Programming Language Semantics Inductive Definitions

1. Programming Language SemanticsInductive Definitions Mooly Sagiv Eran Yahav msagiv@post yahave@post Schrirber 317 Open space 03-640-7606 03-640-5358 html://www.cs.tau.ac.il/~msagiv/courses/sem03.html Textbook:Winskel The Formal Semantics of Programming Languages CS 0368-4348-01@listserv.tau.ac.il

2. Outline • Rule induction • Special rule induction • Proof rules of the operational semantics • Least fixed points

3. Derivations • A set of rule instancesR consists pairs X/y where X is a finite set and y is an element • X/y – rule instance • X – premises • y – conclusion • d R y – d is an R-derivation of y • (/y) R y if (/y)  R • ({d1, …, dn}/y) R y if ({x1, …, xn}/y) R andd1 R x1 & … & dn R xn

4. Derivations Expressions R={(/ <n, >  n) | n  N,   } {(/ <X, > (X)) : X  Loc,   } {({<a0, > n0, <a1, > n1}/<a0+a1, > m) | n0, n1, m  N, m= n0+n1,   } {({<a0, > n0, <a1, > n1}/<a0-a1, > m) | n0, n1, m  N, m= n0-n1,   }{({<a0, > n0, <a1, > n1}/<a0a1, > m) | n0, n1, m  N, m= n0n1,   } (/ <5, [ X  8, Y  7] >  5) R<5, [ X  8, Y  7] >  5 (/ <X, [ X  8, Y  7]  8) RX, [ X  8, Y  7]  8 ({/ <5,[ X  8, Y  7] >  5) , (/ <X, [ X  8, Y  7]  8)}) R(5+X), [ X  8, Y  7]  13 ({(/ <5, [ X  8, Y  7] >  5)}) R (5+5), [ X  8, Y  7]  10

5. Rule induction • A special induction • Define a set by rules • IR={x | Rx} • Examples • <a, , n> of Aexp    N such that <a, >  n • <b, , t> of Bexp    T such that <b, >  t • <c, , ’> of Com     such that <c, >  ’ • Show that the property is true for all elements by induction on the rule application

6. The general principle of rule induction • LetIR={x | Rx} • Let P be a property • x  IR P(X)  for all the rule instances (X/y) in R for which X  IR z X. P(z)  P(y)

7. Justifying the principle of induction • A set Q is closed under rule instances or simply R-closed if for all rule instances X/yX Q  y  Q • Proposition 4.1: • IR is closed and • If Q is an R-closed set then IR  Q • Application • Q = { x  IR | P(x) } • Examples • R = {(/0)} {{n}/{n+1) | n  } • Referential transparency for expressions

8. Expressing Syntax using Rules • a ::= … | a0 + a1| … • a0: Aexp a1: Aexp a0+a1: Aexp

9. Special Rule Induction • Handles rules of different types • BNF • c ::= … | X := a | …| if b then c0 else c1 | … • Rules • X : Loc a : Exp X:=a: Com • b : Bexp c0 : Com c1: Com if b then c0 else c1: Com

10. The special principle of rule induction • LetIR={x | Rx} • A  IR • Let Q be a property • a  A. Q(a)  for all the rule instances (X/y) in R for which X  IR and yAx XA.Q(x)  Q(y)

11. Proof rule for operational semanticsArithmetic Expressions • P(a, , n) is true of all evaluations <a, > n if it is preserved by the expression rules

12. Proof rule for operational semanticsArithmetic Expressions • P(a, , n) is true of all evaluations <a, > n if it is preserved by the expression rules

13. Rule Induction for Arithmetic Expressions aAexp, , n N. <a, > n P(a, , n) iff n N, . P(n, , n) & XLoc, . P(X, , (X)) & a0, a1 Aexp, , n0, n1 N.<a0, >  n0 & P(a0, , n0) &<a1, >  n1& P(a1, , n1) P(a0+a1, , n0+n1) & …

14. Proof rule for operational semanticsBoolean Expressions • P(b, , t) is true of all evaluations <b, > t if it is preserved by the Boolean expression rules • Define a subset of • (AexpN) (BexpT) • Obtained from the special principle of induction for properties P(b, , t) on the subset BexpT

15. Rule Induction for Booleans bBexp, , t T. <b, > t  P(b, , t) iff . P(false, , false) & . P(true, , true) & a0, a1 Aexp, , n0, n1 N.<a0, >  m& <a1, >  n & m=n P(a0=a1, , true) & a0, a1 Aexp, , n0, n1 N.<a0, >  m& <a1, >  n & mn P(a0=a1, ,false) … & b Bexp,  , t T. <b, > t & P(b, , t) P(b, ,  t) &…

16. Proof rule for operational semanticsCommands • P(c,,’) is true of all evaluations <c, >’if it is preserved by the command rules • Define a subset of • (AexpN) (BexpT) (Com) • Obtained from the special principle of induction for properties P(c, , ’) on the subset Com

17. Rule Induction for Commands cCom, , ’. <c, > ’  P(c, , ’) iff . P(skip, , ) & X Loc, a Bexp, . <a0, >  m P(X:=a, , [m/X]) & c0, c1 Com, , ’, ’’.<c0, >  ’’& P(c0, , ’) & <c1, ’’>  ’ &P(c1, ’’, ’) P(c0;c1, , ’) & …

18. Proposition 4.7 Define LocL(c) to be the variables which appear on the left side of some assignment in c Let y  Loc For all commands c and states , ’ Y  LocL(c). <c, ,>  ’  (Y) = ’(Y)

19. Operators and their least fixed points • For a set of rule instances R • R(B)={y | X B, X/y  R} • Proposition 4.11 A set B is closed under R if R(B) B • R is monotonic • A  B  R(A)  R(B) • Define the sequence of sets • A0 = R0() =  • A1 = R1() =R() • A2 = R2() =R(R()) • … • An = Rn() • Define A = n  An

20. Proposition 4.12 • A is R-closed • R(A) = A • A is the least R-closed set Let fix(R) denote the least fixed point of R fix(R)= nRn()

21. Summary • Induction allows to prove properties of the programming language • Example properties • Deterministic • Referential transparency • Equivalent of small step and natural semantics