Semantics with Applications 3. More on Operational Semantics

Transcription

1 Semantics with Applications 3. More on Operational Semantics Hanne Riis Nielson, Flemming Nielson (thanks to Henrik Pilegaard) [SwA] Hanne Riis Nielson, Flemming Nielson Semantics with Applications: An Appetizer Springer, / 40

2 Recap Natural Semantics Structural Operational Semantics 2 / 40

3 Language While: Syntactic Categories Numerals n Num n ::= 0 1 n0 n1 Variables x Var e.g. x, y, z,... Arithmetic expressions a Aexp a ::= n x a 1 + a 2 a 1 a 2 a 1 a 2 Boolean expressions b Bexp b ::= true false a 1 = a 2 a 1 a 2 b b 1 b 2 Statements S Stm S ::= x := a skip S 1 ; S 2 if b then S 1 else S 2 while b do S 3 / 40

4 Natural semantics for While 4 / 40

5 The Semantic Function for Statements The meaning of statements can be summarised as a partial function from State to State S ns : Stm (State State) Partial function: not necessarily defined for all elements of the domain Definition: { s if S, s s S ns [[S]]s = undef otherwise Need for partiality: non-terminating statements such as while true do skip 5 / 40

6 Induction on the Shape of Derivation Trees In the proof of the previous lemma we were inspecting the structure of the derivation tree for certain transitions This can be generalized to the following proof technique: Prove that the property holds for all the axioms Prove that the property holds for all other rules : Assume that the property holds for its premises (this is called the induction hypothesis, sometimes abbreviated IH) Prove that it holds for the conclusion (provided the side conditions are satisfied) 6 / 40

7 Structural Operational Semantics for While 7 / 40

8 Derivation Sequences A derivation sequence of a statement S starting in state s is either a finite sequence γ0, γ 1,..., γ k γ 0 γ 1... γ k and γ k is final (of the form s) an infinite sequence γ0, γ 1, γ 2,... γ 0 γ 1 γ 2... where γ 0 = S, s and γ i γ i+1 for 0 i(< k) We write γ 0 i γ i to indicate i derivation steps We write γ 0 γ i to indicate finitely many derivation steps 8 / 40

9 The Semantic Function for Statements As we did in the case for Natural Semantics, the meaning of statements can be summarised as a partial function from State to State S sos : Stm (State State) Definition: { s if S, s S sos [[S]]s = s undef otherwise 9 / 40

10 Concepts for NS and SOS The following concepts can be defined both for Natural Semantics and Structural Operational Semantics Termination Looping Semantic equivalence Determinism The formalisations differ however we will compare them in the following 10 / 40

11 Termination Natural Semantics: The execution of S from state s terminates if and only if there is a state s such that S, s s. Structural Operational Semantics: The execution of S from state s terminates if and only if there is a finite derivation sequence starting with S, s, i.e. and γ k is final. S, s γ 1... γ k 11 / 40

12 Looping Natural Semantics: We say that the execution of S from state s loops if and only if there is no state s such that S, s s. Structural Operational Semantics: We say that the execution of S from state s loops if and only if there is an infinite derivation sequence starting with S, s, i.e. S, s γ 1 γ / 40

13 Semantic Equivalence Natural Semantics: Two statements S 1 and S 2 are semantically equivalent if for all states s and s S 1, s s if and only if S 2, s s Structural Operational Semantics: Two statements S 1 and S 2 are semantically equivalent if for all states s: S 1, s γ if and only if S 2, s γ, whenever γ is final there is an infinite derivation sequence starting with S 1, s if and only if there is one starting in S 2, s 13 / 40

14 Determinism Natural Semantics: The semantics is deterministic if for all statements S and states s, s, and s we have that S, s s and S, s s imply s = s Structural Operational Semantics: The semantics is deterministic if for all S and s, γ, and γ we have that S, s γ and S, s γ imply γ = γ 14 / 40

15 Induction on the Length of Derivation Sequences For Structural Operational Semantics it is often useful to conduct proofs by the length of derivation sequences Prove that the property holds for all derivation sequences of length 0 Prove that the property holds for all other derivation sequences Assume that the property holds for all derivation sequences of length at most k (this is called the induction hypothesis) Prove that it holds for derivation sequences of length k+1 15 / 40

16 Three Proof Principles Structural Induction Induction on the Shape of Derivation Trees Especially important for Natural Semantics Induction on the Length of Derivation Sequences Especially important for Structural Operational Semantics 16 / 40

17 Non-sequential Language Extensions Reading material: Section 3.1 of SwA 17 / 40

18 Abortion We extend the syntax of While with the statement abort S ::= x := a skip S 1 ; S 2 if b then S 1 else S 2 while b do S abort Intuition: abort stops the execution of the complete program We model this by ensuring that configurations abort, s are stuck Neither for NS nor for SOS we change the set of transition rules! Then there is no rule to apply when the abort-statement is encountered, ensuring that the respective configuration is stuck. 18 / 40

19 abort NS: no derivation tree Abortion: Three examples SOS: finite derivation sequence abort, s skip NS: derivation tree skip, s s SOS: finite derivation sequence skip, s s Both NS and SOS can establish that abort and skip are not semantically equivalent. 19 / 40

20 abort NS: no derivation tree Abortion: Three examples SOS: finite derivation sequence abort, s skip NS: derivation tree skip, s s SOS: finite derivation sequence skip, s s Both NS and SOS can establish that abort and skip are not semantically equivalent. while true do skip NS: no derivation tree SOS: infinite derivation sequence 20 / 40

21 abort NS: no derivation tree Abortion: Three examples SOS: finite derivation sequence abort, s skip NS: derivation tree skip, s s SOS: finite derivation sequence skip, s s Both NS and SOS can establish that abort and skip are not semantically equivalent. while true do skip NS: no derivation tree SOS: infinite derivation sequence abort and while true do skip are semantically equivalent in NS! SOS however can distinguish between abnormal termination and looping. 21 / 40

22 Abortion Natural semantics: we cannot distinguish between looping and abnormal termination SOS: looping is reflected by infinite derivation sequence and abnormal termination by finite derivation sequence ending in a stuck configuration 22 / 40

23 Non-determinism We extend the syntax of While with the statement or S ::=... S 1 or S 2 Intuition: For S 1 or S 2 we can non-deterministically choose to execute either S 1 or S 2 For both NS and SOS, new rules have to be introduced to handle the statement 23 / 40

24 Rules for non-determinism in Natural Semantics 24 / 40

25 Rules for non-determinism in SOS 25 / 40

26 Non-determinism: Two examples x := 1 or (x := 2; x := x + 2) NS: x := 1 or (x := 2; x := x + 2), s (while true do skip) or (x := 2; x := x + 2) 26 / 40

27 Non-determinism: Two examples x := 1 or (x := 2; x := x + 2) NS: x := 1 or (x := 2; x := x + 2), s s[x 1] (while true do skip) or (x := 2; x := x + 2) 27 / 40

28 Non-determinism: Two examples x := 1 or (x := 2; x := x + 2) NS: x := 1 or (x := 2; x := x + 2), s s[x 1] NS: x := 1 or (x := 2; x := x + 2), s s[x 4] Similar in SOS (while true do skip) or (x := 2; x := x + 2) 28 / 40

29 Non-determinism: Two examples x := 1 or (x := 2; x := x + 2) NS: x := 1 or (x := 2; x := x + 2), s s[x 1] NS: x := 1 or (x := 2; x := x + 2), s s[x 4] Similar in SOS (while true do skip) or (x := 2; x := x + 2) NS: (while true do skip) or (x := 2; x := x + 2), s s[x 4] is the only derivation tree! 29 / 40

30 Non-determinism: Two examples x := 1 or (x := 2; x := x + 2) NS: x := 1 or (x := 2; x := x + 2), s s[x 1] NS: x := 1 or (x := 2; x := x + 2), s s[x 4] Similar in SOS (while true do skip) or (x := 2; x := x + 2) NS: (while true do skip) or (x := 2; x := x + 2), s s[x 4] is the only derivation tree! SOS: Infinite derivation sequence, if the left branch is chosen; finite derivation sequence ending in s[x 4] if the right branch is chosen 30 / 40

31 Non-determinism: Two examples x := 1 or (x := 2; x := x + 2) NS: x := 1 or (x := 2; x := x + 2), s s[x 1] NS: x := 1 or (x := 2; x := x + 2), s s[x 4] Similar in SOS (while true do skip) or (x := 2; x := x + 2) NS: (while true do skip) or (x := 2; x := x + 2), s s[x 4] is the only derivation tree! SOS: Infinite derivation sequence, if the left branch is chosen; finite derivation sequence ending in s[x 4] if the right branch is chosen NS suppresses looping! (Never chooses the wrong branch.) SOS: Looping is not suppressed. 31 / 40

32 Parallelism We extend the syntax of While with the statement par S ::=... S 1 par S 2 Intuition: For S 1 par S 2, both branches have to be executed, but their execution can be interleaved For example: x := 1 par x := 2; x := x / 40

33 Parallelism We extend the syntax of While with the statement par S ::=... S 1 par S 2 Intuition: For S 1 par S 2, both branches have to be executed, but their execution can be interleaved For example: x := 1 par x := 2; x := x + 2 can be executed as either: x := 1; x := 2; x := x + 2 x := 2; x := 1; x := x + 2 x := 2; x := x + 2; x := 1 For SOS new rules can be introduced to handle the statement For NS? 33 / 40

34 Rules for parallelism in SOS 34 / 40

35 An example of parallelism in SOS (1/3) 35 / 40

36 An example of parallelism in SOS (2/3) 36 / 40

37 An example of parallelism in SOS (3/3) 37 / 40

38 Attempt at rules for parallelism in Natural Semantics Unfortunately these are erroneous as they do not express interleaving so Natural Semantics cannot express parallelism 38 / 40

39 Summary: Non-sequential Language Constructs Natural semantics Structural Operational Semantics Non-determinism suppresses looping does not suppress looping Parallelism cannot express interlea- can express interleaving ving of computations of computations 39 / 40

40 Summary Recap Extensions of operational semantics non-determinism, parallelism Exercise Class Exercises 3.1, 3.3, 3,4, 3.5, 3.6 from SwA. 40 / 40