CSE 3341.03 Winter 2008 Introduction to Program Verification January 31

1. CSE 3341.03 Winter 2008Introduction to Program VerificationJanuary 31 proofs through simplification

2. propositions and proofs • they’re different animals • "P implies Q" is not the same thing as "from P infer/deduce Q" • rules of inference are different from tautologies, but in prop. logic, they’re closely related • tautologies always have proofs. Why? • example: truth-table = proof from alist of 2n cases.messy from human point of view but perfectly effective as a logic tool

3. 3.7 The "Deduction Theorem" • if P implies Q is a tautology, then Q can be proved from the assumption that P is true. • (To prove this rigorously, we would need to formalize concept of proof.) • Idea: look at all rows of the truth table for which P is true. • Informally, saying that Q can be proved from P just means that Q can be shown (calculated) to be true in all these rows.

4. the converse • if Q has a (valid) proof, given P, then if P is true, Q can't be false, so P implies Q is a tautology. • (this follows from the definition of valid proof)

5. getting a proof from wang? • implement a trace feature: • sequence of logically equivalent sequents, terminating in an overlap = true, or not = false. • use the fact that the rewrite rules are logical equivalences • but if wang is working correctly, a derivation is not very useful: • like intermediate steps in a multiplication. We don't need to check them if we trust the algorithm.

6. preprocess wang input • use simplification to prepare input for Wang's algorithm, in the hope that what we want proved becomes a tautology • example from SVT: • x > 0 implies a+a = 2*a. • simplification uses mathematical theory of + to simplify a+a to 2*a, and logic to simplify 2*a = 2*a to true • up to us to find an appropriate theory

7. simplification adds semantics to logic • simplification = mechanism for taking meanings of terms into account simplification rules are used to represent mathematical knowledge ("truths") • mathematical truths are relative to a system of axioms and inference rules

8. axioms and inference rules determine what the symbols mean (in that system) • typically, mathematical and logical truths are representable by equations: • a+a = 2*a, where a is an integer • (P implies true ) = truewhere P is a proposition.

9. truths as equations • in general: mathematical truth is an equation you learned in school, or a mathematical 'fact' from a book • something you or someone else has proved • something assumed to be true (0-length proof) = axiom to use these ‘facts’, axioms, etc., we put them into the form of equations, and give them an orientation. cf. 4.1: what makes a valid rule

10. given the “theory” X - X = 0X + 0 = XX = X is true • then a + (a - a) = a simplifies to true. • note how the theory implicitly specifies the meaning of the functors

11. simplification shortens expressions • eliminate redundancy from mathematical expressionsx + 0 = x 1 + x + 1 = x + 2 • use it also to eliminate redundancies from logical descriptions A and A = A

12. "x < 0 and x <= 0" doesn't say any more than "x < 0" what lets us simplify this to x < 0? the general logical equation A and (A or B) = A i. e., A and (A or B) iff A is a tautology together with a mathematical "truth" (here a definition): ? (notice that definition rules don't simplify (shorten))

13. theory files • theory files = collection of rules = "programs" for the simplify "interpreter" • available in /cs/course/3341 • example: equality.simp max(A,C) = C ->> A <= C.max(B,C) = B ->> C <= B.X <=Y and not Y <= X ->> X < Y.X <= Y and not X = Y ->> X < Y.X <= Y and not Y = X ->> X < Y.X <= Y or Y < X ->> true.X = Y and X <= Y ->> X = Y.

14. variables • Note the difference between rule (pattern) variables and mathematical variables we use lower case for mathematical variablesupper case for pattern or rule variablesthese match arbitrary terms in the input • suppose we had a rule X/X ->> 1.2+(x<0)/(x<0) ->> 3 ??

15. why individual theory files? • theory files in /cs/course/3341 arithmetic.simp, equality.simp, logic.simp • why not have one huge theory file covering everything? • same advantage as modules in constructing a program • e. g., the theory of ‘+’ is independent of the theory of stacks

16. implementing simplification • simplification means finding a simplification rule whose left-side matches the structure of some sub-term and then rewriting (replace match with right-side of rule) then repeat this until no rule applies. • usually, simplification makes an expression shorter, but for definitions, we want expansion A < B < C ->> A < B and B < C.

17. the algorithm • simplify(Expr) = Result if path_arg(Path, Expr) = Lhs, % (there is a path in Expr to the sub-expression LHS) and Lhs ->> Rhs, and Modified = change_path_arg(Path, Expr, Rhs), and Result = simplify(Modified) • otherwisesimplify(Expr) = Expr.

18. entering rules • How do we get the ->> rules into this algorithm? • enter from the terminal or from a file. • simplify supplements rewrite rules with special code for arithmetic expressions

19. arithmetic problems • some operators are commutative :X + Y = Y + X (but not X**Y = Y**X) • simplify to canonical form to detect identity: let x + y ->> y + x then given Y + X - X ->> Y, x + y - x ->> y

20. canonical form • suppose you had to handle date calculation in a variety of formats:February 1, 2007, Feb 1 07, 1/2/2007 (Can.) 2/1/2007 (US) etc. • use canonical form for date calculation example: seconds after Jan 1, 1904. • canonical form allows us to recognize equivalences between terms with the same commutative functors

21. associativity • associativity • difference between syntactic associativity and semantic associativity • semantic: X op (Y op Z) = (X op Y) op Z • syntactic: (left) X op Y op Z = (X op Y) op Z (right) X op Y op Z = X op (Y op Z) • simplification algorithm chooses left associativity as a canonical form (if term is not parenthesized)

22. simplifying with canonical forms • if A op ( B op C) = (A op B) op C) pick one as a canonical form create an additional rule for the other case. • canonical forms for relations and their converseswhat’s the converse of a relation? • what's the converse of >= ? simplify x >= y ->> y <=x. x > y ->> y < x.

23. cancellation • cancellation: rewrite rules don't do this easily • current version of simplify: a + b + c + . . - a ->> . . c+banda - b - c + b ->> a - cbut • a - b - c - a ->> a - b - c - a • a - b - a - c ->> a - b - a- c