S ::= x := a | skip | S1;S2. | if b then S1 else S2. | while b do S ii.1. This slide is
derived from the book & slides by Nielson & Nielson: “Semantics with
applications” ...
x=1
let x = 1 in ... x(1). x.set(1)
!x(1)
Programming Paradigms and Formal Semantics
Big-Step Operational Semantics Ralf Lämmel
A big-step operational semantics for While
© Ralf Lämmel, 2009-2011 unless noted otherwise
33
n ∈ Num This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
• variables x ∈ Var
Syntactic categories • arithmetic expressions Aexp of the Whileaa ∈::=language n | x | a +a
Syntactic Categories for While language • numerals n ∈ Num • variables x ∈ Var • arithmetic expressions a ∈ Aexp a ::= n | x | a1 + a2 | a1 ∗ a2 | a1 − a2 • booleans expressions b ∈ Bexp b ::= true | false | a1 = a2 | a1 ≤ a2 noted | ¬otherwise b | b1 ∧ b2 © Ralf Lämmel, 2009-2011 unless
1
2
| a1 ∗ a2 | a1 − a2
• booleans expressions b ∈ Bexp b ::= true | false | a1 = a2 | a1 ≤ a2 | ¬ b | b1 ∧ b2 • statements S ∈ Stm S ::= x := a | skip | S1 ; S2 | if b then S1 else S2 | while b do S II.1
34
N = {0, 1, 2, · · ·}
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Semantic categories of the While language Truth values T = {tt, ff} States
Semantic Categories Semantic Categories
• State = Var → N • State" = (Var × N)∗
Natural numbers Natural numbers
• State"" = Var∗ × N∗
N=N {0,=1,{0, 2, ·1,· ·} 2, · · ·}
Lookup in a state: s x
TruthTruth values values
Update a state: s" = s[y %→ v]
= ff} {tt, ff} T=T {tt,
StatesStates
s" x =
• State =→ VarN→ N • State = Var
II.2
" " • State = (Var ×∗ N)∗ • State = (Var × N) "" ∗ ∗ • State =∗ Var "" • State = Var × N×∗ N Lookup in a state: s x Lookup in a state: s x © Ralf Lämmel, 2009-2011 unless noted Update a state: s" =otherwise s[y %→
"
Update a state: s = s[y %→ v]
s x if x &= y v if x = y
v]
35
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Meanings of syntactic categories Meanings of the syntactic categories
Numerals N : Num → N Variables s ∈ State = Var → N Arithmetic expressions A : Aexp → (State → N) Boolean expressions B : Bexp → (State → T) Statements S : Stm → (State !→ State) © Ralf Lämmel, 2009-2011 unless noted otherwise
36
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Semantics of arithmetic expressions A : Aexp → State → N
A[n]s
= N [n]
A[x]s
= sx
A[a1 + a2 ]s
= A[a1 ]s + A[a2 ]s
A[a1 ∗ a2 ]s = A[a1 ]s ∗ A[a2 ]s A[a1 − a2 ]s
© Ralf Lämmel, 2009-2011 unless noted otherwise
= A[a1 ]s − A[a2 ]s
37
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Semantics of boolean expressions B : Bexp → State → T
B[true]s
= tt
B[false]s
= ff
B[a1 = a2 ]s =
B[a1 ≤ a2 ]s =
B[¬b]s
=
B[b1 ∧ b2 ]s =
tt if A[a1 ]s = A[a2 ]s
ff if A[a1 ]s "= A[a2 ]s
tt if A[a1 ]s ≤ A[a2 ]s
ff if A[a1 ]s "≤ A[a2 ]s
tt if B[b]s = ff
ff if B[b]s = tt
tt if B[b1]s = tt and B[b2]s = tt
© Ralf Lämmel, 2009-2011 unless noted otherwise
ff if B[b1]s = ff or B[b2 ]s = ff 38
II.5
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Semantics of statements
© Ralf Lämmel, 2009-2011 unless noted otherwise
39
Transition systems in semantics
© Ralf Lämmel, 2009-2011 unless noted otherwise
40
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Semantics of statements Statements
Syntactic Category S ::= x := a | skip | S1 ; S2 | if b then S1 else S2 | while b do S Meaning of the syntactic category: S : Stm → (State !→ State) Two operational semantics • Natural Semantics • Structural Operational Semantics specified by transition systems
© Ralf Lämmel, 2009-2011 unless noted otherwise
41
II.7
Transition System
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Transition systems
(Γ, T, >) • Γ: a set of configurations • T : a set of terminal configurations T ⊆Γ • >: a transition relation >⊆Γ × Γ
© Ralf Lämmel, 2009-2011 unless noted otherwise
42
Small step vs. big step • Small-step semantics !
aka Structured Operational Semantics (SOS)
!
Many transitions
!
Computation steps modeled by transitions
• Big-step semantics !
aka Natural semantics
!
One (fewer) transition(s)
!
Computation steps modeled by derivation tree
© Ralf Lämmel, 2009-2011 unless noted otherwise
43
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Natural semantics
Natural semantics: describe how the “final” describe how the overall result of the result ofIdea: the computation is obtained. computation is obtained Transition system: (Γ, T, →) • Γ = {(S, s) | S ∈ While, s ∈ State} ∪ State • T = State • → ⊆ {(S, s) | S ∈ While, s ∈ State} × State Typical transition: (S, s) → s& where S is the program s is the initial state s& is the final state © Ralf Lämmel, 2009-2011 unless noted otherwise
44
II.9
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Small step vs. big step Program configuration Transitions = small steps
Final state © Ralf Lämmel, 2009-2011 unless noted otherwise
45
Variable assignments (States)
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Small step vs. big step
Derivation tree
© Ralf Lämmel, 2009-2011 unless noted otherwise
Transition = big step
46
Compositionality
© Ralf Lämmel, 2009-2011 unless noted otherwise
47
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Recall semantics of arithmetic expressions A : Aexp → State → N
A[n]s
= N [n]
A[x]s
= sx
A[a1 + a2 ]s
= A[a1 ]s + A[a2 ]s
A[a1 ∗ a2 ]s = A[a1 ]s ∗ A[a2 ]s A[a1 − a2 ]s
© Ralf Lämmel, 2009-2011 unless noted otherwise
= A[a1 ]s − A[a2 ]s
48
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
© Ralf Lämmel, 2009-2011 unless noted otherwise
49
Properties of semantics and induction proofs
© Ralf Lämmel, 2009-2011 unless noted otherwise
50
Free variables in arithmetic expressions
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
FV(n)
= ∅
One property of the semantics
FV(x)
= {x}
Proof by Structural Induction
FV(a1 + a2 ) = FV(a1 ) ∪ FV(a2 ) FV(a1 ∗ a2)
Intuitively: The value of an arithmetic expression only depends on the values of the variables that occur in it.
= FV(a1 ) ∪ FV(a2 )
FV(a1 − a2 ) = FV(a1 ) ∪ FV(a2 )
Lemma [1.11] Lemma 1.11:
Free variables in arithmetic expressions
Let s and s% be two states satisfying s x = s% x for all x ∈ FV(a). Then % A[a]s Proof = byA[a]s Structural Induction
FV(n)
= ∅
FV(x)
= {x}
FV(a1 + a2 ) = FV(a1 ) ∪ FV(a2 )
IV.5
FV(a1 ∗ a2)
Intuitively: The value of an arithmetic expression only depends on the values of the variables that occur in it.
= FV(a1 ) ∪ FV(a2 )
FV(a1 − a2 ) = FV(a1 ) ∪ FV(a2 ) Lemma 1.11:
Free variables in arithmetic expressions
Proof
FV(n)
= ∅
FV(x)
= {x}
Let s and s% be two states satisfying s x = s% x by structural induction for all xon∈ the FV(a). Then arithmetic expressionsA[a]s = A[a]s%
FV(a1 + a2 ) = FV(a1 ) ∪ FV(a2 )
© Ralf Lämmel, 2009-2011 unless noted otherwise FV(a1 ∗ a2) = FV(a1 ) ∪ FV(a2 )
IV.5
51
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
© Ralf Lämmel, 2009-2011 unless noted otherwise
52
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Let s and s% be two states satisfying s x = s% x for all x ∈ FV(a). Then A[a]s = A[a]s%
Proofs for basis elements
Proof by structural induction on the arithmetic expressions © Ralf Lämmel, 2009-2011 unless noted otherwise
53
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Let s and s% be two states satisfying s x = s% x for all x ∈ FV(a). Then A[a]s = A[a]s%
Proofs for composite elements
Proof by structural induction on the arithmetic expressions © Ralf Lämmel, 2009-2011 unless noted otherwise
54
Proof by Ind. on Shape of Derivation Trees This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Another property Theorem 2.9: of the semantics Theorem [2.9]
Proof
Theorem 2.9: The natural semantics of While is determinThe natural of WhileSis of deterministic, that is semantics for all statements While ! statements istic,allthat is for S of While and states s, sall and s!! if (S, s) → and all states s, ss!! and and (S, s!! s) → s!! ! then =→ s!!s. ! and (S, s) → s!! if (S,ss) then s! = s!! . Proof: Proof: We assume (S, s) → s! . !! ! !! We that We prove assume (S,ifs)(S, →s)s!→ . s then s = s . prove that (S, s) →ons!!the then s! = s!! . We proceed by ifinduction inference ! of (S, s) → s . induction on the inference We proceed by
of (S, s) → s! .
Proof by induction on the shape of derivation trees IV.7 © Ralf Lämmel, 2009-2011 unless noted otherwise
55
IV.7
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Induction on the shape of derivation trees
Structural induction on syntactical categories is not applicable because of the non-compositional semantics of while!
© Ralf Lämmel, 2009-2011 unless noted otherwise
56
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
© Ralf Lämmel, 2009-2011 unless noted otherwise
57
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Theorem 2.9:
Theorem [2.9]
The natural semantics of While is deterministic, that is for all statements S of While and all states s, s! and s!! if (S, s) → s! and (S, s) → s!! then s! = s!! . Proof: We assume (S, s) → s! . We prove that if (S, s) → s!! then s! = s!! . We proceed by induction on the inference of (S, s) → s! .
IV.7
Proof by induction on the shape of derivation trees © Ralf Lämmel, 2009-2011 unless noted otherwise
58
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Proof by induction on the shape of derivation trees © Ralf Lämmel, 2009-2011 unless noted otherwise
59
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Proof by induction on the shape of derivation trees © Ralf Lämmel, 2009-2011 unless noted otherwise
60
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Non-compositional semantics is Ok for this proof scheme.
Proof by induction on the shape of derivation trees © Ralf Lämmel, 2009-2011 unless noted otherwise
61
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Yet another property of the semantics Lemma [2.5]
Proof
Part 1: (*)
(**)
Part II: (**)
(*)
© Ralf Lämmel, 2009-2011 unless noted otherwise
62
This slide is derived from the book & slides by Nielson & Nielson: “Semantics with applications” (1991 & 1999+).
Proof only (*)
(**)
only for tt
No induction needed here.
© Ralf Lämmel, 2009-2011 unless noted otherwise
63
• Summary: Big-step operational semantics ! Models relations between syntax, states, values. ! Uses deduction rules (conclusion, premises). ! Computations are derivation trees. ! Induction proofs are the key tool in semantics. • Prepping: “Semantics with applications” ! Chapter 1 and Chapter 2.1 • Outlook: ! Small-step semantics ! Type systems © Ralf Lämmel, 2009-2011 unless noted otherwise
64
