Some Fundamental Properties of Boolean Ring Normal Forms

Series Logo

Volume 00 xx

Some Fundamental Prop erties of Bo olean Ring Normal

Forms

Jieh Hsiang and Guan Shieng Huang

Abstract Bo olean ring is an algebraic structure which uses exclusiv e or

instead of the usual or It yields a unique normal form for every Bo olean

function In this pap er we present several fundamental prop erties concerning

Bo olean rings We present a simple metho d for deriving the Bo olean ring nor

mal form directly from a truth table We also describ e a notion of normal form

of a Bo olean function with a dontcare condition and show an algorithm for

generating such a normal form We then discuss two Bo olean ring based theo

rem proving metho ds for prop ositional logic Finally we give some arguments

on why the Bo olean ring representation had not b een used more extensively

and how it can b e used in computing

Intro duction

Bo olean ring is an algebraic structure which is equivalent to Bo olean algebra

The ma jor representational dierences are that Bo olean ring uses exclusiveor

instead of or to represent Bo olean functions and that there is no need for

negation in Bo olean ring Furthermore there is a unique Bo olean ring normal form

for every Bo olean function It is curious however that in spite of its long history

and elegant algebraic prop erties the Bo olean ring representation has rarely b een

used in the computational context

In this pap er we present several fundamental results concerning Bo olean rings

Some of the results are so elementary and simple that they can and should b e

taught in a rstyear logic design course It is surprising that most of them seem

not known at least to our knowledge

In Section we give an overview of some known results ab out Bo olean ring

normal form notably the term rewriting metho d for pro ducing the normal form

through simplication

In Section we show a metho d for generating the Bo olean ring normal form

directly from the truth table of a Bo olean function This metho d is conceptually

very simple and is straightforward to implement Unlike the wellknown Karnaugh

Mathematics Subject Classication Primary C E Secondary E C

This research is supp orted in part by grant NSC E of the National Science

Council of the Republic of China

American Mathematical So ciety

p er page

JIEH HSIANG AND GUAN SHIENG HUANG

metho d which may pro duce dierent conjunctive normal forms dep ending on which

prime implicants are chosen our metho d is completely deterministic

We then present a notion of normal form for a Bo olean function with a dont

care condition and an eective way for generating this normal form

A Bo olean function with a dontcare condition A is a partial Bo olean function

whose values on truth assignments in A are undened This concept is useful

in circuit design in which the task of sp ecication can b e simplied by ignoring

truth assignments that are inconsequential However during the actual design and

fabrication of such a sp ecication the truth assignments in A must b e given some

value Dep ending on the designs the values may b e lled dierently Thus even if

two designs satisfy the same sp ecication they may represent dierent functions

This makes the circuit verication problem signicantly more complicated

From an algebraic p oint of view a truth table with a contcare condition rep

resents a class of Bo olean functions Thus if there is a mechanical way of cho osing

a normal form out of the entire class of functions then checking the equivalence

of two functions under the same dontcare condition b ecomes reducing the two

functions into the same normal form we present such a metho d in section Our

metho d is based on the Buchb erger algorithm for generating the Grobner basis of

the ideal dened by A and uses this Grobner basis to pro duce a unique normal form

for each equivalent class of Bo olean functions Our metho d should make verifying

correct implementations of sp ecication with a dontcare condition considerably

easier

In Section we show two ways of p erforming automated theorem proving for

prop ositional logic in the Bo olean ring framework The rst one is a resolution

style pro cedure based on Buchb erger algorithm This pro cedure was rst describ ed

in KN and is dierent from resolution in several ways First the input is not

restricted to clausal form Second it employs the inference rule of simplication

which has no natural counterpart in the resolution framework Simplication is a

p owerful way of reducing the search space in theorem proving BH

The second metho d we present is a DavisPutnam like pro cedure In addition to

splitting an inference rule employed in DavisPutnam DP DLL it also utilizes

the inference rule of simplication Since simplication is more natural and has

much more reduction p ower than the unit clause rule of DavisPutnam we feel

that it may have some advantage over DavisPutnam as a basis for an ecient

satisability checker

In the last section of the pap er we give some reasons on why Bo olean ring has

not b een used more extensively in logic or computer science We also p oint out

areas where it can b e used pro ductively

The Bo olean ring normal form

A Boolean ring is a commutative ring B in which is idempotent

ie x x x and is nilpotent ie x x The op erator is known in

The rst approach to automated theorem proving using the Bo olean ring representation is

to our knowledge H

It is easy to show that the nilp otence of is a consequence of the idemp otence of

SOME FUNDAMENTAL PROPERTIES OF BOOLEAN RING NORMAL FORMS

logic design as exclusiveor By intro ducing the relationship

x y x y

x y x y xy

x x

one can show that the corresp onding algebraic structure B is a Bo olean

algebra S The isomorphic relationship b etween Bo olean algebras and Bo olean

rings was found in by Stone S and could probably date back to by

Zhegalkin Z

A Bo olean function of n variables is a mapping from f g to f g In the

rest of the pap er we reserve B for the set f g and F for the set of Bo olean

functions with n variables

The op erators and can b e extended to the functional level Let f g F

we dene f g as a function h such that hx f x g x for all x B The

op erator is dened similarly Since a Bo olean ring is also a eld we know that

F is a commutative ring where and are the constant functions

and

Let x x F b e the pro jection functions such that the value of x dep ends

n i

only on the i argument it is easy to see that the set fx x g generates

the entire ring F In other words F can also b e regarded as a polynomial ring

B x x

A Bo olean function m is a monomial if it can b e represented as a conjunction

x where V fx x x g

If V is empty then m is the unity function In this denition as in the rest of the

Q P

pap er we use as a shorthand for a chain of conjunctions and for a chain of

Note that since is idemp otent each Bo olean variable app ears in a monomial

only once

A Bo olean function can b e expressed as a sum of monomials Such a represen

tation is called a Boolean polynomial By the nilp otence of an identical pair of

monomials in a Bo olean p olynomial can b e deleted Thus each monomial can ap

p ear at most once in a Bo olean p olynomial With these simplications a Bo olean

function can b e represented by a unique Bo olean p olynomial normal form which

is either or a sum of distinct monomials This normal form is the Boolean

ring normal form B RN F We emphasize that unlike the wellknown disjunctive

normal form DNF BRNF is unique for any Bo olean function

Theorem Stone There exists a unique BRNF for each Boolean

function with n variables

Given a Bo olean function we can derive its Bo olean ring normal form by re

duction using a canonical set of rewrite rules This metho d was rst presented in

H We describ e it here briey

A rewrite rule is an oriented equation A rewrite rule can b e applied to reduce a

term via equational replacement in the lefttoright fashion This pro cess is called

simplication We require that the simplication relation b e wel lfounded That

is no term can b e simplied indenitely The wellfoundedness requirement can

usually b e ensured by imp osing a simplication ordering when orienting equations

into rules D If a term cannot b e simplied by a set of rewrite rules R then we

JIEH HSIANG AND GUAN SHIENG HUANG

say that the term is Rirreducible If a term s is simplied by R to a term t and t is

Rirreducible then we say that t is an Rnormal form of s Note that the normal

form of a term may not b e unique A set of rewrite rules R is called a canonical

rewrite system if the Rnormal form of every term is unique

The following is a canonical rewrite system called BA for Bo olean algebra

x y x y x y

x y x y

x x

x y x y x

x x

x x x

x x

x y z x y x z

We remark that the op erators and are commutative and asso ciative

Given a Bo olean function one can apply the rules of BA to simplify it in

arbitrary order until no more simplication is p ossible The resulting unique

normal form is the BRNF of the Bo olean function

For example given p p q it can b e transformed into it BRNF as follows

p p q ppq p q

ppq pp pq

pq pq p

Generating BRNF from a truth table

In this section we describ e a metho d for generating the Bo olean ring normal

form of a Bo olean function represented by a truth table Our metho d works on the

truth table directly and do es not need auxiliary notions such as prime implicants

in Karnaugh map Furthermore since BRNF is unique our metho d is also more

deterministic

Let D denote f g the domain of the Bo olean function where n is assumed

to b e a xed integer throughout this section Given a truth assignment s of the n

variables x x we use s to denote the value of x in s

n i i

Definition Let s and t b e two truth assignments We say that s is a

positive extension of t if

for all i such that t s

i i

there exists an i such that t and s

i i

The set of p ositive extensions of s is denoted pexs

Definition Let s b e a truth assignment We use

jsj to denote the numb er of s which is

br s to denote the monomial x For the truth assignment s which

assigns to all Bo olean variables br s is

SOME FUNDAMENTAL PROPERTIES OF BOOLEAN RING NORMAL FORMS

For example if s then jsj pexs f g

and br s z

We are now ready to give the iptag algorithm which pro duces the BRNF from

a truth table

Fliptag algorithm

Input a truth table f

Output the BRNF of f

For each s D let tag s f s

For i from to n for each s such that jsj i if tag s

then for every t in pexs tag t tag t

output br s

tag s

Example Consider the following truth table

x y z f

At the b eginning the tag function of the truth assignments is the same as f

The Fliptag algorithm dictates that the tag b e examined from top to b ottom

Whenever a is encountered the tag of all the p ositive extensions of that truth

assignment is reversed For the given function the algorithm works as the following

table shows

x y z tag s f inal tag s br s

y z

xy z

By collecting br s of those truth assignments s whose nal tags are we get

y z xy xy z as the BRNF of f

For the ease of demonstrating the example we created a new tag column when

ever a in the tag of a truth assignment s is encountered In this case the tag of

each the p ositive extension of s is changed in the same column

We now show the correctness of the iptag algorithm Before we start we

need a few denitions and simple lemmas

Definition Let s b e a truth assignment we use r eps to denote the

Q Q

x where x is x or equivalently x x Bo olean expression

j i

s s

j i

JIEH HSIANG AND GUAN SHIENG HUANG

For example if s then r eps x y z

Proposition Let V be a set of variables and f be a Boolean function

represented by a truth table then

P Q Q

x x

j j

U V x U x V

j j

f r eps

s w her e f s

These are two basic Bo olean prop erties and we skip the pro ofs

Lemma Let s be a truth assignment then r eps br s br t

tpexs

Proof We rst observe that for each t pexs

Y Y

x x br t

i j

t s

j j

br s x

t s

j j

Let V b e the set of Bo olean variables such that s Then

Y Y

r eps x x

i j

s s

i j

br s x

X Y

br s x

x U

U V

X Y

br s x

t s

tpexs

j j

X Y

br s br s x

t s

tpexs

j j

Y X

br s x br s

t s

tpexs

j j

br t br s

tpexs

Thus by Prop osition and Lemma we have

Lemma Let f be a Boolean function given as a truth table then f

P P

br t br s

tpexs s w her e f s

The correctness of the f l ip tag algorithm follows from Lemma For each

of truth assignment s whose value is a copy of br s and one of each of br t

where t pexs need to b e added to the Bo olean expression of f However

since is nilp otent any two identical copies of monomials can b e eliminated By

working the main lo op from the less dened fewest s to the more dened truth

assignments we are ensured that the monomial representing less dened truth

assignments will not b e reconsidered later in the algorithm The tag function is

then used to keep track of whether a truth assignment s really has the value when

all truth assignments less dened than s have already b een considered

SOME FUNDAMENTAL PROPERTIES OF BOOLEAN RING NORMAL FORMS

Normal form of a Bo olean function with a dontcare condition

Let A b e a subset of B A partial Bo olean function f is said to b e with the

dontcare condition A if the values of f on truth assignments in A are undened

Such a partial Bo olean function can also b e regarded as a class of total Bo olean

functions with identical values for truth assignments not in the set A From this

viewp oint we say that two total Bo olean functions f and g are equivalent under

A denoted f g if f s g s for all s A It is obvious that denes an

A A

equivalent relation We use f to denote the equivalent class of f under

A A

In this section we describ e a metho d for deriving a unique normal form for

an equivalent class of This problem is interesting at least for the following

reason In circuit design one often encounters a circuit sp ecied as a truth table

with a dontcare condition say A since one may not b e concerned with some of

the output values During actually design one needs to assign values to the truth

assignments in A in order to have a working circuit The values assigned however

may dier due to dierent design techniques Thus two dierent designers may

pro duce circuits representing dierent functions although b oth are correct with

resp ect to the original design In a mathematical formulation it simply means that

the two functions designed f and g b elong to the same equivalent class of

A challenge that arises is how one may verify the correctness of such two circuits

Most of the known metho ds do not apply since they usually assume that the two

circuits under investigation represent the same function

The metho d which we are going to present here provides a solution to this

problem

Generating set of an ideal Let F x x b e a p olynomial

ring and A b e a dontcare condition Let F ff jf g It is easy to see that

A A

F is an ideal of F

The equivalence of the two relations F and F F is established by the

A A

following lemma whose pro of is trivial

Lemma f g if and only if f g F

A A

This lemma suggests a scenario for solving our problem That is if there is an

eective way for checking the memb ership of F then the equivalence of f and g

can b e easily decided

We present such a metho d based on a notion of generating set dcalled Grobner

basis Informally a Grobner basis is a set of rewrite rules which reduces all memb ers

of the same equivalent class to a unique normal form The normal form for F

under a Grobner basis is obviously

Definition Given a p olynomial ring F x x a dontcare

condition A and its asso ciated ideal F a generating set G of F is a set of

A A

p olynomials fg g g such that

for every p F there exists p olynomials p p such that p g p

A n i i

and

for every p p p g F

n i i A

Lemma The singleton set fg jg s if f s Ag is a generating set of

The pro of is obvious

JIEH HSIANG AND GUAN SHIENG HUANG

Example Let A f g Then the set fxy z xg is

a generating set of F The expression xy z x can b e generated using the iptag

algorithm given in the previous section

Grobner basis Most of the results in this section apply to any p oly

nomial ring over a eld However for simplicity we present them only in terms of

Bo olean p olynomial rings which is sucient for our purp ose

Continuing from Example since xy z x we can treat it as a rewrite

rule xy z x By imp osing a total ordering on the set of monomials we can orient

all p olynomials into rewrite rules of the form m where m is a monomial and

is the rest of the p olynomial Such an ordering can always b e obtained by rst

imp osing a total ordering on the set of Bo olean variables say x x then

compare two monomials rst by their sizes then by the memb ers of the monomials

We remark that the resulting ordering is wellfounded

A p olynomial p when treated as a rule m induces a reduction dened

as follows Given two Bo olean p olynomials p and p p p if p m m

for some monomial m and p olynomial and that p m For simplicity we

shall drop the subscript p unless confusion may o ccur Since the ordering used to

orient p olynomials into rules is wellfounded the asso ciated reduction relation

is irreexive antisymmetric and wellfounded We call such a reduction relation a

noetherian relation

Let R b e a set of Bo olean p olynomial rewrite rules and b e its asso ciated

reduction relation a Bo olean p olynomial p is said to b e in Rnormal form or R

irreducible if p is a BRNF which is not reducible using rules in R Let and

b e the trnasitive and reexivetransitive closures of then is Church

Rosser if for every p olynomials p and q such that p q there is an r such that

p r q It is wellknown that if is no etherian and ChurchRosser then

every p has a unique normal form see eg DJ

Definition A Grobner basis of an ideal F is a generating set G such

A A

that when oriented as rules the reduction relation is no etherian and Church

Rosser

Theorem Buchb erger For every ideal F of a Boolean polynomial ring

there is a Grobner basis G Furthermore a polynomial p is in F if and only if

A A

An immediate consequence of the theorem is that

Corollary Let F be an ideal and f be an equivalent class Then al l

A A

Boolean functions in f have the same G normal form

A A

In other words once we nd the Grobner basis of F we can pro duce all the

intended normal forms

Buchb erger algorithm for generating Grobner bases An algo

rithm for generating the Grobner basis of an ideal was given by Buchb erger B

As is consistent with the rest of the pap er we simplify the algorithm to work only

with Bo olean p olynomials

Other total orderings can b e obtained in similar ways For instance by comparing two

monomials using the multiset ordering DM one can order monomials in a dierent way

SOME FUNDAMENTAL PROPERTIES OF BOOLEAN RING NORMAL FORMS

Let R b e a set of rewrite rules converted from Bo olean p olynomials and let

m m and m m b e two rules in R where m then m m is an

Rcritical polynomial An Rcritical p olynomial is nontrivial if its Rnormal form

is not We use CP R to denote the set of rewrite rules obtained by orienting all

nontrivial Rcritical p olynomials derived from rules in R

A set of rewrite rules R is interreduced if every rule r R is irreducible with

resp ect to R n fr g By reducing rules in R among each other one can always

making R into an interreduced one We call the resulting interreduced set of

rules r educedR

With these denitions in mind we intro duce a version of the Buchb erger algo

rithm for generating a Grobner basis for a set of p olynomials

Input A set of p olynomials P and a total ordering on the set of variables

Output a Grobner basis G of P

Initiate R to b e the set of rules converted from the p olynomials

in P

While CP R fxx xg do

a R R CP R fxx xg

b R r educedR

G R

This version of the algorithm is clearly not the most ecient but it is sucient

for exp ository purp oses

Including the idemp otence rules xx x is essential for the completeness of

the metho d

Example Take the ideal in Example in which A f g

and fxy z xg is a generating set of F Assume that x y z then the starting

rewrite system is

xx x

y y y

z z z

xy z x

Rules and pro duces a critical p olynomial xy z xz which after reduction

b ecomes xz x It is then made into a rule

xz x

The new rule then reduces rule into

xy x

Since there is no more nontrivial critical p olynomials we obtain G I fxy

x xz xg

Normal form of a Bo olean function with a dontcare condition

Now we are ready to give a pro cedure for pro ducing the normal form of a Bo olean

function with a dontcare condition

Let f b e a Bo olean function with the dontcare condition A We may assume

that f is represented by a truth table We pro ceed as follows

We should note that we could not nd any literature on Grobner basis in which the idem

p otence rules are explicitly included in the generation pro cess although it may b e due to our

unfamiliarity with the literature

JIEH HSIANG AND GUAN SHIENG HUANG

Find a generating set of F and call it I This can b e easily done by

applying Lemma

Construct the Grobner basis of F G using the aforementioned Buch

A A

b erger algorithm and I

Cho ose a total Bo olean function g from the equivalence class f This

can b e done by assigning to all truthassignments in A and apply the

iptag algorithm

Reduce g to its Bo olean ring normal form with resp ect to G The resulting

normal form is a unique normal form for f

Example Consider the function f dened by the following truth table

x y z f

In Example we have already derived the Grobner basis for the dontcare

condition in this example G fxy x xz xg By replacing the X in the

denition of f by we obtain a function g whose BRNF by iptag algorithm

is y z xy xz y z xy z A nal stage of normalization using G yields

x y z y z which is the unique normal form of f

Bo olean ringbased prop ositional reasoning

Buchb erger algorithm for SATUNSAT The Buchb erger algorithm

describ ed ab ove can b e easily adopted to b ecome a theorem proving pro cedure for

prop ositional logic

Given a set of prop ositional formulas S f g Since logically it

means that each of the formulas is true we convert the s into Bo olean

i i

ring rewrite rules and carry out the Buchb erger algorithm If S is unsatisable

then the ideal represented by S will b e the ring itself In other words the equation

will b e pro duced by the Buchb erger algorithm with S fx x x g if and

i i i i

only if S is unsatisable

When converting formulas into rules heuristics can b e applied to pro duce

shorter rules For instance an equation of the form

can b e transformed into n equations

b efore converting into rules Similarly an equation

can b e transformed into equations

SOME FUNDAMENTAL PROPERTIES OF BOOLEAN RING NORMAL FORMS

We remark that the pro cedure we are describ ed here do es not require that the

input formulas are in clausal form

Example Given a set of clauses S fp q t s p t p s q

t q sg S can b e transformed into

pq p q

st s t

q t

q s

Rules and pro duce a critical equation pt q t t which after further

simplication b ecomes rule

Rule immediately deletes rules and and simplies rule into

Rule then simplies rule into

and rule into

As the last step of the pro cedure rules and simplies rule into

which means that S is unsatisable

Applying Bo olean ring formalism to theorem proving was rst describ ed in

H H in which a complete pro cedure for inputs in clausal form was pre

sented The main purp ose there was aimed at studying complete theorem proving

metho ds for rst order logic and the prop ositional pro cedure came as a side result

The rst theorem proving metho d which used b oth Bo olean ring and Buch

b erger algorithm was given in KN This metho d although complete for prop osi

tional logic was not complete for rst order logic It was later made into a complete

metho d in BD

DavisPutnam a la Bo olean ring We may regard the pro cedures

mentioned in the previous section as resolutionlike pro cedures in the framework

of Bo olean ring While critical p olynomial generation roughly corresp onds to res

olution these pro cedures have the advantage of p owerful simplication inferences

which have no equivalent notion in resolution However the pro cedure has its

shortcoming For instance one needs to include the idemp otence rules when gen

erating critical p olynomials

In resolution framework there are simplication inference rules such as subsumption and

clausal simplication as describ ed in eg L but they are not as natural as simplication with rewrite rules

JIEH HSIANG AND GUAN SHIENG HUANG

In this section we intro duce a DavisPutnam like pro cedure which do es not

need the generation of critical p olynomials This pro cedure has two inference rules

The rst one is reduce which reduces a set of Bo olean rules in an interreduced set

r educedR This inference rule is the same as the one in the Buchb erger algorithm

The second inference rule is split which cho oses a Bo olean variable say x and split

R into two sets R R fx g and R R fx g

The pro cedure works as follows Given an input set of rules R we rst reduce

it to r educedR If the resulting set contains contradiction then R is

inconsistent Otherwise cho ose a Bo olean variable and split r educedR accordingly

The two resulting sets of rules can b e treated recursively The input set R is

inconsistent if and only if all of the resulting sets lead to contradiction If R is

consistent then each resulting set that do es not contain contradiction will contain

a truth assignment which satises R

In the following we put the ab ove informal description into a recursive pro ce

dure We call a Bo olean variable x splittable in R if neither x nor x is a

rule in R

pro cedure DPBRR S

input a set of Bo olean rules R

output a collection of truth assignments S

R r educedR

if R

then stop

else if there is a splittable variable x in R

then R R fx g

call DPBRR S

R R fx g

call DPBRR S

else add R as a truth assignment to S

If the input set of rules is R then the pro cedure starts from DPBRR

It is obvious that this pro cedure is complete It is dierent from DavisPutnam

in several asp ects First the input set needs not b e in clausal form Thus it

may have some advantage if the input formulas cannot b e readily transformed into

clausal form Second r educe is strictly more p owerful than the unit clause rule of

DavisPutnam in terms of reduction p ower Therefore we feel that the numb er of

splits necessary in the average case should b e smaller than DavisPutnam although

we have not yet done any rigorous studies

Example Now we redo Example using the DavisPutnam approach

Since no reduction can b e done we cho ose an arbitrary variable say p to split

Then the two new sets are

SOME FUNDAMENTAL PROPERTIES OF BOOLEAN RING NORMAL FORMS

pq p q

st s t

q t

q s

and

pq p q

st s t

q t

q s

It is easy to see that in b oth cases the set pro duces after one round of

simplication

As another example we demonstrate that our metho d may use fewer applica

tions of splitting than DavisPutnam

Example Let S b e the set of clause

p q r

JIEH HSIANG AND GUAN SHIENG HUANG

It is easy to see that DavisPutnam needs two steps of splitting In our metho d the

clauses are transformed into

pq r

pq r pq

pq r q r

pq r pr

pq r pq pr p

pq r pq q r q

pq r pr q r r

pq r pq pr q r p q r

Note that the contraction through simplication alone without using any

splitting

Discussion

Bo olean ring is an alternative representation to Bo olean algebra Instead of

it uses exclusiveor Consequently there is a unique normal form for every

Bo olean function in which negation is not necessary

In this pap er we presented several pro cedures for various op erations based on

Bo olean rings We describ ed a simple metho d for deriving the Bo olean ring normal

form directly from a truth table We also describ ed a notion of normal form of

a Bo olean function with a dontcare condition and showed an algorithm based

on Grobner basis for generating such a normal form Finally we discussed two

Bo olean ring based theorem proving metho ds for prop ositional logic Although the

two metho ds to some extent resemble ground resolution and DavisPutnam they

have more simplication p ower and seem to b e quite eective

Despite its extreme simplicity the Bo olean ring representation has not b een

used extensively b oth in logical reasoning and in computation It is interesting to

investigate why it is the case

Op erationally the main dierence b etween excl usiv e or and or is that the

former is nilp otent Consequently negation do es not app ear in the normal form

This makes Bo olean ring formulas hard to read for human since one cannot tell

which predicate symb ol is negated and which is not When a formula is long it

b ecomes imp ossible to make a natural interpretation of its meaning This problem

may partially explain why logicians have not used Bo olean ring in actual reasoning

The same problem may also explain why Bo olean ring has not b een more

widely used in automated deduction Bo olean ring based rst order theorem proving

metho ds eg H have b een demonstrated to b e quite favorable when compared

with other metho ds HJ PR WS BB KZ However due to the normalization

pro cess it is not likely that one can reconstruct the generated pro of back to human

readable form once the pro of is found Thus when one is interested in deriving a

convincing pro of rather than just demonstrating that the theorem is correct then

Bo olean ring is not a go o d choice

Bo olean ring is not used more widely in circuit design for a similar reason

One of the more p opular circuit design metho dologies is programmable logic array

PLA It is conceivable that the OR gates in PLA can b e replaced by XOR

However once an input to the OR gate is true the output is decided For XOR

SOME FUNDAMENTAL PROPERTIES OF BOOLEAN RING NORMAL FORMS

on the other hand all inputs to XOR must b e evaluated b efore an output can b e

decided The reason is once again due to the nilp otence of XOR

The question then is whether the Bo olean ring representation is go o d for any

practical applications We feel that the answer is a resounding yes Basically one

can regard Bo olean ring as an ecient internal data structure that provides a uni

form representation and fast basic op erations Thus for any problem for which one

only cares ab out the inputoutput relationship but not how the computation is p er

formed Bo olean ring is a feasible candidate Satisabilty problems see GPFW

for a survey such as constraint solving is an example In addition to its eec

tiveness our metho d also allows a more exible input format Pro ofchecking is

another In pro ofchecking one often only cares ab out having a exible and ecient

pro cedure to check the correctness of simple to mo derately dicult theorems but

not what the pro ofs lo ok like A third example is circuit verication in particular

when dontcare conditions are involved since Bo olean algebra cannot provide a

satisfactory algebraic framework for eectively handle these problems

References

B B Buchb erger Ein algorithmisches Kriterium fur die Losbarkeit eines algebraischen Gle

ichungssystems Aequ Math

BB F Ba j MP Bonacina M Bruschi and A Zanzi Another term rewritingbased proof of the

nonobvious theorem Asso ciation of Automated Reasoning Newsletter

BD L Bachmair and N Dershowitz Inference rules for rewritebased rstorder theorem prov

ing Second Symp osium on Logic in Computer Science Ithaca NY

BH MP Bonacina and J Hsiang Towards a Foundation of Completion Procedures as Semide

cision Procedures Theoretical Computer Science

D N Dershowitz Orderings for termrewriting systems Theoretical Computer Science

DJ N Dershowitz and JP Jouannaud Rewrite systems Handb o ok of Theoretical Computer

Science volume B Jan van Leeuwen ed

DLL M Davis G Logeman and D Loveland A machine program for theorem proving Comm

of ACM

DM N Dershowitz and Z Manna Proving termination with multiset orderings Comm ACM

DP M Davis and H Putnam A computing procedure for quantication theory J ACM

GPFW J Gu PW Purdom J Franco and BW Wah Algorithms for Satisability SAT Prob

lem A Survey DIMACS Volume Series on Discrete Mathematics and Theoretical Computer

Science The Satisability SAT Problem American Mathematical So ciety

H J Hsiang Topics in automated theorem proving and program generation PhD Thesis

Univ of Illinois Urbana IL

H J Hsiang Refutational theorem proving using termrewriting systems Articial Intelligence

HJ J Hsiang and NA Josephson TeRSe A Term Rewriting Theorem prover The Rewrite Rule

Lab oratory Workshop General Electric Research and Development Center Schenectady NY

KN D Kapur and P Narendran An equational approach to theorem proving in rstorder

predicate calculus Ninth International Joint Conference on Articial Intelligence LA

KZ D Kapur and H Zhang Nonobviousness last time Asso ciation of Automated Reasoning

Newsletter

L DW Loveland Automated theorem proving a logical basis North Holland New York

PR FJ Pelletier and Piotr Rudnicki Nonobviousness Asso ciation of Automated Reasoning

Newsletter

S M Stone The theory of representation for Boolean algebra Trans Amer Math So c

JIEH HSIANG AND GUAN SHIENG HUANG

WS R Wilkerson and B Smith Nonobviousness again Asso ciation of Automated Reasoning

Newsletter

Z II Zhegalkin On a technique of evaluation of propositions in symbolic logic Mat Sb

Department of Computer Science and Information Engineering National Taiwan

University Taipei TAIWAN

Email address hsiangcsientuedutw