for every element of chain it also

holds for its least upp er b ound

All universally quantied combinations of

conjunctionsdisjunctions that use only v

over functional expressions are inclusive

Mainly useful for showing equivalences of pro

gram constructs

Wolfgang Schreiner

Domain Theory I

Reasoning ab out Least Fixed Points

Crep eat C until B xf s

0 0 0 0

let s CCs in BBs s f s

?

CC while B do C = Crep eat C until B

Pro of P (f g ) = sf (C[[C]]s) = (g s)

P holds obviously

Prove P (F (f ) G(g ))

s B Bs f CCs s F = (f

0 0

s let s CCs in BBs G = (f

0 0

s f s

a F f CC Gg

b s =

0 0

i CCs = F f let s in BBs

0 0

s g s Gg

ii CCs = s = F f s B Bs

0 0 0

f CCs s BBs s f CCs

0 0 0 0 0

BBs s f g s Gg s

0 0 0

Wolfgang Schreiner