CSE 3341.03 Winter 2008 Introduction to Program Verification January 8 Boolean operators

CSE 3341.03 Winter 2008Introduction 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 http://www.cse.yorku.ca/course/3341/lecture notes/3341 2008-1-3 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:

http://www.cse.yorku.ca/course/3341/small-large.html

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 54.43 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 BooleBoolean 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 0x 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

http://www.home.gil.com.au/~bredshaw/boole.htm

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 argumentsthey 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

http://www.cse.yorku.ca/course/3341/lost-logic.html

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