The While language
Big-step
Meaning
Small-step
The While programming language Matthew Hennessy
January 28, 2015
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
The language While
B ∈ Bool ::= true | false | E = E | B&B | ¬ B | · · · E ∈ Arith ::= l ∈ Locs | n ∈ Nums | (E + E ) | · · · C ∈ Com ::= l := E | if B then C else C |
C ; C | skip | while B do C
l, k from a set of locations Locs
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
Example program
l2 := 1; l3 := 0; while ¬ (l1 = l2 ) do l2 := l2 + 1; l3 := l3 + 1; How do we describe the behaviour of these programs?
How can we prescribe how these programs should be executed?
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
Dependencies Behaviour of commands in Com depend on behaviour of Boolean expressions in Bool I
if l = k then l1 := k else l2 := l
I
while ¬ (l1 = l2 ) do l2 := l2 + 1 ; l3 := l3 + 1
Behaviour of Boolean in Bool depend on behaviour of expressions in Arith I
(l + 1) = (k + 2)
I
(l2 + l) = k
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
Evaluating expressions
Value of expressions depend on current values in locations k+l−1
I
Value depends on current values of locations k and l
Values stored at locations change as programs are executed
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
States I
A state (of the memory) is a function from locations to numerals, s : Locs → Nums.
I
The state s[k 7→ n] is defined by n if k = l s[k 7→ n](l) = s(l) otherwise
I
Behaviour of commands is relative to a state
I
The state changes as the execution of a command proceeds
I
Complete execution of a command transforms an initial state into a terminal state
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
Big-step semantics of arithmetic expressions Judgements: hE , si ⇓ n meaning: value of expression E relative to the state s is n
Alternative: hE , si ⇓ hn, s 0 i meaning: I
E relative to the state s evaluates to n
I
the evaluation changes the state from s to s 0
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
Big-step semantics of arithmetics
(b-num)
hn, si ⇓ n (b-add)
hE1 , si ⇓ n1
hE2 , si ⇓ n2
hE1 + E2 , si ⇓ n3
n3 = add(n1 , n2 )
(b-loc)
hl, si ⇓ s(l)
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
Assignments Evaluate the command (l := E ) relative to state s? Intuition: (1) evaluate E relative to state s to some value n (2) update location l with new value n
Inference rule: (b-assign)
hE , si ⇓ n hl := E , si ⇓ s[l 7→ n]
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
Sequential composition Evaluate command C1 ; C2 relative to state s? Intuition: (1) evaluate C1 relative to state s, to get new state s1 (2) then evaluate C2 relative to new state s1
Rule: (b-seq.s)
hC1 , si ⇓ s1
hC2 , s1 i ⇓ s 0
hC1 ; C2 , si ⇓ s 0
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
If commands Evaluate command (if B then C1 else C2 ) relative to state s? Intuition: (1) first evaluate B to some boolean value bv (2) if true evaluate C1 relative to state s (3) if false evaluate C2 relative to state s
Rules: (b-if.t)
(b-if.f)
hB, si ⇓ true hC1 , si ⇓ s 0
hB, si ⇓ false hC2 , si ⇓ s 0
hif B then C1 else C2 , si ⇓ s 0
hif B then C1 else C2 , si ⇓ s 0
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
While commands Evaluate command (while B do C ) relative to state s? Intuition: (1) first evaluate B to some boolean value bv (2) if false nothing to be done (3) if true evaluate C relative to state s to get new state s1 (4) then evaluate original (while B do C ) relative to s1
Rules:
hB, si ⇓ false
hB, si ⇓ true hC , si ⇓ s1 hwhile B do C , s1 i ⇓ s2
hwhile B do C , si ⇓ s
hwhile B do C , si ⇓ s2
(b-while.f)
While language
(b-while.t)
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
The skip command Evaluate command skip relative to state s? Intuition: (1) nothing to do
Rule:
(b-skip)
hskip, si ⇓ s
While language
The While language
Matthew Hennessy
Big-step
Meaning
Small-step
Properties of big-step semantics
Normalisation: For every state s and every command C there exists some state s 0 such that `big hC , si ⇓ s 0 False
Determinacy: If `big hC , si ⇓ s1 and `big hC , si ⇓ s2 then s1 = s2
True
Proof requires rule induction
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
Non-termination in big-step semantics
I
Let C be while ¬ (l = 0) do l := l + 1
I
Let s(l) = 1
I
How can we derive hC , si ⇓ s 0 for any s 0 when s(l) > 0?
I
What is the shortest proof of judgement of the form hC , si ⇓ s 0 ? where s(l) > 0
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
The meaning of commands l2 := 1; l3 := 0; while ¬ (l1 = l2 ) do l2 := l2 + 1; l3 := l3 + 1; l1 := l3 What does this program do?
I
A program transforms an initial state in a terminal state
I
For some initial states there may be no terminal state
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
Partial functions f :A*B
Meaning: f calculates an element of B for some elements of A
Notation: I
A is the domain of f
I
B is the range of f
Note: f (a) may not be defined for some a in A
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
The meaning of commands
[[−]] : Com → (States * States) [[C ]] transforms an initial state s into a terminal state
Definition:( [[C ]](s) =
s 0, undefined,
if hC , si ⇓ s 0 otherwise
Determinacy ensures this is a proper definition
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
Example Let C denote l2 := 1; l3 := 0; while ¬ (l1 = l2 ) do l2 := l2 + 1; l3 := l3 + 1; l1 := l3 How do we describe [[C ]] : (States * States) ? [[C ]](s) is only defined when s(l1 ) > 0: [[C ]](s)(l1 ) = s(l1 ) − 1 [[C ]](s)(l2 ) = s(l1 ) [[C ]](s)(l3 ) = s(l1 ) − 1 [[C ]](s)(l) = s(l) otherwise While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
Small-step semantics for While Judgements: hC , si → hC 0 , s 0 i
Meaning: I
starting from state s
I
when executing command C
one step of computation leads to I
state s 0
I
with command C 0 remaining to be executed
What is a step? Depends While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
What is in a step? Decision: I I
Ignore how expressions, Booleans, are evaluated One step consists of: I I
memory update or branching decision
Concentrate on execution of commands
Terminal configurations: I
hskip, si is terminal
I
hskip, si → hC , s 0 i not possible
for every s
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
Assignment How to execute one step of command (l := E )
relative to the state s
?
Intuition: I
Evaluate E relative to state s
I
Update state s with resulting value
Inference rule: (b-ass)
hE , si ⇓ n hl := E , si → hskip, s[l 7→ n]i One step suffices for entire execution – ignoring evaluation of E While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
Conditional How to execute one step of (if B then C1 else C2 )
relative to state s
Intuition: I
Evaluate B relative to state s
I
If true start evaluating command C1
I
If false start evaluating command C2
Inference rule: (b-cond.t)
hB, si ⇓ true hif B then C1 else C2 , si → hC1 , si (b-cond.f)
hB, si ⇓ false hif B then C1 else C2 , si → hC2 , si While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
Sequential composition How to execute one step of command (C1 ; C2 )
relative to state s
Intuition: I
Execute one step of C1 relative to state s
I
If C1 has terminated start executing C2
skip indicates termination
Inference rule: (b-seq.left)
hC1 , si → hC10 , s 0 i hC1 ; C2 , si → hC10 ; C2 , s 0 i (b-seq.skip)
hskip ; C2 , si → hC2 , si While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
While commands How to execute one step of command (while B do C )
relative to state s
Intuition: I
Evaluate B relative to s
I
If false then terminate
I
if true then execute one step of C . . . . . .
Inference rule: (b-while.f)
hB, si ⇓ false hwhile B do C , si → hskip, si (b-while.t)
hB, si ⇓ true hwhile B do C , si → hC ; while B do C , si While language
Matthew Hennessy
The While language
Big-step
While loops:
alternative
Meaning
Small-step
the unwinding rule
How to execute one step of command (while B do C )
relative to state s
Intuition: I
combination of (if B then C else . . .) and sequential composition
Inference rule:
(b-while)
hwhile B do C , si→ hif B then (C ; while B do C ) else skip, si
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
Running commands
To run command C from state s: Find state s 0 such that hC , si →∗ hskip, s 0 i
Example: I
See Course Notes page 49
I
See McGusker notes, slide 50
Configurations hskip, si are terminal
While language
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
Running commands: Problems can occur Infinite loops: Let C be command while true do skip I
hC , si →3 hC , si →3 hC , si →3 hC , si → . . .
I
No state s 0 such that C →∗ hskip, s 0 i
Progress property: I
Configurations hskip, si are terminal
I
Either hC , si is terminal or hC , si → hC 0 , s 0 i
While language
for some configuration hC 0 , s 0 i
Matthew Hennessy
The While language
Big-step
Meaning
Small-step
Questions Questions Questions I
Determinacy: I
I
hC , si →∗ hskip, s1 i and hC , si →∗ hskip, s2 i implies s1 = s2 ?
Consistency with big-step semantics: I I
hC , si ⇓ s 0 implies hC , si →∗ hskip, s 0 i ? hC , si →∗ hskip, s 0 i implies hC , si ⇓ s 0 ?
Proof strategy: Similiar to that used for expression language Exp More powerful proof principle required
While language
Matthew Hennessy
