Denotational Semantics of Programs. : SimpleExp N.

Transcription

1 Models of Computation, Denotational Semantics of Programs Denotational Semantics of SimpleExp We will define the denotational semantics of simple expressions using a function : SimpleExp N. Denotational Semantics of While We will later discuss the denotational semantics of our while programs.

2 Models of Computation, Denotation of Simple Expressions We define : SimpleExp N by induction on the structure of expressions: n = n (E 1 + E 2 ) = E 1 + E 2 [Recall the function den from the previous lectures.]

3 Models of Computation, Calculating Semantics (1 + (2 + 3)) = 1 + (2 + 3) = 1 + (2 + 3) = 1 + ( ) = 1 + (2 + 3) = 6.

4 Models of Computation, Associativity of addition Theorem For all E 1, E 2 and E 3, (E 1 + (E 2 + E 3 )) = ((E 1 + E 2 ) + E 3 ) Proof (E 1 + (E 2 + E 3 ) = E 1 + (E 2 + E 3 ) = E 1 + ( E 2 + E 3 ) = ( E 1 + E 2 ) + E 3 = (E 1 + E 2 ) + E 3 = (E 1 + E 2 ) + E 3 )

5 Models of Computation, Some SimpleExp Contexts C 1 [ ] = C 2 [ ] = ( + 2) C 3 [ ] = (( + 1) + )

6 Models of Computation, Filling the Holes C 1 [(3 + 4)] = (3 + 4) C 2 [(3 + 4)] = ((3 + 4) + 2) C 3 [(3 + 4)] = (((3 + 4) + 1) + (3 + 4))

7 Models of Computation, Contextual Equivalence Expressions E 1 and E 2 are contextually equivalent with respect to the big-step semantics if and only if, for all contexts C[ ] and all numerals n, C[E 1 ] n C[E 2 ] n.

8 Models of Computation, Compositionality and Contextual Equivalence Theorem For arbitrary expression E, E = n if and only if E n. By compositionality of, expressions E 1 and E 2 are contextually equivalent if and only if E 1 = E 2.

9 Models of Computation, State Transformers The set of state transformers is defined to be ST = [Σ Σ ] : that is, the set of (total) functions which take a starting state and return either, to indicate that the computation got stuck or looped for ever, or a final state.

10 Models of Computation, Domains for Booleans and Expressions The domain of predicates is defined by P = [Σ B ] where B = {tt, ff}. The domain of expressions is E = [Σ N ]

11 Models of Computation, Semantic Functions for While C : Com ST E : Exp E B : Bool P

12 Models of Computation, Variable Lookup The semantics of a variable is simple. The store is examined to see if there is a value for the variable: if so, this value is returned; and if not, the expression is stuck and we return : E x (s) = s(x) if s(x) is defined = otherwise

13 Models of Computation, Assignment This is easy. An assignment x := E transforms store s by updating x to contain the value of E: C x := E (s) = s[x E E (s)] if E E (s) = otherwise Note that, in this definition, E is evaluated in store s.

14 Models of Computation, skip skip is the easiest command of all. It simply leaves the store alone: C skip (s) = s

15 Models of Computation, Sequential Composition How does C 1 ;C 2 transform a store? Intuitively, first C 1 transforms the original state s to some s, then C 2 starts running in state s, leaving some s, which is the outcome of the whole command. If C 1 gets stuck or into an infinite loop, so does the whole command; similarly for C 2. We define a state transformer for C 1 ;C 2 which reflects this intuition.

16 Models of Computation, Semantics of Sequential Composition The state transformer C C 1 ;C 2 is defined by C C 1 ;C 2 (s) = if C C 1 (s) = = C C 2 (C C 1 (s)) otherwise Notice that this second line is well-typed, because if C C 1 (s) then C C 1 (s) Σ, so we can indeed apply C C 2 to it.

17 Models of Computation, Conditional A command (if B then C 1 else C 2 ) transforms a state s as follows: work out if B is true or false in state s if true, transform the state s by running C 1 if false, transform the state s by running C 2

18 Models of Computation, Conditional. continued We therefore define C if B then C 1 else C 2 (s) to be C C 1 (s) if B B (s) = tt C C 2 (s) if B B (s) = ff otherwise

19 Models of Computation, Semantics of the Sequential Composition Operator We define the function by seq : ST ST ST seq(f,g)(s) = if f(s) = = g(f(s)) otherwise

20 Models of Computation, Semantics of the Conditional Operator We define the function by cond : P ST ST ST cond(p,f,g)(s) = f(s) if p(s) = tt = g(s) if p(s) = ff = otherwise

21 Models of Computation, An Equation for while C while B do C = C if B then (C; while B do C) else skip.

22 Models of Computation, Introducing Fixed Points What we need to do is to find some f ST such that f = cond( B,(C C ;f), id) where id, the identity function, is the semantic function for skip we have already defined. We can then use f as the semantics of while B do C.

23 Models of Computation, A Helper Function To put it another way, define a function F : ST ST by F(f) = cond(b B,(C C ;f), id).

24 Models of Computation, A First Approximation of while If B is false in state s, it does nothing: that is, it returns the state s. In this particular case, the transformation is the same as that given by if B then anything else skip

25 Models of Computation, A Sneaky Step Since the anything above could be anything (!!), let us replace it with the phrase (C; anything). This gives us if B then (C; anything) else skip. The semantics of this is now F(anything).

26 Models of Computation, A Second Approximant If B is true in state s but becomes false after running the loop body C once, then the loop transforms the state in the same way as if B then C;(if B then C;anything else skip) else skip The semantics of this is F(F(anything)).

27 Models of Computation, A Sequence of Approximants Give a starting state s: if, starting in state s, the loop body would be executed less than n times, then for any state transformer f, F n (f)(s) gives the same final state that the loop would give; if more than n executions of the loop body would be required, F n (f)(s) is right on more and more starting states.

28 Models of Computation, Better Approximants In the above, set f to be the state transformer which gives for any starting state s. Write this state transformer as too! Then if, starting in state s, the loop body would be executed less than n times, then F n ( )(s) gives the same final state that the loop would give; if more that n executions of the loop body would be required, F n (f)( ) gives, which may be incorrect. Note that if F n ( )(s), we know it must be the right answer.

29 Models of Computation, We ve Got a Fixed Point Define a state transformer f as follows: f(s) = F n (s) if F n (s) for some n = otherwise This is well-defined and is a fixed point of F.

30 Models of Computation, Approximating a While Loop We define the approximants of while B do C as follows: C 0 = diverge C 1 = if B then (C; diverge) else skip. C n+1 = if B then (C;C n ) else skip This is an inductive definition (mathematical induction on the subscript i of C).

31 Models of Computation, Semantics of while C while B do C (s) = s, if any C C k (s) = s =, otherwise