Overview
Transition Semantics
Configurations and the transition relation
Executions and computation
Inference rules for small-step structural operational semantics for
the simple imperative language
Transition semantics of failure
Labeled transition semantics of input and output
Relationship with (direct) denotational semantics
Operational (or Transition) Semantics
Idea: Define the execution of a program as a sequence γ0, γ1, . . .
of configurations γi ∈ Γ.
Configurations are either terminal or nonterminal:
Γ = ΓT ∪ Γ N
ΓT ∩ ΓN = {}
e.g. for the SIL
ΓT = Σ
ΓN = comm × Σ
[x : 42] ∈ ΓT
hx:= x+1, [x : 41]i ∈ ΓN
Define a transition relation 7→ from ΓN to Γ:
informally, γ 7→ γ 0 if γ 0 is obtained “in one step” from γ, e.g.
hx:= x+1, [x : 41]i 7→ [x : 42]
Executions and Computation
An execution is a (finite or infinite) sequence of configurations γ0, γ1, . . .
such that γi 7→ γi+1 whenever γi and γi+1 are in the sequence.
The relation of computation 7→∗
is the reflexive and transitive closure of 7→
γ 7→∗ γ 0 if there is a finite execution starting with γ and ending with γ 0.
For the SIL we will define 7→ which is a total function from ΓN to Γ,
⇒ for every γ ∈ Γ there is a longest execution starting with γ;
if it is infinite, then γ diverges: γ ↑;
otherwise there is a unique γ 0 ∈ ΓT such that γ 7→∗ γ 0.
Plotkin Style Small-Step Structural Operational Semantics
for the SIL
We define the relation 7→ in terms of inference rules.
(skip)
(assgn)
hskip, σi 7→ σ
hv:= e, σi 7→ [σ | v : [[e]]intexp σ]
hc0, σi 7→ σ 0
(seq t)
hc0 ; c1, σi 7→ hc1, σ 0i
hc0, σi 7→ hc00, σ 0i
(seq s)
hc0 ; c1, σi →
7 hc00 ; c1, σ 0i
Example:
(by (assgn))
hx:= x+1, [x : 4 | y : 6]i 7→ [x : 5 | y : 6]
(by (seq t))
hx:= x+1 ; y:= y+x, [x : 4 | y : 6]i 7→ hy:= y+x, [x : 5 | y : 6]i
(by (seq s))
hx:= x+1 ; y:= y+x ; skip, [x : 4 | y : 6]i 7→ hy:= y+x ; skip, [x : 5 | y : 6]i
More SOS Rules
(cond t)
(cond f)
(while t)
(while f)
hif
b then c else c0, σi 7→ hc, σi
hif
b then c else c0, σi 7→ hc0, σi
hwhile b do c, σi 7→ hc ; while b do c, σi
hwhile b do c, σi 7→ σ
when [[b]]boolexp σ = true
when [[b]]boolexp σ = false
when [[b]]boolexp σ = true
when [[b]]boolexp σ = false
However the naı̈ve rule for variable declaration
hnewvar v:=e in c, σi 7→ hc ; v:= n, [σ | v : [[e]]intexp σ]i where n = σ v
exposes the local variable name in the result,
which becomes a problem when we extend the language.
SOS Rule for Local Variable Declaration
Idea: Use the declaration to reflect changes in the value of the variable.
(decl t)
(decl s)
hc, [σ | v : [[e]]intexp σ]i 7→ σ 0
hnewvar v:=e in c, σi 7→ [σ 0 | v : σ v]
hc, [σ | v : [[e]]intexp σ]i 7→ hc0, σ 0i
hnewvar v:=e in c, σi 7→ hnewvar v:=σ 0 v in c0, [σ 0 | v : σ v]i
(assgn)
(seq t)
(decl s)
hx:= x+1, [x : 24 | y : 10]i 7→ [x : 25 | y : 10]
hx:= x+1 ; y:= x+2, [x : 24 | y : 10]i 7→ hy:= x+2, [x : 25 | y : 10]i
hnewvar x:=x+3 in x:= x+1 ; y:= x+2, [x : 21 | y : 10]i
7→ hnewvar x:=25 in y:= x+2, [x : 21 | y : 10]i
Inference Rules for the Computation
The reflexive and transitive closure of 7→ can also be defined
using inference rules:
γ 7→ γ 0
(incl)
γ 7→∗ γ 0
(refl)
γ 7→∗ γ
γ 7→∗ γ 0 γ 0 7→∗ γ 00
(trans)
γ 7→∗ γ 00
Meaning of Commands
7→ ∈ ΓN → Γ (total function)
⇒ ∀γ ∈ Γ there is a longest execution starting from γ,
either infinite or ending with a γ 0 ∈ ΓT = Σ.


⊥,
if hc, σi ↑
[[c]]comm σ =  0
σ , if hc, σi 7→∗ σ 0
Transition Semantics of Failure
Define ΓT = Σ ∪ ({abort} × Σ). Then
(fail)
hfail, σi 7→ habort, σi
Propagation of failure:
hc0, σi 7→ habort, σi
(seq x)
hc0 ; c1, σi 7→ habort, σ 0i
(decl x)
hc, [σ | v : [[e]]intexp σ]i 7→ habort, σ 0i
hnewvar v:=e in c, σi 7→ habort, [σ 0 | v : σ v]i
The semantics of commands becomes


⊥,



[[c]]comm σ = σ 0,



habort σ 0i,
if hc, σi ↑
if hc, σi 7→∗ σ 0
if hc, σi 7→∗ habort σ 0i
Labeled Transition Semantics of Input and Output
Informally: Write labels on transitions to show input or output.
Rules:
(output)
(input)
!n
h!e, σi 7→ σ
when n = [[e]]intexp σ
?n
h?v, σi 7→ [σ | v : n]
Formally, the transition “relation” becomes ternary:
7→ ⊆ ΓN × Λ × Γ, where
Λ = {} ∪ {?n | n ∈ Z} ∪ {!n | n ∈ Z}
λ
and hc, σi 7→ γ stands for hhc, σi, λ, γi ∈ 7→.
( is silent )
Labeled Transition Semantics cont’d
The other rules are generalized to propagate the labels, e.g.
λ
(seq t)
(seq s)
hc0, σi 7→ σ 0
λ
hc0 ; c1, σi 7→ hc1, σ 0i
λ
hc0, σi 7→ hc00, σ 0i
λ
hc0 ; c1, σi 7→ hc00 ; c1, σ 0i
λ
(seq x)
hc0, σi 7→ habort, σi
λ
hc0 ; c1, σi 7→ habort, σ 0i
Properties of the Labeled Transition Semantics
If γ = hc, σi ∈ ΓN , then exactly one of these holds:
∃!γ 0 ∈ Γ such that γ 7→ γ 0 (silent transition)
!n
∃!γ 0 ∈ Γ, n ∈ Z such that γ 7→ γ 0
λ
{λ ∈ Λ | γ 0 ∈ Γ and γ 7→ γ 0} = {?n | n ∈ Z}
Hence for every γ ∈ Γ there is a longest sequence of silent transitions which is either
infinite
ends with a γ 0 ∈ ΓT
!n
ends with a γ 0 ∈ ΓN such that ∃!γ 00 ∈ Γ, n ∈ Z such that γ 0 7→ γ 00
?n
ends with a γ 0 ∈ ΓN such that ∀n ∈ Z. ∃γ 00 ∈ Γ. γ 0 7→ γ 00.
Relationship with the Denotational Semantics
So, for every γ ∈ Γ there is a longest sequence of silent transitions which is either
infinite
ends with a γ 0 ∈ ΓT
!n
ends with a γ 0 ∈ ΓN such that ∃!γ 00 ∈ Γ, n ∈ Z such that γ 0 7→ γ 00
?n
ends with a γ 0 ∈ ΓN such that ∀n ∈ Z. ∃γ 00 ∈ Γ. γ 0 7→ γ 00.
∼ (Σ̂ + (Z × Ω) + [Z → Ω]) and F ∈ [Γ → Ω] is the least solution of
If Ω =
⊥


⊥,




ιterm σ 0,



0
F γ = ιabort σ ,


00 i,

ι
hn,
F
γ

out



ι (λn ∈ Z. F γ ),
n
in
then [[c]]comm σ = F hc, σi.
if γ ↑
if γ →
7 ∗ σ0
if γ 7→∗ habort, σ 0i
!n
if ∃γ 0. γ 7→∗ γ 0 and γ 0 7→ γ 00
?n
if ∃γ 0 ∈ Γ. ∀n ∈ Z. γ 7→∗ γ 0 and γ 0 7→ γn