CSC302 2005F, Class 3: The Science of Programming: Boolean Logic

Admin
* No reading for Monday
  * Guy Steele on video - Helped develop Scheme; Helped design Java
* Reading for Wednesday
  * Code is intended to mean something
* Reading for next Friday
  * Gries, Chapter 2
* Apologies: Some of you will find this stuff easy, some of you will find it difficult
* When to submit questions!  (By 8 p.m.)
* FAQ

Overview
* Background: 
  * What is "The Science of Programming"? (SOP)
  * Why study SOP?
* Boolean logic basics
* A grammar for Boolean propositions
* Reasoning with Booleans

Historical background:
* Early programming (pre 1980's) variable
  * Difficult to ensure correctness
  * Sometimes difficult to say "this is what I want"
* Focus is still on "computation"
* Dijsktra: A "discipline" of programming
  * To contrast with "art"
* Develop and apply careful techniques
  * To the development of programs
  * To the specification of programming languages
* How do/did you specify the meaning of a programming language?
* Our goal: Specify enough about the meaning of the language that a programmer can write a program and have some reasonable expectation about what it will do.
* Example: cons
  * A procedure of two parameters (x and y) that prepends x to y
    * Natural language (English)
      * Time flies like an arrow
      * Fruit flies like a banana
  * (cons x y) is like new Pair(x,y)
    * Demonstrate reduction of new programming language to known programming language
  * Try it, you'll like it!
    * Operationally
  * Define it with mathematics and the rules of logic!
    (car (cons x y)) = x
    (cdr (cons x y)) = y
    (pairP (cons x y)) = #true
    (nullP (cons x y)) = #false
* Dijsktra, Gries, and company wanted to apply similar care to the definition of imperative languages

Gries will tell us:
* How to reason with logic
* How to use logic to carefully specify the meaning of operations
* How to use those meanings to more carefully design programs

Why study Gries in CSC302?
* Useful to see how to define the semantics (meaning) of a programming language
* We think it's important that you study it somewhere, as the "discipline" of programming will make you better programmers (now in 201)
  151-152-201 or 151-152-223-201
* Chapter one is about "the basics of Boolean logic"
  * Defining a "new" language
    * Syntax
    * Semantics
    * Examples of use

Defining the basics of Boolean logic
* Goal: Reasoning about "utterences" (propositions) that may have "truth" associated with them
* Basic values in the language: true and false

What are the "legal" utterences (propositions)

Basic propositions
* true is a proposition
* false is a proposition
* a variable is a proposition (e.g. x is a proposition)
  * Worry: How do we discuss the "truthfulness" of variables?
  * Foreshadowing: States

Recursive propositions: Assume p1 and p2 are propositions
* ( not p1 ) is a proposition
* ( p1 or p2 ) is a proposition
* ( p1 and p2 ) is a proposition
* ( p1 implies p2 ) is a proposition
* ( p1 equals p2 ) is a proposition

NOTHING ELSE IS A PROPOSITION

Which of these are propositions?
* x  - YES
* x and y - NO
* (x implies true) -  YES
* ((x) implies true) - NO

Observation: More parentheses than Scheme

	x && y || z

Language design goal: Change the syntax of the utterences so that we don't have to write so many parens
* Check Gries
  * A grammar that formally specifies association and precedence
* Association and precedence
  parenthesization
  not
  and
  or
  implies
  equals
* Associativity: Given two instances of the same operation, which way do you group?
  * a OP b OP c 
    * (a OP b) OP c - LEFT ASSOCIATIVE
    * a OP (b OP c) - RIGHT ASSOCIATIVE
  * And does it matter?
    and - NOT FOR THE TYPICAL MEANING OF AND
    or - NOT FOR THE TYPICAL MEANIGN OF OR
    implies
    equals

How do we assign meanings to utterances so that we can answer that question?

* Truth table for basic operations
* Assumption of recursive evaluation for non-base subterms

x y	x implies y	x equals y
F F	T		T
F T	T		F
T F	F		F
T T	T		T

Do there exist a b and c for whic
  (a implies b) implies c != a implies (b implies c)
  Let a be false, b be false, c be true
  (false  implies false) implies true != false implies (false implies true)
  true implies true != false implies true
  true == true