CS486 Spring 1997

Lecture 3 Tuesday, Jan. 28, 1997

More comments on

short-circuit and parallel evaluation tables (see over)
proving termination of by ``Formula-induction''

Comparing Smullyan's book to lectures

where is the computational content in Smullyan?
aspects of the ``architecture of formal knowledge''

Smullyan discusses valuations , they are Boolean iff
conditions hold.

Fact. For fixed is a valuation in F.
Fact. There is at most one Boolean valuation consistent with .
Fact. There is at least one Boolean valuation consistent with .

Def: A set is saturated iff (p.12)
Also call a truth set.
Let .

Fact: is saturated iff is a Boolean evaluation. (see p.12)

Fact: .



Note, this defines Tautologies without reference to or . We see that Smullyan's more general approach opens new possibilities. This is an advantage of abstraction and generalization.

Look at all the CS we used.
Recursive data types-from courses 211, 280, 410
Recursive procedures-from courses 211, 280
Termination proofs
Proofs by induction-from course 280
Def of Taut in 2 ways, one using of sets-CS280

New level of concern
The architecture of knowledge expressed in theories
Lecture versus Smullyan

We will see another major example on next Thursday-duality principle.

Extra credit basis of project, type up proofs of these statements in html following Smullyan's arguments.

Introduction to analytic tableaux-organize on goal not on truth-values

Method of tableaux-systematic search for a counter-example

We examine a proof procedure that is superior to truth tables
more efficient than truth-table
generalizes to almost all logics
signed formulas
example of tableau proof

The idea is to systematically search for a counter-example. A proof is a failed search. Let's consider the same examples as last time.







Why don't we need to consider 3 possibilities for -from the truth table?

circle.eps

This gives us something to worry about for completeness. Clearly the tableau method is based on the parallel evaluation idea and this keeps the minimal information.
Is this method equivalent to the BDD with parallel eval?
My proofs are Smullyan's ``modified Block Tableaux'' from Chapter XI.

Rules for building tableaux-Smullyan p.17

Systematic tableaux

Unifying notation- rules and rules

Precise definition of a tableau-Smullyan p.24

Examples

Exercises-solve all exercises on p.24, write up (3) and (5)

Notes on lecture 3

We did not symbolize Smullyan p.10-11 completely because we do not have the concepts yet to do that, but later we will return to this topic. But in a similar spirit to him, we filled in details of his inductive argument and drew out the correspondence to eval.
Here is a more refined (detailed) account of what Smullyan said on p.10-11.
Theorem on Existence of a Boolean Valuation.

Consider a single formula and a state .
There is only one way to assign a truth value to each subformula , of such that the atomic subformulas (i.e. the variables, ) are assigned value and such that the truth value of each compound subformula of is determined from the truth value of immediate subformulas, , by the truth-table rules . Call that value for each subformula .
Proof by Formula-induction
Base case (atomic formulas).
In this case the value of is just which is a unique value since is a function.
Induction case: assume that the theorem is true for all proper subformulas of . Proceed by cases on the structure of .
If is then by the induction hypothesis applied to proper subformulas there are unique truth values . Now let . This assignment is unique. It establishes the theorem for in this case.
If is then argue as above taking .

If is then argue as above taking .

If is then argue as above taking .
Qed
Notice, this proof is constructive. It shows us how to build the truth value for each subformula of and hence for itself. In fact, we can see in this proof the same computational structure as our definition of ! Thus, Smullyan has introduced the same computational content but by implicit means rather than explicit ones. This is typical of mathematics as opposed to computer science. [Indeed, to stress the computational meaning, Smullyan adds a procedure description on page 11 in square brackets; this is essentially a description of ).]
Note also that the form of this theorem is essentially For all for all Formulas there is a unique such that .

Definition of

1. 

   is
   is
   
   is
   
   
   as Unary

   Here assume are defined as in a programming language that requires evaluation of all arguments.

2. is as in 1. except we use a short-circuit evaluation of
   -say
   

   

   

   

3. is as in 1. except we use a parallel evaluation of binary operators called p-and, p-or, p-imp
   

   

   

   

   When needed we distinguish among these as .

Evaluation Procedures
To evaluate s-and first , if the result is then return , if the result is then
, if it is ,then , if it is then .

If the result of evaluating or is not a truth value then abort.

We express this succinctly with these rules of evaluation. Exercise: write the informal version of the other rules.

To evaluate start evaluating and concurrently : if either terminates with value
, then the value is , if both terminate with then the value is .

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