Definitions and Proofs Structural Induction Three approaches to - PowerPoint PPT Presentation

Definitions and Proofs Structural Induction Three approaches to semantics • compositional definitions • Prove that the property holds for all the basis elements of the syntactic cat- • natural semantics egory. • structural operational semantics • Prove that the property holds for all Three proof principles the composite elements by assuming that the property holds for the imme- • structural induction diate constituents of the element (this • induction on the shape of derivation is called the induction hypothesis) and proving that it also holds for the ele- trees ment itself. • induction on the length of derivation sequences IV.1 IV.2

Ind. on Shape of Derivation Trees Ind. on Length of Derivation Seq. • Prove that the property holds for all the simple derivation trees by showing • Prove that the property holds for all that it holds for all the axioms of the derivation sequences of length 0 transition system • Prove that the property holds for all • Prove that the property holds for all other derivation sequences: Assume the composite derivation trees: For that the property holds for derivation each rule assume that the property sequences of length at most k (this is holds for its premises (this is called called the induction hypothesis) and the induction hypothesis) and prove prove that it holds for derivation se- that it also holds for the conclusion of quences of length k + 1 the rule provided that the conditions of the rule are satisfied IV.3 IV.4

Proof by Structural Induction A : Aexp → State → N Intuitively: The value of an arithmetic ex- pression only depends on the values of the variables that occur in it. Free variables in arithmetic expressions ∅ FV( n ) = A [ n ] s N [ n ] = FV( x ) = { x } A [ x ] s = s x FV( a 1 ) ∪ FV( a 2 ) FV( a 1 + a 2 ) = A [ a 1 + a 2 ] s = A [ a 1 ] s + A [ a 2 ] s FV( a 1 ∗ a 2 ) = FV( a 1 ) ∪ FV( a 2 ) A [ a 1 ∗ a 2 ] s = A [ a 1 ] s ∗ A [ a 2 ] s FV( a 1 − a 2 ) = FV( a 1 ) ∪ FV( a 2 ) A [ a 1 − a 2 ] s A [ a 1 ] s − A [ a 2 ] s = Lemma 1.11: Let s and s ′ be two states satisfying s x = s ′ x for all x ∈ FV( a ). Then A [ a ] s = A [ a ] s ′ IV.5 IV.6

Proof by Ind. on Shape of Derivation Trees Natural Semantics ( x := a, s ) → s [ x �→ A [ a ] s ] ( skip , s ) → s ( S 1 , s ) → s ′ , ( S 2 , s ′ ) → s ′′ Theorem 2.9: ( S 1 ; S 2 , s ) → s ′′ The natural semantics of While is determin- ( S 1 , s ) → s ′ istic, that is for all statements S of While ( if b then S 1 else S 2 , s ) → s ′ and all states s , s ′ and s ′′ if ( S, s ) → s ′ and ( S, s ) → s ′′ if B [ b ] s = tt then s ′ = s ′′ . ( S 2 , s ) → s ′ ( if b then S 1 else S 2 , s ) → s ′ Proof: if B [ b ] s = ff We assume ( S, s ) → s ′ . We prove that if ( S, s ) → s ′′ then s ′ = s ′′ . ( S, s ) → s ′ , ( while b do S, s ′ ) → s ′′ ( while b do S, s ) → s ′′ We proceed by induction on the inference of ( S, s ) → s ′ . if B [ b ] s = tt ( while b do S, s ) → s if B [ b ] s = ff IV.7 IV.8

Proof by Ind. on Length of Derivation Seq. Structural Operational Semantics ( x := a, s ) ⇒ s [ x �→ A [ a ] s ] ( skip , s ) ⇒ s ( S 1 , s ) ⇒ ( S ′ 1 , s ′ ) Lemma 2.19: ( S 1 ; S 2 , s ) ⇒ ( S ′ 1 ; S 2 , s ′ ) If ( S 1 ; S 2 , s ) ⇒ k s ′′ then ( S 1 , s ) ⇒ s ′ there exists s ′ , k 1 and k 2 such that ( S 1 ; S 2 , s ) ⇒ ( S 2 , s ′ ) ( S 1 , s ) ⇒ k 1 s ′ , ( S 2 , s ′ ) ⇒ k 2 s ′′ and ( if b then S 1 else S 2 , s ) ⇒ ( S 1 , s ) k = k 1 + k 2 if B [ b ] s = tt Proof: ( if b then S 1 else S 2 , s ) ⇒ ( S 2 , s ) We proceed by induction on the number k . if B [ b ] s = ff ( while b do S, s ) ⇒ ( if b then ( S ; while b do S ) else skip , s ) IV.9 IV.10