SOME INTERPOLATION THEOREMS FOR
FIRST-ORDER FORMULAS I N WHICH
ALL DISIUNCTIONS ARE BINARY (
I
M. R. KROM
)
The atomic formulas of a first-order predicate calculus are the
predicate letters with attached individual symbols, and the signed
atomic formulas are the atomic formulas and the negations of
atomic formulas. A first-order formula is in prenex conjunctive
normal form (cf. page 1 of [ 1
quantifiers
(called a prefix) followed by a quantifier free part
]
(called
a
matrix)
) i n
c a swhich
e is a conjunction of disjunctions of signed
atomic
i
t formulas. We establish properties of the formulas of any
first-order
without identity and without funcc o n predicate
s i s tcalculus
s
tion
symbols
which
are
in
prenex
conjunctive normal form and
o
f
in which the disjunctions are binary (have just two terms). The
a
condition that the disjunctions be binary is not so limiting that
sit restricts
t
r toi triv ial
n classes of formulas; in fact there is an
us
g
apparently s t ill unsolved decision problem f o r these classes
o
of formulas( f c f. the discussion following Theorem 3 o f [51).
However, as we w ill show, there are no results analogous to
ours for classes o f formulas defined by restricting to ternary
disjunctions.
Most of the properties that we establish are expressed in the
form of interpolation theorems. We begin with one such theorem
for formulas in a statement calculus whic h completely characterizes when a formula is logically equivalent to a conjunction
of binary disjunctions of signed statement letters. We have three
interpolation theorems for formulas in first-order languages and
one of them is a modification of Craig's Theorem (Theorem 5
page 267 of l21) which yields a modified form of Beth's Theorem
on definability (Theorem 5.2.1 page 118 of [6]).
(') Th e research report ed here was supported b y Nat ional Science
Foundation Grant number GP-6358.
404
M
.
R. KRO M
For any two formulas I ' , A o f a statement calculus or of a
first-order predicate calculus we use 1'1— A to indicate that A is a
logical consequence of W e use V , A , and T . respectively,
for the logical connectives, disjunction, conjunction, and negation.
§ 1. Binary Disjunctions in Statement Calculus
We will use u, 13, a
letters
t and negated statement letters o f a statement calculus.
For, any
a u we let u' be the negation of u if u is a statement letter
and2the statement letter occurring in u if u is a negated statement
letter.
, A binary disjunction is an expression of the form W O .
For( any
3 a,2 13 we . let aVI3 be either one o f the two logically
equivalent expressions uV13 and 13Va. A chain from u to (3 is
. . t
a set of the form { a
o
l =
Vaa and
- 2 , T h e following lemma is a restatement o f
d e 2.2n of o[4].
Corollary
a ' 2 V a 3 ,
t e
c e :
a r 1.
b i any set S of binary disjunctions, S is incon1 Lemma
V a 4 For
,
t
r
a
r if there is a statement letter a and there are
sistent if and only
V
a
twoy chains formed with members of S such that one chain is
nfrom a to u and one chain is from a' to a'.
s
t
a
} We w ill briefly indicate a proof of this lemma and refer the
t
e
m
w
i
reader
e ton [4] for
t complete details. Any set of binary disjunctions
tforming ahchain from u to 13 obviously has the disjunction JaV13
as a logical consequence. Thus any set S fr om whic h chains
can be formed, one from u to a and one from a' to a', has both
u Vu and u'Va' as logical consequences and so it is inconsistent.
Conversely, if S is a set of binary disjunctions for which there
are no two chains as indicated in the lemma then we can show
by induction that S can be extended to a set S' of binary disjunctions such that for any predicate letter u occurring in members of S' there is, with members of S', a chain from a to a or a
chain from u' to a' but not both. This determines an assignment
of truth values which shows that S is consistent.
We say that a binary disjunction is a singleton in case its two
disjuncts are identical.
I NTERPO LATI O N THEOREMS FO R FI RS T-O RDE R FO RMULAS 4 0 5
Lemma 2. For any inconsistent set S of binary disjunctions,
the two chains required by Lemma I may be formed so that
neither one of them has more than one singleton occurring in it.
Proof. Let c be a chain from a to a in whic h at least two
singletons occur. We may assume that a V u is not a singleton
that occurs in c otherwise we could replace c with the chain
{a Va}. Let ( 3
is,
1 (3
chain
1
V 1 3 c' of c from a to ti'
withh (31
ttogether
i nb ew
i W
c h
V
11
occurs,
and c' together with c* forms a chain from a
I3
f
o
r
m
s
nsingleton
o
a n
to
a
in
whic
h
no
a
s1
e in nd gm l e tmore than one singleton occurs. Hence c
be replaced
by a chain in which no more than one singleton
ci nsh
ocan
ta i n
occurs.
* cg u
ocs i c n
r
r
o
sfa
.
l Theorem
e t oI . I f F is a conjunction of binary disjunctions o f
m
T
h
sn
signed statement letters and A is a disjunction of signed stateea
n
io
c
c
ment letters such that I ' 1- A then there is a subdisjunction
ctn
u
r
r
of A o f no more than two terms such that I ' 1- A and A i - A .
'o
g
i
n
g
[li Proof.
3 Let I' and A be formulas as described in the hypothesis
of the lemma. Let S
1
e
n
in
F e
and let
I b
t hS2ebe the set of singletons of the form (3' V(3' for
itc
each
(3 o f A. Since S2 is logically equivalent t o the
s edisjunct
t
n
o
,onegation
of
A,
S
f
w
h
n
tbstatement
letter,
i Ui nS 2a rsay a, such that there are two chains, one from
io
c one from a' to a', formed with the elements of S
h
i tosa and
ya
h
caBy
Lemma 2, no more than two elements of S, need occur in
di nUi c so nj su
o
ctthese
et ni t Iot. follows that there is a subset S.
ni s ct chains.
n
umore
2
o
f
Stwo
2 elements
w i such
t hthat S
than
nB s
e
rbe
oS
a subdisjunction
i ofs A of no more than two disjuncts, a disjunct
y U
tin
h 9
ri(L3 for
n
c
o
n
s
i
s t(VeV(3'
n tof. each
e t singleton
a
iS
ec with t m
m
consistent
oL
nA.
N
-a A a un d
c1
g2
A There is no result corresponding to Theorem 1 which applies to
r1
iternary
., T - hdisjunctions
e n
I instead
'
of binary disjunctions. In fact, for any
1
npositive
tA
1
- hinteger- n there is a conjunction F of ternary disjunctions
cof
letters and a disjunction A of signed statement
e signed
A
b
e cr statement
a
sb
e
u se ec a u s e
uA
i
F
cis
s
hsia
n
ta
hs u b d
ai s j u
406
M
.
R. K RO M
ietters with n terms such that 1' 1--A and such that no proper
subdisjunction of A is a consequence of F. For such an example
let I ' = (1
...A
f l Q V P , , 2VQ ,, 2) A (
3
(PI
V PP2 V V P H ) .
-1 V
I2Q 2
1.VForPany „formula
I ' of a statement calculus the
V Corollary
)
0P „
following
a1
n two dconditions are equivalent:
A
)
I ' is logically equivalent t o a conjunction o f binary dis=(1)
A
junctions of signed statement letters.
F l
o
t2) Fa o r every disjunction A o f signed statement letters such
l
that I ' A,ther e exists a subdisjunction o f A of no more
V Pthan two terms such that
3 V
Q 2Proof. By Theorem I i t is sufficient to show that (2) im) / plies ( I) .
\ Let I ' be a formula satisfying (2) and let Q be a formula in
tconjunctive
7
normal for m that is equivalent t o F . Then each
4
conjunct
o f Q is a consequence o f F and by ( 2) there is a
:lormula
1
Q obtained from Q by deleting all but two disjuncts
2
from each conjunct of Q, and such that F i .
V
- . P
construction,
so I ' is logically equivalent to
4
T hen
6 1 — Q
V
b Q y
3
§ 2. Binary Disjunctions in First-order Predicate Calculus.
)
A Formulas referred to in this section w i l l be assumed to be
formulas of first-order languages without identity and without
function symbols. We w ill briefly describe the proof theory of
Linear Reasoning which we w ill use (vid. [2]). For any formulas
E and C2, an L-deduction of CHrom E is an ordered (n+1)-tuple
El, —,E„ ) where E„ = E and E
specification
o f how, for m < n , E
r
application
of
an
The reader is referred to pages 252 and
11 , =
Q L-rule.
,
253
of
[2]
for
the
definitions
,,
t o g e t h e r of the eleven L-rules. An L-deduc1 r w
e s ui l t t s h
f ra o m
E
l l
,
b
y
a
n
I NTERPO LATI O N THEOREMS FO R FI RS T-O RDE R FO RMULAS 4 0 7
tion is symmetric if and only if the order in which the different
kinds of L-rules are applied satisfies conditions (iii) through (vi)
on page 257 o f 121. I n addition, for convenience, we require
that exactly one application o f the L-rule matrix change occur
in any symmetric L-deduction. Theorem 2 o f 121 says that for
any prenex formulas F and A such that l ' t h e r e is a symmetric L-deduction of A from
We first show to what extent Theorem I above generalizes to
first-order languages.
Theorem 2. Let F and A be prenex formulas such that A .
Assume that the matrix of F is a conjunction of binary disjunctions of signed atomic formulas, that the matrix of A is a conjunction o f arbitrary disjunctions o f signed atomic formulas,
and that the prefix of A contains no existential quantifiers. Then
there is a formula o b t a in e d from A by deleting all but at most
two disjuncts from each conjunct of the matrix of A and such
that F i
— Proof.
s
Let I ' and A be formulas satisfying the hypotheses of
a ntheorem and let A* be a formula obtained from A by replacing
the
d universally quantified variables w i t h dis tinc t indiv idual
the
A
constant
symbols that do not occur in l ' or A and by deleting
. universal quantifiers. Then A * , so there exists a symthe
metric L-deduction D o f A* fr om B y properties o f L-rules
that may be applied after the application of matrix change in
a symmetric L-deduction and since A " is quantifier free, we
may assume that A* is the matrix of the formula resulting from
the application of matrix change in D . Let b e the formula of
D to which matrix change is applied. By properties of the L-rules
that may be applied before the application of matrix change in
a symmetric L-deduction, it follows that the matrix o f E is a
conjunction o f binary disjunctions o f signed atomic formulas.
By the definition o f the L-rule matrix change, A* is a logical
consequence of the matrix of a n d by Theorem I above there
is a formula 2
-two disjuncts from each conjunct of A* such that 7\* is a logical
ʻ
consequence
o f the matrix o f E. Thus E and E z
* ',
V
s o
o b t a i n
e d
f r
o
m
A
408
M
.
R. K RO M
1)-- A*. Let A be the formula obtained fr om A* by replacing
individual constant symbols, that were introduced to obtain A*
as indicated above, with their corresponding variables and by
attaching the prefix of A. Since A * and the constant symbols
replaced in N* to form A do not occur in F, it follows that A .
Also Ai--A by its construction, so A has the properties required
in the theorem.
We give an example which shows that the requirement that
the prefix o f A contains no existential quantifiers is essential
in the above theorem. Let 1' = Hagb((Fa. VGb) A (Ga L b ) A
(Ma V Hb) A (NaVMb)) and A = Ha((FaVGaVHa) A (La V
Ma VNa)). Then I ' A but for any formula by
disjunct
o f the matrix o f
A deleting
o b t a one
i n e
d
ffr om
r each
o conjunct
m
A,
not
(
I'
1
—
A).
A
The requirement that the prefix of A contains no existential
quantifiers can be deleted from this theorem, however, i f one
adds the requirement that the conjunction forming the matrix
of A has exactly one term. This result can also be established
with Linear Reasoning and Theorem 1 above using the fact that
the formula resulting from matrix change in any symmetric Ldeduction of such a formula A would have a matrix which is a
disjunction of signed atomic formulas.
Theorem 3. Let F and A be prenex formulas such that A .
Assume that the matrix of F is a conjunction of binary disjunctions of signed atomic formulas and that F and A have a least
one predicate letter i n common. Then there exists a prenex
formula E such that the matrix of E is a conjunction of binary
disjunctions, such that all predicate letters in E also occur both
in F and in A, and such that I' E and A .
Proof. Observe that the theorem is a statement of a well known
interpolation theorem (c f. Theorem 5 o f [2]) w it h an added
feature corresponding to an assumption that disjunctions are
binary in F. Let F and A be as described in the hypotheses of
this theorem. Then by the known interpolation theorem cited,
there is a formula Q such that all predicate letters occurring in
Q also occur both in F and in A and such that FI—Q and Q A .
I NTERPO LATI O N THEOREMS FO R FI RS T-O RDE R FO RMULAS 4 0 9
Thus there is a symmetric L-deduction o f Q from L e t 0
and (I), respectively, be the formulas just before and just after
matrix change in B y properties of L-rules which occur after
matrix change in a symmetric L-deduction, the predicate letters
that occur in 4:1) are exactly those that occur in Q. Let it be a
formula obtained from (I) by writing its matrix in conjunctive
normal form. The matrix of It is a logical consequence of the
matrix of 0 (which is also a conjunction of binary disjunctions
of signed atomic formulas) and we may apply Theorem 1 above
to the matrix of 0 together with each conjunct o f the matrix
of a. Thus there is a formula o b t a in e d from it by deleting all
but at most two disjuncts from each conjunct of the matrix of it
and such that 01—. Since, by construction, 1,1— a, i t follows
that i s a formula as required in the theorem.
We observe that Theorem 3 can be used to establish a correspondingly modified version o f a theorem known as Beth's
Theorem on Definability. I n particular, in Theorem 5.2.1 page
118 of [6], if we add an assumption that the sentence defining
the relation implic itly is in prenex conjunctive form in whic h
all disjunctions are binary then we may conclude also that the
sentence defining the relation explicitly may be assumed to be
in prenex conjunctive normal for m in whic h a ll disjunctions
are binary.
Theorem 3 has a consequence concerning the existence o f
reduction classes for satisfiability of formulas in pure first-order
languages. Let L
ilet a91nbed a set of formulas of L
cl
Li ass2f o r satisfiability o f formulas o f L., i n case there is an
effective
e n
a fort finding, f o r any formula X o f L2, an
b. T eh procedure
associated
formula
0(X) in al such that X is satisfiable (has a
sn
ai
y
model)
i
f
and
only
tc a w l l e d i f 0(X) is satisfiable (cf. page 32 o f [71).
Now
suppose that all predicate letters that occur i n L2 also
oa
occur in L
fr i e r ds u c t i
the
I similarity type o f L
o -n
tobtained
from b y deleting all predicates for which there is
I.
o
r d e
no
corresponding
predicate letter in L2. Also, if at is an L
,T h
t he e L
rof
a
structure
an
of the similarity type of L
2
n
lT1
a
n
I-f rr e
isoedan
u
d cuLt c t
u
,g
t
h
ef na
w
e
Io
r
g
e
s
t
h
a
t
-sa an ny
a
iy x p a ns
e
n
st i ot n r h
d
e
u
o cf
st u t1 r r
u
9
ce
t
u
r
.
410
M
.
R. K RO M
auction class for satisfiability of formulas of L
X of L., and any model 9:11
2reduction
i n
cclass
a such
s e that 5for any
I
of
v(X)
the
L
i
s
a
2 of X, 91 has an L
91
observe
i- r e d uthat
c t the Skolem normal form for satisfiability (cf. Satz
\/
I
I
page
-o e xf p a n49s of
i o dn e t e r m i n e s a strong reduction class. I f L2
iw
s
any
pure
first-order
language and L
9 3
h i1 c h
many new predicate letters of all ranks and
l i ss s infinitely
a n
iincluding
is the
class
o f i the
formulas o f L
e
x
t
e
n
s
o prenex
n
a
normal
form
prefix,
then
91
is
a strong reduction class for satisI
w
i
t
h
t
h
e
o
m
oo f
d
tiability
of
formulas
o
f
L
S k 2o l e m
dL
e
el
2
l Theorem 4. For any pure first-order languages L
o
.
fothatI La n d
L 2
s u c h
f2
Q
extension
of L2
matrices
,(X
h ta h eXare conjunctions of binary disjunctions of signed atomic
c)aformulas
s l a . s issnot a strong reduction class for satisfiability of the
nformulas
W
o
f of L2.
a
d Proof.
te
h Let L
I
fL
e
l e a n d
conjunctive
oIa L r 2 e formulas of L
p
strong
in
h i c hclass for L
ris
e ewx reduction
s aib n
2
a
l r the associated formula of 9
st po(X)
falet
ol ube
teduction,
3.
B
y
nt i Theorem
t i o n 3 above there is a formula E
.d
L
e
t
s e j 9(XI1—
n f X.
ci By
na
m
l ud eX
t ri u
of
9
b
a
ia
of sein f rs
n
yo
h
letters
occurring
inn lE also
3
s
t
r
o
fa
o
r
m
u
ag occur in X. But any model 91 of X
e
t m
o
rhass some
L
seo
o r df91' is a model of E and thus also the L.,-reduct 91 of
foxe
Iuv(X)I—E,
2n
b
a
te eis ari model
dL
L
of E. Thus Xi—E and we conclude that for any
-a
cr'A'
n
d
y
e
l L.)athere is a logically equivalent formula which is in prenex
IX
p xof
pans
he
n
lconjunctive
w
ri onn g normal f o r m and i n w hic h a l l disjunetions ar e
ts
a we w ill show that this is not true for first-order
hbinary.
e u 1 But
h9
id g
e L2 w ith at least three predicate letters. Suppose L2
olanguages
a'o
language in which three predicate letters u, (3, and
sis
i as first-order
h
tw
y
occur.
s
Let
u,
(3, and y, respectively, be the formulas obtained
en
ci
c
Q
o
from
a u these predicate letters by introducing distinct indiv idual
(h
ft c
variables
into each argument place and then universally quani
X
L
tifying
e h the variables. Let X be °N O Vy. Suppose that E is a
)s2
l t
ia
a
h
e
m
—
n
to a
E
d
t t
,d
se
e
E
u
r
1lp
so
—
I NTERPO LATI O N THEOREMS FO R FI RS T-O RDE R FO RMULAS 4 1 1
and
p such that all predicate letters that occur in I also occur
in
r X. By considering various one element structures as possible
models,
we can see that I can not be logically equivalent to X.
e
In
particular,
either i s satisfied by all one element structures
n
or
there
is
a
conjunct
of I, in which the two disjuncts are not
e
a negated and an unnegated instance o f the same predicate
x
letter. But if there is such a conjunct then the weaker formula
c
I obtained by deleting all but that conjunct from the matrix of
o
I is not a logical consequence of X.
n
Notice that there are instances of Theorem 4 in which L2 is a
jstatement calculus, i.e. when L2 is a language with jus t three
upredicate letters each one being o f rank zero. T he statement
n
obtained from Theorem 4 by replacing "three predicate letters"
cwith "one binary predicate letter" is also true. That follows
t
from
the first part of the above proof and Theorem 1 o f [31.
i Finally we observe that statements corresponding to Theorems
3
v and 4 above with "binary disjunctions" replaced by "ternary
disjunctions"
are false. In particular there is a simple procedure
e
for
strongly
reducing
first-order formulas to formulas in prenex
f
conjunctive
normal
form
in which disjunctions are ternary. For
o
an
r illustration consider the replacement of P
(P
im P
I2distinct statement letters. For further details o f how this reu
duction can be carried out for arbitrary first-order formulas see
V
lthePproof of Theorem 1 in [51.
P
3
a
2
V
t
iUniversityPof California, Davis
M
.
R. KROM
V
w
i
t
h
n
Q
w
)
h
A
BIBLIOGRAPHY
i
(
c
T C. C. CHANG and H. Jerome KEISLER, A n I mprov ed Prenex Normal
h
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V
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l][
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io
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o
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oir
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.
R. K RO M
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