Wolfgang Slany
http://slany.org/wolfgang/
http://www.dbai.tuwien.ac.at/proj/pf2html/
April 23, 2002
Abstract:
HTML version of a sample
proof
that uses
Lamport's
proof
style
[
1], illustrating how structured proofs can be
converted to
HTML
pages via
L^{A}TEX2HTML
enriched with
extensions for Lamport's proof
style. Note that we try
on purpose to carry out Lamport's rule of thumb to ``expand the proof
until the lowest level statements are obvious, and then continue for one
more level'' in order to illustrate the principles of structured
hyperproofs.
What is the disjunctive normal
form
of
The resulting disjunctive normal form is exponentially blown up
compared to the size of the original conjunctive normal form.
The resulting disjunctive normal form is exponentially blown up
compared to the size of the original conjunctive normal form.

Proof Sketch: Consider the directed graph
shown in Figure 1.
Observe that
paths from to can be described either as going through
nodes ( or ) and ( or ) and ... and
( or ) (cf. the conjunctive normal form), or
alternatively as going through nodes ( and and ... and
and ) or ( and and ... and
and ) or ( and and ... and and )
or ( and and ... and and ) or ... or ( and and ... and and ) (cf. the
disjunctive normal form). The formal proof makes this precise.
Let: Denote the left side of Equation 2 by
,
and its right side by
(the variablenames are reused
in Figure 1 to label the top and bottom nodes).
Prove:
Proof: 1: There is a onetoone correspondence (a
bijection)
between the
paths from node to node and the minimal satisfying truth
assignments of (the conjunctive normal form).

1.1: Every path from node to node
corresponds to a unique minimal satisfying truth assigment of .

Let:
1.1.1: Let a variable of be set to true
whenever the path goes through a node with the same name as the
variable.
Proof: 1.1.2: The truth assignments induced by paths from
to are satisfying truth assigments of .

1.1.2.1: Every path from to must go through either or
for all
.

Proof: Structure of the graph in Figure 1.

1.1.2.2: Q.E.D.

1.1.3: The truth assignments induced by paths from
to are minimal satisfying truth assigments of .

1.1.3.1: Flipping any variable from true to false in
such a truth assignment makes unsatisfied.

Proof: Structure of (every conjunct has only one variable
that makes it true in any minimal truth assignment of
).

1.1.3.2: Q.E.D.

1.1.4: These minimal satisfying truth assigments are unique.

Proof: 1.1.3, the structure of the graph, the
structure of , and because the order of variables in truth
assigments does not matter.

1.1.5: Q.E.D.

1.2: Conversely, every minimal satisfying truth
assignment of corresponds to a unique path from node to
node .

Let:
1.2.1: Let a variable of be set to
true whenever the path goes through a node with the same name as
the variable.
Proof: 1.2.2: Every minimal satisfying truth assignment of
corresponds to a path from node to node .

Proof: From the structure of , exactly one variable in each
conjunct must be true in any minimal satisfying truth
assignment of . This defines a path from node to node
by the structure of the graph in Figure 1 and
1.2.1.

1.2.3: The thus induced path is unique.

Proof: Structure of and the construction of the graph in
Figure 1.

1.2.4: Q.E.D.

1.3: Q.E.D.

2: There is a onetoone correspondence between
the paths from node to node and the minimal satisfying truth
assignments of (the disjunctive normal form).

2.1: Every path from node to node
corresponds to a unique minimal satisfying truth assigment of .

Let:
2.1.1: Let a variable of be set to true
whenever the path goes through a node with the same name as the
variable.
Proof: 2.1.2: The truth assignments induced by paths from
to are satisfying truth assigments of .

2.1.2.1: Every path from to must for all
either go through or .

Proof: Structure of the graph in Figure 1.

2.1.2.2: Q.E.D.

2.1.3: The truth assignments induced by paths from
to are minimal satisfying truth assigments of .

2.1.3.1: Flipping any variable from true to false in
such a truth assignment makes unsatisfied.

Proof: Structure of (only one disjunct is made true
by any minimal truth assignment of ).

2.1.3.2: Q.E.D.

2.1.4: These minimal satisfying truth assigments are unique.

Proof: 2.1.3, the structure of the graph, the
structure of , and because the order of variables in truth
assigments does not matter.

2.1.5: Q.E.D.

2.2: Conversely, every minimal satisfying truth
assignment of corresponds to a unique path from node to
node .

Let:
2.2.1: Let a variable of be set to
true whenever the path traverses a node with the same name as
the variable.
Proof: 2.2.2: Every minimal satisfying truth assignment of
corresponds to a path from node to node .

Proof: From the structure of , exactly one variable in each
disjunct must be true in any minimal satisfying truth
assignment of . This defines a path from node to node
by the structure of the graph in Figure 1 and
2.2.1.

2.2.3: The thus induced path is unique.

Proof: Structure of and the construction of the graph in
Figure 1.

2.2.4: Q.E.D.

2.3: Q.E.D.

3: Q.E.D.

Proof: Since each satisfying truth assignment of a boolean
formula must contain a minimal satisfying truth assignment (by definition
of the latter), and since by proof steps 1 and
2 there must be a onetoone
correspondence between the minimal satisfying truth assignment of
and of , a truth assignment satisfies
iff
it also satisfies .


The help and comments of Marcus Carotta, Wolfgang Goedl, Martina
Osztovits, and Helmut Veith are gratefully acknowledged.

 1
 Leslie Lamport, 1993, How to write a proof. In
Global Analysis of Modern Mathematics, pp. 311321. Publish or
Perish, Houston, Texas, February 1993. A symposium in honor of
Richard Palais' sixtieth birthday (also published as SRC Research
Report 94).
http://research.microsoft.com/users/lamport/proofs/src94.ps.Z
 2
 Christos H. Papadimitriou, 1994, Problem
4.4.5. In Computational Complexity, p. 84. AddisonWesley.
Exponential
blowup
from conjunctive
to disjunctive normal
form
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