Formal Semantics of Programming Language s

1. Formal Semantics of Programming Languages Topic 2: Operational Semantics 虞慧群 yhq@ecust.edu.cn

2. IMP: A Simple Imperative Language • numbers N • Positive and negative numbers • n, m  N • truth values T={true, false} • locations Loc • X, Y  Loc • arithmetic Aexp • a  Aexp • boolean expressions Bexp • b  Bexp • commands Com • c  Com

3. (3+5)  3 + 5 3 + 5  5+ 3 Abstract Syntax for IMP • Aexp • a ::= n | X | a0 + a1 | a0 – a1 | a0 a1 • Bexp • b ::= true | false | a0 = a1 | a0 a1 | b | b0 b1 | b0  b1 • Com • c ::= skip | X := a | c0 ; c1 | if b then c0else c1| while b do c 2+34-5 (2+(34))-5 ((2+3)4))-5

4. Example Program Y := 1; while (X=1) do Y := Y * X; X := X - 1

5. But what about semantics

6. Expression Evaluation • States • Mapping locations to values •  - The set of states •  : Loc  N • (X)= X=value of X in  •  = [ X  5, Y  7] • The value of X is 5 • The value of Y is 7 • The value of Z is undefined • For a  Exp,  , n  N, • <a, >  n • a is evaluated in  to n

7. Evaluating (a0 + a1) at  • Evaluate a0 to get a number n0 at  • Evaluate a1 to get a number n1 at  • Add n0 and n1

8. Expression Evaluation Rules • Numbers • <n, >  n • Locations • <X, > (X) • Sums • Subtractions • Products Axioms

9. Derivations • A rule instance • Instantiating meta variables with corresponding values

10. Derivation (Tree) • Axioms in the leafs • Rule instances at internal nodes

11. Computing a derivation • We write <a, >  n when there exists a derivation tree whose root is <a, >  n • Can be computed in a top-down manner • At every node try all derivations “in parallel” 5 16 21

12. Recap • Operational Semantics • The rules can be implemented easily • Define interpreter • Structural Operational Semantics • Syntax directed • Natural semantics

13. Equivalence of IMP expressions iff a0 a1

14. Boolean Expression Evaluation Rules • <true, >  true • <false, >  false

15. Boolean Expression Evaluation Rules(cont)

16. Equivalence of Boolean expressions iff b0b1

17. Extensions • Shortcut evaluation of Boolean expressions • “Parallell” evaluation of Boolean expressions • Other data types

18. The execution of commands • <c, >  ’ • c terminates on  in a final state ’ • Initial state 0 • 0(X)=0 for all X • Handling assignments <X:=5, >  ’ A notation: <X:=5, >  [5/X]

19. Rules for commands Atomic commands • <skip, >   • Sequencing: • Conditionals:

20. Rules for commands (while)

21. Example Program Y := 1; while (X=1) do Y := Y * X; X := X - 1

22. Equivalence of commands iff c0c1

23. Proposition 2.8 while b do c  if then (c; while b do c) else skip

24. Small Step Operational Semantics • The natural semantics define evaluation in large steps • Abstracts “computation time” • It is possible to define a small step operational semantics • <a, > 1 <a’, ’> • “one” step of executing a in a state  yields a’ in a state ’

25. Small Step Semantics for Additions

26. Summary • Operational semantics enables to naturally express program behavior • Can handle • Non determinism • Concurrency • Procedures • Object oriented • Pointers and dynamically allocated structures • But remains very closed to the implementation • Two programs which compute the same functions are not necessarily equivalent

27. Exercise 2 (1) Evaluate a  (X4) + 3 in a state  s.t. (X)=0. (2) Write down rules which express the “parallel” evaluation of b0 and b1 in b0 V b1 so that evaluates to true if either b0 evaluates to true, and b1 is unevaluated, or b1 evaluates to true, and b0 is unevaluated.