CSC302 2005S, Class 6: Gries 2: Boolean Reasoning

Admin:
* Late: SAM, Arjun
* Please don't forget to submit questions.
* Homework for Monday: Sections 1-5 of the Scheme Report.  
  * Scheme Report available this afternoon.
* Homework for Friday: Questions from Gries.

Overview:
* How to prove
* Some basic laws
* Terminology
* "What is the alternative?"

Goals of Gries's book:
* To teach you a mechanism for proving programs correct
* To teach you a new mechanism for *developing* programs
* To describe the semantics of a simple language

Focus of this chapter: Issues in proof
* Need to look at proof a little bit more formally (ooh, semantics) 

What kinds of things might you want to prove in Boolean logic (at this point)?
* Proposition is a tautology
* Tautology: True for every state in which it is well-defined
* State: Mapping of variables in a statement to true and false
* Can a proposition every be ill-defined?
  * When some variables are not defined
  * When some variables have definitions other than true or false

What techniques do you know for proving that E is a tautology?
* Make a truth table

Example
	not (a and b) = (not a) or (not b)
a  b 1:(a and b) 2:(not 1) 3:(not a) 4:(not b) 5:(3 or 4) 6:(2 = 5)
F  F 1:F 2:T 3:T 4:T 5:T 6:T
F  T 1:F 2:T 3:T 4:F 5:T 6:T
T  F 1:F 2:T 3:F 4:T 6:T 6:T
T  T 1:T 2:F 3:F 4:F 5:F 6:T

What permitted the intermediate computations of (intermeidate value 1) and (intermediate value 2) and ...
* Definitions of and, or, not
* Rule of variable substitution
* Rule of recursive computation

Proof that E is a tautology, v1: Truth table
Strength
* Easy to read proof (intuitive)
* Confidence
* AUTOMATABLE
Weaknesses
* Time-consuming 2^#variables*time-to-eval-expression
* Errors can crop up in carelessly-constructed truth tables
* Limited help of proving other tautologies

Other ways to prove that E is a tautology?
* Reduce to something simpler and equivalent, E', and prove that E' is a tautology
* Normalize (automatically) and hope that normal form helps

Need a system (Calculus) which describes the valid transformations

Valid transformations? (Rules for transforming equations)
* Substitution
  * IF E1 = S then L(E1) = L(S)
* Transitivity
  if E1 = S and S = E2 then E1 = E2
  Goal: E1 = E2; prove E1 = S and S = E2
  * Human intelligence normally helps us identify a good intermediate S
Alternate view of substitution
  If E1 = E2 then E1(S) = E2(S)
Laws
* Reflexivity: E1 = E2 then E2 = E1

Question: If I have proven that L(E1) = L(S) have I also proven that E1 = S?
* Yes: 4
* No: 9
* Doesn't think: 1

Suppose L(x) = (F and x)
L(T) = F
L(F) = F
but T != F

SUBSTITUTION IS NOT BI-DIRECTIONAL
* General facts let us prove more specific facts
* More-specific facts do not let us prove more general facts (unless we accumulate a lot of them into a truth table)

* Law is a specific proposition you can use in substitution and transitivity

Open questions for Monday:
* Suppose I want to prove that E1 = E2.  How does substitution help?
* Are there systems in which some of Gries's laws don't hold?