Domain Theory I
Domain Theory I
Wolfgang Schreiner
Research Institute for Symbolic Computation (RISC-Linz)
Johannes Kepler University, A-4040 Linz, Austria
[email protected]
http://www.risc.uni-linz.ac.at/people/schreine
Wolfgang Schreiner
RISC-Linz
Domain Theory I
Semantics of Loops
B: Boolean-expression
C: Command
...
C ::= . . . j while B do C j . . .
...
C[[while B do C]] =
s: B[[B]]s ! C[[while B do C]](C[[C]]s) [] s
Problem: meaning of a syntax phrase may be
dened only in terms of its proper subparts.
C[[while B do C]] = w
! Store
s:B[[B]]s ! w(C[[C]]s) [] s
where w:
w=
Store
?
?
Recursion in syntax exchanged for recursion
in function notation!
Wolfgang Schreiner
1
Domain Theory I
Recursive Function Denitions
Function
Denition
q = n:n equals zero
Possible
! one
[]
q (n plus one)
Function Graphs:
f(zero, one)g
f(zero, one), (one, ?), (two, ?), . . . g
{ f(zero, one), (one, six), (two, six), . . . g
{ f(zero, one), (one, k ), (two, k ), . . . g
{
=
Several functions satisfy specication; which
one shall we choose?
Wolfgang Schreiner
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Domain Theory I
Least Fixed Point Semantics
Theory that establishes meaning of recursive
specications:
1. Guarantees that every specication has a
function satisfying it.
2. Provides means for choosing the \best"
function out of the set of possibilities.
3. Ensures that the selected function corresponds to the conventional operational
treatment of recursion.
Argument is mapped to dened answer i simplication
of the specication yields a result in a nite number of
recursive invocations.
Wolfgang Schreiner
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Domain Theory I
The Factorial Function
fac(n) = n equals zero ! one
[] n times (fac(n minus one))
Only one function satises specication:
graph(factorial) =
f(zero, one), (one, one),
(two, two), (three, six),
. . . , (i, i!), . . . g
Wolfgang Schreiner
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Domain Theory I
Simplication
fac(three)
! three equals zero
! one [] three times fac(three minus one)
= three times fac(three minus one)
= three times fac(two)
! three times (two equals zero
! one [] two times fac(two minus one))
= three times (two times fac(one))
! three times (two times (one equals zero
! one [] one times fac(one minus one)))
= three times (two times one (times fac(zero)))
! three times (two times (one times (zero equals zero
! one [] zero times fac(zero minus one))))
= three times (two times one (times one))
= six
Wolfgang Schreiner
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Domain Theory I
Partial Functions
Answer
is produced in a nite number of
unfolding steps.
Idea: place limit on number of unfoldings
and investigate resulting graphs
zero: fg
{ one: f(zero, one)g
{ two: f(zero, one), (one, one)g
{ i
: f(zero, one), (one, one), . . . (i, i )g
{
+ 1
Graph
!
at stage i denes function fac i.
Consistency with each other:
graph fac graph fac +1
{ Consistency with ultimate solution:
graph fac graph factorial
{ Consequently
S
=0 graph fac graph factorial
{
(
i)
(
(
i)
(
1
i
Wolfgang Schreiner
(
i)
)
i
)
(
)
6
Domain Theory I
Partial Functions
Any
result is computed in a nite number
of unfoldings.
a; b) 2 graph(factorial)
! (a; b) 2 graph(faci) (for some i)
(
Consequently
S =0 graph fac
graph factorial
)
graph factorial
) =
Thus
(
(
1
i
S
(
i)
=0 graph(faci)
1
i
Factorial function can be totally understood
in terms of the nite subfunctions faci!
Wolfgang Schreiner
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Domain Theory I
Partial Functions
Representations
of sub-functions
fac0 = n: ?
fac +1 = n: n equals zero ! one
[] n times fac (n minus one)
i
Each
i
denition is non-recursive
Recursive specication can be understood in terms of
a family of non-recursive ones.
Common
format can be 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 functional!
Wolfgang Schreiner
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Domain Theory I
Functional and Fixed Point
Partial
functions
fac +1 = F(fac ) = F (?)
?
i
n:?)
:= (
Function
graph
graph factorial
Fixed
(
i
i
) =
S
=0 graph(F
1
i
point property
i
?
(
))
graph(F(factorial)) = graph(factorial)
F(factorial) = factorial
The function factorial is a xed point of the
functional F!
Wolfgang Schreiner
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Domain Theory I
q Function
Q = q:n:n equals zero
! one [] q (n plus one)
Q0(?) = n:?
graph(Q0(?)) = fg
(
)
Q1(?) = n:n equals zero
! one
n:?)(n plus one)
[] (
= n:n equals zero ! one ?
graph(Q1(?)) = f(zero, one)g
[]
Q2(?) = Q(Q1(?)) = n:n equals zero
! one n plus one equals zero ! one
graph(Q2(?)) = f(zero, one)g
[] ((
Wolfgang Schreiner
)
[]
?
)
10
Domain Theory I
q Function
Convergence
has occured
graph(Q (?)) = f(zero, one)g, i i
Resulting
S
graph
1
i
=0 graph(Q (?)) = f(zero; one)g
1
Fix
i
point property
Q(qlimit)
Still
=
qlimit
many solutions possible
graph(q ) =
f (zero, one), (one, k), . . . , (i, k), . . . g
k
Least
xed point property
graph(qlimit) graph(q )
k
The function qlimit is the least xed point of
the functional Q!
Wolfgang Schreiner
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Domain Theory I
Recursive Specications
The meaning of a recursive specication f
F f is
taken to be x(F ), the least xed point of the functional
denoted by F .
S
graph x F
=0 graph F ?
=
(
The
cpo
) =
1
i
(
i
(
(
)
))
domain D of F must be a pointed
partial ordering on D,
{ every chain in D has a least upper bound in D ,
{ D has a least element.
{
F
must be continuous
Preserves limits of chains.
Semantic domains are cpos and their oper-
ations are continuous.
Pointed cpos are created from primitive domains and
union domains by lifting.
Wolfgang Schreiner
12
Domain Theory I
Factorial Function
F = f. n. n equals zero ! one
[] n times (f(n minus one))
Simplication rule
x F = F(x F)
(x F)(three)
= (F (x F))(three)
= ( f. n. n equals zero ! one
[] n times (f(n minus one))(x F))(three)
= ( n. n equals zero ! one
[] n times (x F)(n minus one))(three)
= three equals zero ! one
[] three times (x F)(three minus one)
= three times (x F)(two)
= three times (F (x F))(two)
= ...
= three times (two times (x F)(two))
Fixed point property justies rec. unfolding!
Wolfgang Schreiner
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Domain Theory I
Double Recursion
g = n. n equals zero ! one
[] (g(n minus one) plus
g(n minus one)) minus one
graph(F0(?)) = fg
graph(F1(?)) = f(zero, one)g
graph(F1(?)) = f(zero, one), (one, one)g
graph(F2(?)) = f(zero, one), (one, one), (two, one)g
...
graph(F +1(?)) = f(zero, one), . . . , (i, one)g
i
x F = n. one
Stepwise construction of graph yields insight!
Wolfgang Schreiner
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Domain Theory I
Simultaneous Denitions
f, g: Nat ! Nat
f = x. x equals zero ! g(zero)
[] f(g(x minus one)) plus two
g = y. y equals zero ! zero
[] y times f(y minus one)
T = Nat ! Nat
F: (T T) ! (T T)
F = (f,g).(. . . , . . . )
F0(?) = (fg, fg)
F1(?) = (fg, f(zero, zero)g)
F2(?) = (f(zero,zero)g, f(zero, zero)g)
...
F5(?) = (f(zero,zero), (one, two), (two, two)g,
f(zero, zero), (one, zero),
(two, four), (three, six)g)
F (?) = F5 (?), i >
x(F) = (f,g)
?
?
i
Wolfgang Schreiner
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15
Domain Theory I
While Loops
C[[while B do C]] =
x(f:s. B[[B]]s ! f (C[[C]]s) [] s)
Function: Store ? ! Store ?
Example:
C[[while A>0 do (A:=A-1; B:=B+1)]]
= x F where
F = f:s: test s ! f (adjust s) [] s
test = B[[A > 0]]
adjust = C[[A:=A-1; B:=B+1]]
Partial function graphs:
Each pair in graph shows store prior to loop
entry and after loop exit.
Each graph F i+1(?) contains those pairs
whose input stores nish processing in at
most i iterations.
Wolfgang Schreiner
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Domain Theory I
Example
graph(F0(?))=fg
graph(F1(?))=f
( f ([[A]],zero), ([[B]],zero), . . . g,
f ([[A]],zero), ([[B]],zero), . . . g ),
...
( f ([[A]],zero), ([[B]],four), . . . g,
f ([[A]],zero), ([[B]],four), . . . g ), . . . g
graph(F2(?))=f
( f ([[A]],zero), ([[B]],zero), . . . g,
f ([[A]],zero), ([[B]],zero), . . . g ),
...
( f ([[A]],zero), ([[B]],four), . . . g,
f ([[A]],zero), ([[B]],four), . . . g ),
...
( f ([[A]],one), ([[B]],zero), . . . g,
f ([[A]],zero), ([[B]],one), . . . g ),
...
( f ([[A]],one), ([[B]],four), . . . g,
f ([[A]],zero), ([[B]],ve), . . . g ), . . . g
Wolfgang Schreiner
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Domain Theory I
While Loops
Representation by nite subfunctions
C[[while B do C]] = Ff
s:?,
s.B[[B]]s ! ? [] s,
s.B[[B]]s ! (B[[B]](C[[C]]s) ! ? [] C[[C]]s)
[] s,
s.B[[B]]s ! (B[[B]](C[[C]]s) !
(B[[B]](C[[C]](C[[C]]s)) ! ?
[] C[[C]](C[[C]]s)) [] C[[C]]s) [] s, . . . g
= Ff
C[[diverge]],
C[[if B then diverge else skip]],
C[[if B then (C; if B then diverge else skip)
else skip]],
C[[if B then (C; if B then
(C; if B then diverge else skip)
else skip) else skip]], . . . g
Loop iteration can be understood by sequence
of non-iterating programs.
Wolfgang Schreiner
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Domain Theory I
Reasoning about Least Fixed Points
Fixed
Point Induction Principle:
To prove P x F , it suces to prove
1. P ?
2. P d ! P F d , for arbitrary d 2 D
(
(
)
)
( )
(
( ))
for pointed cpo D, continuous functional F D
inclusive predicate P D ! B.
:
Inclusiveness
! D, and
:
of predicates
If predicate holds for every element of chain, it also
holds for its least upper bound.
All
universally quantied combinations of
conjunctions/disjunctions that use only v
over functional expressions are inclusive.
Mainly useful for showing equivalences of program constructs.
Wolfgang Schreiner
19
Domain Theory I
Reasoning about Least Fixed Points
C[[repeat C until B]] = x(f:s:
let s = C[[C]]s in B[[B]]s ! s [] (fs ))
C[[C; while :B do C]] ? C[[repeat C until B]]
0
0
0
0
=
Proof: P (f; g ) = 8s:f (C[[C]]s) = (gs)
1. P(?, ?) holds obviously.
2. Prove P (F (f ); G(g ))
F = (f:s: B[[: B]]s ! f (C[[C]]s) [] s)
G = (f:s: let s = C[[C]]s in B[[B]]s
! s [] (fs ))
0
0
0
0
(a) F (f )(C[[C]]?) = ? = G(g )(?).
(b) s 6 ?:
i. C[[C]]s ?: F (f )(?)=?=(let s = ? in B[[B]]s !
s [] (gs )) = G(g )(?).
ii. C[[C]]s
s0 6 ?: F (f )(s0 )=B[[: B]]s0 !
f (C[[C]]s0)[]s0 = B[[B]]s0 ! s0 [] f (C[[C]]s0) =
B[[B]]s0 ! s0 [] f (gs0) = G(g)(s)
=
0
=
0
0
=
Wolfgang Schreiner
0
=
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