Operational Semantics

Transcription

1 University of Science and Technology of China (USTC) 07/19/2011

2 Transition Semantics Program configurations: γ Γ def = Commands Σ Transitions between configurations: Γ ˆΓ where ˆΓ def = Γ {abort} Σ The transition relation describes how programs run step by step. The corresponding transition semantics is also called small-step semantics.

3 Transition rules Assignment: (x := e,σ) (Skip,σ{x e intexp σ}) Here Skip plays the rule of a flag recording the end of the program. Other rules: (c 0,σ) (c 0,σ ) (c 0 ; c 1,σ) (c 0 ; c 1,σ ) (Skip ; c 1,σ) (c 1,σ) b boolexp σ = true (if b then c 0 else c 1,σ) (c 0,σ) b boolexp σ = false (if b then c 0 else c 1,σ) (c 1,σ)

4 Much simpler semantics for while b boolexp σ = true (while b do c, σ) (c ; while b do c, σ) b boolexp σ = false (while b do c, σ) (Skip, σ) An alternative definition: (while b do c, σ) (if b then (c ; while b do c) else Skip, σ)

5 Semantics for newvar An unsatisfactory attempt: (newvar x := e in c, σ) (x := e ; c ; x := n, σ) where n = σ x Unsatisfactory because the value of local variable x could be exposed to external observers while c is executing. This is a problem when we have concurrency. Solution (due to Eugene Fink): (c,σ{x e intexp σ}) (c,σ ) (newvar x := e in c, σ) (newvar x := n in c,σ {x σ x}) where n = σ x (c,σ{x e intexp σ}) (Skip,σ ) (newvar x := e in c, σ) (Skip, σ {x σ x})

6 Semantics for fail (fail, σ) (abort, σ) Need two new rules: (c 0,σ) (abort, σ ) (c 0 ; c 1,σ) (abort, σ ) (c,σ{x e intexp σ}) (abort,σ ) (newvar x := e in c, σ) (abort, σ {x σ x})

7 Properties Determinism: For all γ Γ and γ,γ ˆΓ, ifγ γ and γ γ, then γ = γ. In other words, the binary relation is a partial function. Multi-step transitions: γ γ γ γ γ γ γ γ That is, is the reflexive transitive closure of. n-step transitions: γ 0 γ γ γ γ n γ γ n+1 γ We have γ γ iff n.γ n γ.

8 Consistency with Denotational Semantics We say γ diverges (γ ) iffforallγ such that γ γ, there exists γ such that γ γ. We say c gets stuck at the state σ iff there s no γ ˆΓ such that (c,σ) γ. Skip gets stuck at any state. For all c and σ, c comm σ = iff (c,σ) c comm σ =(abort,σ ) iff (c,σ) (abort,σ ) c comm σ = σ iff (c,σ) (Skip,σ )

9 A Variation Skip is overloaded as a flag for termination. (x := e,σ) (Skip,σ{x e intexp σ}) One more identity step is introduced after every commands: consider x := x + 1 ; y := y + 2. Reynolds takes a different approach: Γ (Γ Σ {abort} Σ) New rules: (x := e,σ) σ{x e intexp σ} (Skip,σ) σ (c 0,σ) (c 0,σ ) (c 0 ; c 1,σ) (c 0 ; c 1,σ ) (c 0,σ) σ (c 0 ; c 1,σ) (c 1,σ )

10 Big-Step Semantics Big-step transitions: Γ ˆΣ (x := e,σ) σ{x e intexp σ} (Skip,σ) σ (c 0,σ) σ (c 1,σ ) σ b boolexp σ = true (c 0,σ) σ (c 0 ; c 1,σ) σ (if b then c 0 else c 1,σ) σ b boolexp σ = false (c 1,σ) σ (if b then c 0 else c 1,σ) σ b boolexp σ = false (while b do c, σ) σ b boolexp σ = true (c,σ) σ (while b do c, σ ) σ (while b do c, σ) σ

11 Big-Step Semantics (cont d) (c,σ{x e intexp σ}) σ (newvar x := e in c, σ) σ {x σ x} (fail, σ) (abort,σ) Corresponding rules should be added to handle the abort case. Equivalence between big-step and small-step semantics: For all c and σ, (c σ) (abort,σ ) iff (c,σ) (abort,σ ) (c σ) σ iff (c,σ) (Skip,σ )

12 New Semantics of Program Specifications Syntax of specifications: (spec) φ ::= [p]c[q] total correctness {p}c{q} partial correctness Semantics of specifications: spec spec B φ is valid if φ spec = true. [p]c[q] spec def = σ. p assert σ σ. ((c, σ) (Skip,σ )) q assert σ {p}c{q} spec def = σ. p assert σ σ. ((c, σ) (Skip,σ )) q assert σ