A formal derivation of the decidability of the theory SA

Theoretical Elsevier

Computer

1

Science 127 (1994) 1-23

Fundamental

Study

A formal derivation of the decidability of the theory SA C. Hosono Institute of hformation

Sciences,

University qf Tsukuha,

Tsukuba

Ciry, Ibaraki,

Japan

Y. Ikeda Insritute of hformarion

Processing,

Saitama

Junior

College,

Saitama.

Japan

Communicated by M. Nivat Received March 1992 Revised October 1992

Abstract

Hosono, C. and Y. Ikeda, A formal Computer Science 127 (1994) l-23.

derivation

of the decidability

of the theory

SA, Theoretical

A formal definition of a first-order theory SA, which is an extension of Presburger arithmetic to rational numbers, is introduced and syntactic proof of the decidability of SA is given. This proof has already been outlined by Smorynski, but this work is independent of his work. We give a whole syntactic proof.

Contents 1. Introduction. 2. Theory SA 2.1. Language. 2.2. Axiom 3. Elementary theorems 4. 21 formula. 5. Decidability theorem. 6. Conclusion Acknowledgment. References . Correspondence

Ibaraki,

to: C. Hosono,

305, Japan.

..........................

Institute of Information Email: hosono@,coins.is.tsukuba.ac.jp.

0304-3975/94/$07.00 0 1994-Elsevier SSDI 0304-3975(93)EOOll-R

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.......................... .......................... .......................... .......................... .......................... .......................... .......................... .......................... Science, University

Science B.V. All rights reserved

of Tsukuba,

Tsukuba

City,

2

C. Hosono,

Y. Ikeda

1. Introduction

Presburger with

arithmetic

addition

introduced

is the fragment

It does

of elementary

involve

multiplication,

number but

theory

which deals

an expression

nx is

of x + ... + x. It is well known that every closed formula W is decidable, namely, there is an algorithm which decides the

arithmetic

of a given formula

A computation subclasses

not

as an abbreviation

of Presburger truth

only.

[6, 11.

time

of formulae

or space

of decision

[8] are studied

In this paper, first-order

theory

algorithm

[7, lo]

and

that

of each

after the above investigation.

SA,

which is one of the extensions of Presburger arithmetic to rational numbers described. The function l/n represents division by n and predicate Z represents integer. The idea of this theory comes from Skolem’s arithmetic [9] (L?;Z, <,+,{q.:qE2},L.J

is an

3031).

Skolem used his theory to show the decidability of Presburger arithmetic. However, since any closed formula of the theory SA is decidable, any closed formula of Skolem’s arithmetic is decidable. The basic idea of the proof of the decidability The notation

theorem

of SA is the following.

LlA is introduced, where u is a variable, s and t are terms which do not include U, and A is a formula. A formula Zi:uA represents that there is only one integer u such that S
Decidability

announced in [lo] (in the form of an exercise). Smorynski’s, and we give a syntactic proof. This new theory SA is established

are described.

The definition

Section 4. The decidability Section

theorem

But our work

is independent

as the core part of NU interpreter

an interpreter of v-definable acts [3]. The definition of theory SA is given in Section theorems

3

of the theory SA

2. In Section

[4,5] which is

3, the elementary

and the basic feature of symbol is proved in Section

of

5. Conclusions

u are shown in are given in

6.

2. Theory SA In this section, we show the language and the axioms of the first-order extension of Presburger arithmetic to rational numbers.

theory SA, an

2.1. Language 2.1.1. Notation Constant symbols: Function symbols:

0, 1. binary function +. unary function l/n for any natural Predicate symbols: binary predicates =, < . unary predicate 2. Semantically, l/n is the function which represents division which means “be integer”. 2.1.2. Abbreviation We introduce the following

abbreviations:

1 -x=x n n’

x+x+ .. . + x = nx, Y

n

1 -x+ n

1 1 ...+nx=m-x=~x, n

l+ ..’ +l=n Y

n

and Il, n

. . ++r+;.

n

number

n except 0.

by n, and Z is the predicate

C. Hosono, Y. Ikeda

4

From

these abbreviations,

it follows that

mx+nx=(m+n)x and l+n

1

-x+&-x, m m

m

where (m + n), 1+ n are usual additions of integers. From above, a term which can be written in SA is intuitively whose coefficients

a linear

polynomial

are constants.

And we abbreviate

x < y v .Y= y to

xdy

and abbreviate

x
A y
to

x
2.2. Axiom Axioms of the theory l l

SA are as follows:

Equality axioms. Axioms of comparison: asymmetry

law

.x-CL’ 3 transitive

(01)

1(y
law

X
A

y
3

(02)

x
total order v x=y

x
Axioms

of addition

associative

v y
(abelian

group):

law

x+(y+z)=(x+y)+z, commutative

(AlI

law

x+y=y+x, unit element

(03)

(A-9

0

x+0=x,

(A3)

Decidability

existence

5

of the theory SA

of inverse

vx3y.x + y = 0.

The additive

(A4)

inverse provided

group, we can denote

by (A4) is unique

this inverse

by -x.

and, for any element

Subtraction

is definable

x of an abelian by

x-y=x+(-y). From

this, we define

and 0x=%=0, m

l

Axiom of 0 and 1: O
l

Axiom of comparison x
l

Axioms

and addition

(preservation

of order):

PO)

IJ x+z
of l/n: 1

n-x=x. n

PA)

(There are infinite l Axioms of Z:

axioms.)

subgroup Z(x) existence

A Z(Y)

3

Z(x-y).

(Zl)

y
(Z2)

of integers

V’x3YMY) A Vx(O
3

lZ(X)).

between

(23)

0 and 1.)

3. Elementary theorems In this section, we show some elementary the decidability theorem. The following theorems are evident.

theorems

which are necessary

to prove

C. Hosono,

6

Theorem (4) x
3.1. (1) x=y z) lx=y

=

x-y=O,

and (5) lx
3.2. Suppose O
Y. Ikeda

(2) x
=

X-y
(3) x
=

O<-X,

= ydx.

Then (1) nx=O

E x=0,

(2) O
3

O
and (3)

= x
Proof. If x=0 Assume

then nx=O. that x < 0. Then 2x < x, 3x <2x, . . . are provable

by transitive

from axiom (OP). Hence,

law

nx
from 0

those

preceding,

law).

Proof. 1 1 n-(x+y)=x+y=n-x+n-y=n n n

1 n

1 1 ;x+;y (

Then by n(k(x+y)-(ix+iy))=O and Theorems

3.2 and 3.1. we have the theorem.

Corollary 3.4 (Corollary

of distributive

(1)

;(x+y)=;x+fy,

(2)

;nrx=nl;x,

(3)

-~x=~(-x).

law).

. >

q

x
v

x =0

v

0 < x,

Decidability of the theory SA

7

Proof. We show only a proof for (1).

of order of m/n).

Theorem 3.5 (Preservation x < y 3 m x
for any positive

integer

m.

x
= Z(x-y)
= ;x-:y
Proof. = x-y<0

E Tx
0

Theorem 3.6 (Reduction). nd -x2x. md

m

Proof. m

md

Corollary 3.7. Cx+C’x=(C+C’)x, where

(C + C’) is a usual addition.

For two given rational numbers then it follows that C + C’ = d.

C and C’, suppose

Theorem 3.8 (Order

Suppose

o
Proof. From

3

preservation).

that 0
“.<“I.. m m’

O
n’

n

mn’

m

m

mm

7x--x=~x-mnx=

that d is the addition

it holds that m’n cmn’. ’

md

-

Then

m’n X.

mm’

mm’

Now since 0 < (mn’ - m’n)/mm’, from 0
mm’

Hence, we have the conclusion.

0

3.5, we have

of them,

C.

8

Hosono.Y.Ikeda

Theorem 3.9.

-( 1 np -x m 4

np

=-xx.

mq

Proof. mq(t(Fx)-zx)=qn(Sx)-npx =npx-npx=O.

Hence, we can derive the conclusion.

q

Theorem 3.10.

Z(O). Proof. From From

axiom (Z2), some integer

axiom

x exists. From

(Zl), it follows that an additive

inverse

(Zl), Z(x -x). of integer

0

is an integer.

Theorem 3.11. Z(x) 3 Z( - x). Theorem 3.12. Z(x)=Z(x+

1).

Proof. To prove this, from axiom (Zl) and Theorem 3.11, it suffices to show that Z( 1) holds. We show this by reduction to absurdity. We assume lZ(1). From (23) and lZ(l), it follows that O
IJ

lZ(X).

(1)

From axiom (22) 3y(Z(y) A y< 2
holds

for every integer n.

Decidability

9

of the theory SA

on n.

Proof. Use an induction

In case of 0 < n, it follows that Z(O)=Z(O+

l)=Z(O+

1+ l)= ... =Z(O+

1+ 1+ ... + 1). w

In case of n ~0,

we

can prove immediately

from Z( -n)

and Theorem

3.11.

0

Theorem 3.14. Z(x) A Z(y) Proof.

= Z(x -tY).

- (- y) = y holds. See also Theorem

3.11 and axiom (Zl).

0

Corollary 3.15. Z(x) = z(nx), Z(x) A Z(x+ Y) = Z(Y), 1

(>

lZ(X)IlZ

--x

n

Proof. The first and the second formula contraposition of the first formula. 0

are evident,

and the third formula

is the

Corollary 3.16. Z(X)A From

1Z(x+y)

=

axiom (Z2), the following

lZ(Y).

theorem

holds.

Theorem 3.17. Vx3y(Z(y)

A x
1).

Theorem 3.18. Z(y) A y
1 1

lZ(X).

Proof. We show this by the reduction to absurdity. Suppose that Z(y) A y
follows from axiom (22).

10

C. Hosono,

Y. Ikeda

Theorem 3.19 (Cofinality).

(Cf)

V’x3Y(Z(Y) * X
Z(x)33w,y(Z(w)

A

x=ny+w

A

Odw
Proof. By axiom (22) for any integer x, some integer y exists such that y<(l/n)x< y + 1. Suppose that y, is such an integer. It follows that ny,
Proof. (+) Z(x) E Z(x + n) follows from Theorem By Lemma 3.20, Z(x)33w,y(Z(w)r\Z(y)

And from Theorem Z(w)

A

A

x=ny+w

3.12.

A

Odw
3.18, O
E

w=o

v

..‘V

\v=n-1.

Hence, AZ(y)

3w,y(Z(w) -3y(Z(y) v ..‘V =3y

A x=ny+w

A x=ny+O)

3y(Z(y)

v

3y(Z(y)

A

x=ny+l)

r\x=ny+n-1) v

Z(y)A;x=y

3y

1

( v ... v

A Odw
Z(Y)Ai(X-l)=y (

1

3y Z(y) A ;(x-(n-l))=y (

)

Ez(;x)”z(;(x-*))”...v Z(;(x-(n-l)))

If Z((x- i)/n) then by Corollary follows that Z(x). 0 (6)

We have the next lemma,

adopting

3.15, Z(n(x-

i)/n) holds.

Since Z(x-

the mean value 3(x + y) of x and y.

i), it

Decidability

of the theory SA

Lemma 3.22. x
3

3r(x
3

3u(x
Theorem 3.23. x
A iZ(mIr+n,)

then let d= 1; otherwise,

Proof. If 1
Let r. be an object for r where Z(mlro+nl)

r\...r\7Z(mlr+nl)). let d=y-i(x+y). holds. From

also satisfy Z(m,r +nI). And there is nothing else in allofro,ro+l/m,,ro+2/m,,... the interval [ro, r. + 1) except these. Thus from d < 1, there are at most ml objects which satisfy Z(m,r+nI) in the interval [$(x+y), f(x+y)+d)]. Similarly, in this interval, there are at most m2, . . . , ml objects which satisfy Z(m2r + n2), . . , Z(m[r + nl), respectively. Then there are at most m, +m2 + ... +ml objects which satisfy Z(m,r+ nI) v ... v Z(mlr+nl) in this interval. From this, at least one object of 1 m

1

d+;(x+y),

+...+fq+l

ml+ . . . . m +..:,‘mm; 1

(there are m, + ... + ml + 1 objects)

1 d+;(x+y) 1

satisfies

iZ(m,r+n,)~~~~~iZ(m~r+n~)). Therefore,

we have the conclusion.

0

Theorem 3.24. Let m, n be relatively prime integers. 1z

Ifm# 1 then

0n m

Proof. We can let n=mq+r

(ldr-cm).

(Because these are relatively

prime, r #O holds.) Then

and

From

this and Theorem

3.18,~

Z(n/m) holds.

0

C. Hosono,

Note 1: From

the above discussions

Y. Ikeda

we have

(1) If “a” is a constant or a term which includes the least integer that is greater than “a”. (2) We can decide the truth

no variable,

of any closed primary

then we can calculate

formula.

4. U formula Let us introduce Definition

the following

abbreviation.

4.1. u) s
A

t-l
where s and t are terms which do not include

AA),

u, and A is a formula

in which u may

appear. A formula Z$UA means A between terms s and t.

intuitively

that there is only one integer

which

satisfies

Definition 4.2 (Pure u formula). (1) T or -L are pure u formulae. (2) If A is either T or a conjunction of pure v formulae, which is neither T nor I, then ~Lu.4 is a pure o formula, where s and t are terms which do not include U. Definition 4.3 (Extended v clause, extended v jbrmula). (1) A conjunction of pure v formulae, formulae of the forms of Z(t) or 1 Z(t), and inequalities is called extended v clause. (2) A disjunction of extended u clauses is called an extended v formula. Next proposition

is evident.

Proposition 4.4. Let A[x] be an extended v formula, and t be a term. Then A [t] is also an extended v formula. Evidently, the number of v notations remains unchanged by this substitution. In the following, we assume that each variable And we deal with only a formula which consists constructors.

appears at most once in every term. of only 3, A, v ,l and v as logical

Proposition 4.5. A pure v formula which does not include free variables whether it is equivalent to T or 1.

is decidable

of the theory SA

Decidability

13

Proof. Use induction on construction of a pure u formula. Suppose that I$ uA does not include free variables. For s, t are terms which consist of constants, it is possible to calculate the least integer that is greater than s, by Note 1. Let uO be such an integer. Thus, s < uO ds + 1 holds. By Note 1, we can decide the truth ofbothformulae,u,+l
1 duo<&

A s < x < t) holds. So in either case, it is

the formula

U:UA [u] is equivalent

to A[u,].

of pure v formulae which do not include free variables. the truth of this formula is decidable. 0

We get the below lemma Lemma 4.6. An extended

as a corollary

v formula

which

By the

to this proposition. includes

no free

variable

is decidable.

Proposition 4.7. Zf t - s < 1, then 3u(z(u)

s
A

A A&A

S

holds.

Proof. From t-s
0

Proposition 4.8. Let c be a constant and let t -s=c,

1, it is possible

to decide

the truth of Z(c))

then

3u(Z(u) = -

(by Note

A s
AA)

(Vi:0 ’ l~“ff+‘uA}vZ(s)AVf~~A[s+i] s+l i {VP,’

where Lx J represents

’ v::i+ l uA} v Uf+LcJ~A v Z(s) A V\f{ A[s+i] the greatest

integer

s
VU=S+~VS+~
otherwise

that is less than or equal to x.

Proof. The case of Z(c):

vu=s+l

ifZ(c)

vs+c-l
C. Hosono,Y.Ikeda

14

By Proposition

4.7,

3u(Z(u)

s
A

E3U(Z(U) V”‘V

A

A[u])

A

s
A A[u])

v 3u(Z(u) A u=s+

1 A A[u])

A A[U])

h(z(U)AU=S+c-t

Vh(Z(U)AS+C-l
s+i+l

ix

1

iy

c-l =

c-

s+i+1

3U(zb)A

c-1

iy

iy

(z(S+i)”

Vz(S)A

;~s:;lUAIU]~

In case of 12(c),

u=s+i”A[ul)

A[s+il)

{$‘l[S+i]}.

it follows the similar

proof.

17

Proposition 4.9. Ler s, t be terms which do not include equivalent to 1
u. 3u(Z(u)

A

s
t) is

Proof. From the linear order, 1~ t -s v t -s d 1 follows. In case of 1 < t -s, because s
4.7, 3u(Z(u)

A

s
= ; T.

In the sequel, we may describe

q

s

ax+e

as s(x), because

we may think it is a function

of x. Proposition 4.10. Let si (0~ i
A ...A z(S,)

AlZ(t0)

A ...

A lz(t,)

A B[X].

Here 06 n but m might be negative, i.e. there might be no elementary formula of the form OflZ(ti).

Then there is some positive integer k, and it holds

3x(ACxl)s+ ci A 3Yi(z(Yi) * BC.K’ (Yi)lh i=O

where Ci is a conjunction qfformulae which are of the form of either Z(t) or 1 Z(t), and L(X) = cx + hi (c # 0), if j such that sj = x exists then 0 < c < 1.

oj’ the theory SA

Decidabifity

15

Proof. First we show

where 0
n=O,

we make

of the form Z(t) and does not include x.

of formulae

the coefficient

of x positive,

Z( - so), if necessary. When n>l. Let so=(nox/mo)+eo, sl=(n,x/m,)+el, prime, and l=LCM(mo, m,), aomO=l, alml =l. Let d= GCD(no, nl), no =dod and n1 =dl d. GCD(aodo, aIdI)= 1. Thus, there are some

namely

substitute

Z(Q)

for

llirmi (i=O, I) be relatively From these it integers ko,kl,

follows such

that that

aodoko+ald,kl = 1. Let h,=dx/l+koeo+k,e, and h,=aodoe,--aIdleo. Then h,=k,s,+k,sl and hZ=-aldlso+aodosl.Andalsos,=aodoh,-klh2,s1=aldlhl+kohZ.Fromthese, Z(s,) A Z(s,) is equivalent to Z(h,) A Z(h,), namely, Z(dx/I+koeo+ k,el) A Z(aodoel -aIdleo). In this operation, if no= 1 then d= 1. It is noted that h2 does not include x. show the case of this suffices to operation, it BY repeating A[x]-D A Z(aX+e) AIZ(to) A ... AlZ(t,) AB[x], where if sj=x then O
;i

gY(

z (7)

A ... “lz(k(Y))

Al-%(Y))

A Bt-g-‘(Y)]

and to

A

This is, by defining

...

Alz(t;(i))A

B[g-l(y)]

g: by yi=(y-_)/l=g;(y),

l-l

1Z

t’ i

( O( ))

A

...

~yi(Z(yi)

AlZ(tk(i))A

So we have the desired conclusion.

0

A B[g-‘(g:-‘(yi))]).

5. Decidability Proposition

theorem

5.1.

-+mU(Z(U)

A

s
A

s
Proof. To prove its sufficiency, we should derive ~fulA A s
A

s
A

.s
A

from l~‘,uA,

13u(Z(u)

uds+l).

Also from l!lu(Z(u) A s
When 3u(Z(u) A s
Propositions

Proposition

5.2 and 5.3 are used later in the proof of Lemma

5.2. D [u, x] = S(X) < u < f(x)

f(x) = a’x + e’. Let c0 he a solution Suppose

O
then it

A t(x)

-

5.8.

1 < u
qf s(x) = t(x) with x, c2 a solution

s(x) = ax + e,

of s(x) + 2 = t(x).

jbllows that

D[U,X]~S(CO)
A

D[u,x].

Proof. It is clear that D [u, x] implies t(x) & s(x) + 2. On the other hand, if x ~(x) and x>c’~ implies s(.Y)+ 2 < t(x). Thus, D [u, x] implies co
Decidability

of the theory

17

SA

Proposition 5.3. Suppose m > 0. Then (m/n)x+e’

(rn/fl)X +e

dk(z(k)AA[x+++z,x+$])).

Proof. Sujjiciency. e
We

can

find

x,,,

u.

such

that

(m/n)x,+e’1
A[xo]

A Z[uo]

A (m/n)xo+

1 A B[u,, x0]. Let k. be x0=x1 + nko/m. Then

A m(xl+nko/m)+e’-l~ko~~(xl+nko/m)+e+l n n ~Z[k~]~A[x,+nk~/m]~B[k~,x,+nk~/m]. The part of inequality mx~+e
n

is

n

A

mxl+e’-l
So this is equivalent to -ne’/m
u formula

is the maximum

The elementary transformation below regularizes which variable u appears, in formula %A.

equalities

of heights

of extended

and inequalities,

in

C. Hosono, Y. Ikeda

18

Definition 5.5. The following equivalent transformation of 3uA is called an elementary transformation. Since the result of this is a disjunction of several formulae, we express these results

by S, a set of formulae.

with only a one element Repeat the following fixed):

To show this transformation,

set S= (3uA1, and we show how to change operations

we begin

this set.

while it is possible (note that bounded

variable

u is

(OPI) When 3u(B A (C v D) A E)eS, eliminate this formula from S. Add 3u(B A C A E) and 3u(B A D A E) instead of it. (OP2) When B A 3u(C A D A E)ES and D is a formula which does not include u, substitute (OP3)

B A D A 3u(C A E) for this formula. When B A h(C A s,
this

formula

from

S. Add

B A 3u(C A s < t A D) and B A 3u(C A s = t A D) to S instead of this formula. (OP4) B A h(C A D A E)ES and D is either an equality or an inequality, since this formula includes variable u, let D’ be a formula with u (namely u = s, s
an extended

v formula

which is a result of

of the theory SA

Decidability

19

Proposition 5.6. Let A be an extended

Proof. ~fuA is equivalent to 3u(Z(u) A s
v formula

3u(l
which is equivalent

A z(u)

A z(S,)

A

...

AA). to show that there

to the formula

A z(S,)

A lz(t,)

A

...

A lz(t,)

A B),

where B is a conjunction of pure v formula, s < 1 and h
A BCf

-‘(~)l).

(2)

and O
f(h)-l
(2) is equivalent

to

Lemma 5.7. Let A be a pure v formula. v formula equivalent to 1 A exists. Proof. We show by an induction trivial. So suppose that A s Vfu(B, A 3@(U)

A

S
The procedure

on the construction ...

V (lh(z(

which gives the extended

of A. When A is T or I, it is

From Proposition 5.1,~ A is equivalent to s
u)

A

sible to get an extended v formula Proposition 4.9. VfU(lB, v ... vlB,)isequivalentto~:ulB, v...v v:ulB,,and,frominduction hypothesis, there exists an extended v formula Ci which is equivalent to 1 Bi. By Proposition 5.6, it is possible to calculate the extended v formula which is equivalent t0

VjUCi.

For i 3u(Z(u) l&(z(U)

A s <

A

u <

t), we show that

S
E

h@(U)

A tdudS+

1).

It is clear that h(z(U)

At
So to show the inverse lh(z(U)A

3

lh(z(U)A

S
of this, we show the contraposition, t
3

h(Z(U)AS
20

C. Hosono.

Y. lkrda

From Theorem 3.17, 3u(Z(u) A s < u ds+ order, t < u v u < t follows. From these

*3u(Z(u)

=3u(Z(u)

A

A

(stu
v t
s
v 3u(Z(u)

1) holds.

And from the axiom

of total

1)) A

t,
If we suppose 13u(Z(u) A t
u formula.

0

Lemma 5.8. Let f be 3x(A A B), where A is a conjunction of equalities, inequalities and a formula Z(x) or a conjunction of1 Z(t 1), . . . , lZ(t,) (both Z and its negation do not occur at the same time), und B is a conjunction of pure v formulae. Then there exists a procedure which gives the extended v formula A equivalent to r. It holds that height(d),
of the This is equivalent s+ 1 A C)). 3x(A A s
A

formable into an equivalent t(cz)-s(co)=2n’m/(mn’-m’n)

theory

SA

s(c0)
t-

extended u formula. and is constant. From

formulae can be transformed equivalently, Similarly, we have the desired conclusion

Let a = n/m, a’= n’/m’, then Propositions 4.8 and 5.6, all

and the height condition holds. in the case a < a’ < 0, a < 0 < a’ (either a or

a’ is not 0) O 0 (m, n are relatively prime natural numbers), and the 5.3, this formula is given formula is 3x(A[x] A ZI~~!~’ uC[u, x]). Then by Proposition equivalent

to

3x

n(e+ l)
(

(1) Suppose jx

m

m

m

A does not include -!?<,<-“, m

m

1)

A

m

Z(x). Exchange

quantifier

_~,n(e+ l)
and apply induction

to

1) A

.[,+;]Ac[k,X+;]).

Since the height of this result is K - 1, by induction the formulae (3) can be transformed into an extended u formula equivalently. (2) Suppose A includes Z(x). Z(k) A A [x + nk/m] is equivalent to the formula of the form Z(k) A Z(x + nk/m) A D [x + nk/m]. Let ( uO, u0 ) be one of the solutions of mu + nu = 1. Z(k) A Z(x + nk/m) is equivalent to Z(mx) A Z(k/m + QX). Thus,

F=3w

Z(w) A -ne’
-ne

A

-n(e+l)
where w = mx, y = (k + u. w)/m. By induction hypothesis, we can make an extended u formula, whose height is at most K, and which is equivalent to inner formula -ne A -n(e+l)
22

C. Hosono,

Y. Ikeda

Theorem 5.9. There is a procedure

which gives an extended

v formula equivalent to any

given formula.

Proof. Use an induction For any formula of 3x to disjunction, A[X]-Z(So)

on the number

A, from induction we can assume A

...

of existential

hypothesis

quantifiers.

and Lemma

5.7, and by distribution

...

A

that

A z(S,)

A lZ(t,)

A

A lz(t,)

B[X]

A

c[X],

where B is a conjunction of equalities and inequalities which include the variable x, and C[x] is a conjunction of pure v formula; both Si and ti include the variable x. If the formula Z(s,) exists, then by Proposition 4.10, 3~4 is equivalent to jY(Z(Y) A B[f-‘(Y)]

A C[f’-l(Y)])

and this can be transformed into an extended v formula, by Lemma If Z(s,) does not exist, then we can use Lemma 5.8 immediately. Theorem 5.10 (Decidability Proof. It is evident

theorem).

from Theorem

Any closed formula

5.9 and Lemma

4.6.

ofSA

5.8. 0

is decidable.

0

6. Conclusion We have described a theory which is one of the extensions of Presburger arithmetic to rational numbers, and we have shown the syntactic proof of the decidability of this theory. This decidability has already been outlined by Smorynski. But our work is independent of his. To begin with, this theory was regarded as an extension to rational numbers. But it is evident from the axioms of SA that the universe of this theory can be the set of all real numbers.

Acknowledgment The authors would like to thank Dr. Prof. S. Igarashi and Dr. T. Tsuji for their advice and pointing out some weak points in the proofs of the first draft.

References [l]

M. Davis, A computer program for Presburger’s algorithm, in: J. Siekmann and G. Wrightson, eds., Symbolic Computation Automation ofReasoning,Vol. I (Springer, Berlin, 1957) 41-48. [2] C. Hosono and Y. Ikeda, On a decidability of the rational Presburger arithmetic, in: Compufer Software. Vol. 9 (1992) 54-61 (in Japanese).

De&ability

of the theory SA

23

S. Igarashi, The v-conversion and an analytic semantics, in: R.E.A. Mason, ed., Proc. lnf: 83 (Elsevier, Amsterdam, 1983) IFIP, 657-668. [4] Y. Ikeda, On an interpreter of the higher typed logical programming language NU, Master’s Thesis, i vt’ 1pnctoral Program in Eng&&ring, University of Tsukuba, 1989 (in Japanese). [S] Y. Ikeda, C. Hosono, T. Tsuji and S. Igarashi, A design df NU interpreter, in: Proc. 7th Conf: Japan Societyfor Software Science and Technology (1990) 293-296 (in Japanese). 163 M. Presburger, iiber die Vollstandigkeit eines gewissen Systems der Arithmetik ganzer Zahlen in welchem die Addition als einzige Operation hervortritt, in: Comptes-Rendus du Congres des Mathematicians des Pays Slaves, Warsaw (1930) 92-101; 395. [7] C.R. Reddy and D.W. Loveland, Presburger arithmetic with bounded quantifier alternation, in: Proc. Con& Record of 10th Ann. ACM Symp. on Theory qf Computing, San Diego (ACM, New York, 1978) 32G325. [S] B. Scarpellimi, Complexity of subcase of Presburger arithmetic, Trans. Amer. Math. Sot. 284 (1984) 203-218. [9] T. Skolem, ijber einige Satzfunktionen in der Arithmetik, in: J.E. Fenstad, eds., Selected Works in Logic (Universitetsforlaget, Oslo, 1970) 281-306. [lo] C. Smoryriski, Logical Number Theory I (Springer, Berlin, 1990). p [ll] K. Wiihl, Zur Komplexitlt der Presburger Arit_hejk und des ;iquivalenzproblems einfacher Programme, in: Theoret. Cornput. Sci. 4th GI C?n$y979& ec t ure Notes in Computer Science, Vol. 67 -__ --~ (Springer, Berlin) 310-318. I. ? [3]