CSE Winter 2008 Introduction to Program Verification January 8 Boolean operators

January 3 lecture notes available as a PDF file see link from the table of contents panel on the course home page or notes/ slides.pdf

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.

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

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

"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”

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.

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!

the view from 10,000 ft. verification occupies only a very small part of the field of software development:

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?

multiple notations

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?

it gets worse! (Theorem of Principia Mathematics)

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.

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...

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?

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

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 ?

who was George Boole? - 19th cent. English mathematician after whom Boolean algebras, operators, variables are named The Mathematical Analysis of Logic (1847)

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

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

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

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.

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

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

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

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

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

Boolean values SVT 1.1 assumes that the range of Boolean functions is {true, false} is this important? see

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

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