Understanding Program Verification: Approaches & Tools

1. CSE 3341.03 Winter 2008Introduction to Program VerificationJanuary 8 Boolean operators

2. January 3 lecture notes available as a PDF file • see link from the table of contents panel on the course home page or • http://www.cse.yorku.ca/course/3341/lecture notes/3341 2008-1-3 slides.pdf

3. what does verification give you? • a top-down understanding of a piece of software: • general and rigorous, not based on specific examples • the more perspectives, the better for catching errors. • that's the idea behind the argument that open-source will be more reliable than proprietary code.

4. two main approaches to verification • build a model of a machine that abstractly executes the program • check zillions of cases for key desired properties • works well for hardware, communication protocols • construct and prove a theorem

5. our approach • construct a logical proof based on the mathematical meaning assigned to the program text (i. e. a theorem) • this has been considered to be controversial from a theoretical/philosophical perspective • it’s been argued that verification, for a variety of reasons, is only conditional, not absolute (some truth to this) • but having a proof is still better than claiming a program is correct without one

6. "verification is a pain" • another knock on verification, from a practical perspective, is that even simple proofs are too hard to do, and even when “easy” are very tedious • we’ll deal with this objection head on:use tools to do the “heavy lifting”

7. treat logic as computation • since our tools are computational and concerned with proofs, we will need to develop an understanding of the basics of computational logic • a very different perspective on logic and mathematics than you have been given in your math courses • but inevitable in a computer-based technological society.

8. contrast with "semantic" approach • logic is about truth? • so logic is about distinguishing correct from incorrect arguments? • in the computational approach, these issues essentially disappear!

9. the view from 10,000 ft. • verification occupies only a very small part of the field of software development: • http://www.cse.yorku.ca/course/3341/small-large.html

10. logical operators • chapter 1 • start with a table of logical operators • operator vs operation ? • an operator is a syntactic entity • associated operation is a function • cf. x f y with f(x, y) • is the table of operators complete?

11. multiple notations

12. additional operators • two more logic operators: NAND (sometimes NAN) and NOR • we won’t use them • NAND is a basic building block of digital logical circuits we could replace not,and,or expressions involving only the single operator NAND not A =A NAND A • how do we get OR?

13. it gets worse! (Theorem 54.43 of Principia Mathematics)

14. standardizing our notation • important to be able to recognize various notations not much pressure on a mathematician to adopt a standard notation; that’s why we get so many. but we are required by the software we want to use to standardize on specific English words.

15. examples of logical operators in use • Find results of search for strings with all of the words A,B, C • = A and B and C . . with at least one of the words • = A or B or C . . without the words A or B or C • = not A and not B and not C . . .

16. propositions • propositions are expressions constructed from Boolean operators, the special names true and false, and other names which are assumed to have a fixed truth value • this is a recursive datatype - why? • hint: function composition?

17. Boolean operators • our approach to logic derives from the work of George Boole Boolean operators are syntactically analogous to +, x, etc. Boolean operations are functions of truth-valued Boolean variables truth-valued = true or false

18. arithmetic analogue • example: represent true by 1, false by 0 x and y = x*y where * is ordinary multiplication • how about or? • what's the arithmetic analogue to not ?

19. who was George Boole? - • 19th cent. English mathematician • after whom Boolean algebras, operators, variables are named http://www.home.gil.com.au/~bredshaw/boole.htm • The Mathematical Analysis of Logic (1847)

20. most famous work • An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities • based on a binary approach, processing only two objects:the now famous 1-0, yes-no, true-false, on-off

21. logic as rules for correct argument • which arguments yield TRUTHS? • Greeks discovered that one could have correct arguments about incorrect or nonsensical things; • or imagined, idealized objects such as points or lines • what made the argument correct was independent of what it was about; it depended on relationships (patterns) between propositions

22. logic as a calculus • some 1500 years after the Greeks, Boole invents a radical new perspective: • the arithmetic analogy means that logic can be interpreted as a calculus like arithmetic • propositions can be interpreted as arithmetic expressions

23. a calculus doesn't require meaning • in mathematics, functions evaluate the values of their arguments they are blind to what's inside those arguments (3401 survivors: compare with macros) • so the truth-value of ". . . " and ". . . . " must depend only on the truth-values of the arguments, as far as propositional logic is concerned. • blind to what the names or non-logical expressions may mean.

24. in ordinary/natural language this isn't always true: some language use a double negative for emphasis: not not p = not p

25. operations as binary functions • operator = binary function = 2x2 table • you should be able to construct the 2x2 table for all the binary operators we are concerned with • note: only one non-trivial unary function • more generally, lists of argument -value pairs • if the function has arity n, 2^n entries in the table

26. operator grammar • operators are syntactic, require grammatical properties • operators involve precedence issues and associativity to avoid writing ( ) the choices: • left-associativity: x + y + z = (x+y) + z • right-associativity ? I don't have an example: x op y op z = x op (y op z) • no associativity: x xor y xor z is an error: have to use parentheses because the function xor is not associative

27. operator precedence logic operations compared with arithmetic: • and has the precedence of * • or has the precedence of + • iff has the precedence of =

28. are the boolean values arbitrary? • does it matter what the boolean values are just as long as there are 2?

29. Boolean values • SVT 1.1 assumes that the range of Boolean functions is {true, false} • is this important? see http://www.cse.yorku.ca/course/3341/lost-logic.html

30. can logic tells us which is which? • "we can tell which value functions as true and which as false by looking at and/or" • true and false = false • true or false = true • but look at the "logic tables" page

31. Boolean algebra is self-dual • if we swap the names of Boolean values and the names of and/or? we get exactly the same structure! • you actually knew that, in the form of "De Morgan’s Law" • not(P and Q) = not P or not Q not(P or Q) = not P and not Q