'$�

�

'$

Æ

��

I N F O R M A T I K

� �

De idable fragments of

simultaneous rigid rea hability

Veronique Cortier, Harald Ganzinger,

Florent Ja quemard, and Margus Veanes

MPI{I{1999{2-004 Mar h 1999

FORSCHUNGSBERICHT RESEARCH REPORT

M A X - P L A N C K - I N S T I T U T

F

�

UR

I N F O R M A T I K

Im Stadtwald 66123 Saarbr�u ken Germany

Authors' Addresses

Veronique Cortier

�

E ole Normale Sup�erieure de Ca han, dpt Math�ematiques

61 Avenue du Pr�esident Wilson, 94235 Ca han edex, Fran e

Veronique.Cortier�dptmaths.ens- a han.fr

Harald Ganzinger

Max-Plan k-Institut f�ur Informatik

Im Stadtwald

66123 Saarbr�u ken

hg�mpi-sb.mpg.de

Florent Ja quemard

LORIA and INRIA, 615 rue du Jardin Botanique,

B.P. 101, 54602 Villers-les-Nan y Cedex, Fran e

Florent.Ja quemard�loria.fr

Margus Veanes

Max-Plan k-Institut f�ur Informatik

Im Stadtwald

66123 Saarbr�u ken

veanes�mpi-sb.mpg.de

Publi ation Notes

A short version of this paper appears in the pro eedings of the 26-th Interna-

tional Colloquium on Automata, Languages, and Programming (ICALP'99),

In LNCS.

A knowledgements

We thank J�urgen Stuber for suggestions to improvements and for pointing

out several in onsisten ies in an earlier version on this report.

Abstra t

In this paper we prove de idability results of restri ted fragments of simul-

taneous rigid rea hability or SRR, that is the nonsymmetri al form of simul-

taneous rigid E-uni� ation or SREU. The absen e of symmetry for es us to

use di�erent methods, from the ones that have been su essful in the ontext

of SREU in the past (for example word equations). The methods that we

use instead involve �nite (tree) automata te hniques, and the de idability

proofs provide pre ise omputational omplexity bounds. The main results

are 1) monadi SRR with ground rules is PSPACE- omplete, and 2) balan ed

SRR with ground rules is EXPTIME- omplete. The �rst result indi ates the

di�eren e in omputational power between fragments of SREU with ground

rules and nonground rules, respe tively, due to a straightforward en oding

of word equations in monadi SREU (with nonground rules). The se ond re-

sult establishes the de idability and pre ise omplexity of the largest known

subfragment of nonmonadi SREU.

Keywords

Rigid Uni� ation, Term Rewriting, Rea hability.

1 Introdu tion

Rigid rea hability (RR) is the problem, given a rewrite system R and two

terms s and t, whether there exists a substitution � su h that s�, t�, and R�

are ground, and s� rewrites in some number of steps via R� into t�. The

term \rigid" stems from the fa t that for ea h rule only one instan e an

be used in the rewriting pro ess. Simultaneous rigid rea hability (SRR) is

the problem in whi h a substitution is sought whi h simultaneously solves

ea h member of a system of rea hability onstraints (R

i

; s

i

; t

i

). A spe ial

ase of [simultaneous℄ rigid rea hability arises when the R

i

are symmetri ,

ontaining for ea h rule s! t also its onverse t! s. Su h systems arise for

example by orienting a set of equations in both dire tions. The latter problem

was introdu ed by Gallier, Raatz & Snyder [1987℄ as \simultaneous rigid E-

uni� ation" (SREU) in the ontext of extending tableaux or matrix methods

in automated theorem proving to logi with equality. Rigid rea hability was

initially introdu ed in the ontext of se ond-order uni� ation [Farmer 1991,

Levy 1998℄.

Although the non-simultaneous ase of SREU (rigid E-uni� ation) was

proved NP- omplete by Gallier, Narendran, Plaisted & Snyder [1988℄, SREU

in general was shown by Degtyarev & Voronkov [1995℄ to be unde idable.

Further impli ations of the latter result are dis ussed in [Degtyarev, Gure-

vi h & Voronkov 1996℄. In a series of papers, SREU has been studied ex-

tensively and several sharp boundaries have been laid between its de id-

able and unde idable fragments. Most re ent developments are dis ussed by

Voronkov [1998℄ and Veanes [1998℄. Rigid rea hability was shown unde id-

able by Ganzinger, Ja quemard & Veanes [1998℄.

The, arguably, most diÆ ult remaining open problem regarding SREU

is the de idability of \monadi " SREU, or SREU restri ted to signatures

where all non onstant fun tion symbols are unary. The importan e of this

fragment stems from its lose relation to word equations [Degtyarev, Matiya-

sevi h & Voronkov 1996℄, and to fragments of intuitionisti logi [Degtyarev

& Voronkov 1996℄. What is known about monadi SREU in general is

that it redu es to a nontrivial extension of word equations [Gurevi h &

Voronkov 1997℄. In the ase of ground rules, the de idability of monadi

SREU was established in [Gurevi h & Voronkov 1997℄ by redu ing it to \word

equations with regular onstraints". The de idability of the latter prob-

lem is an extension of Makanin's [1977℄ result by S hulz [1990℄. Conversely,

word equations redu e in polynomial time to monadi SREU [Degtyarev,

Matiyasevi h & Voronkov 1996℄. The �rst main result of this paper (in Se -

tion 3) is that monadi SRR with ground rules is in PSPACE, improving the

EXPTIME result in Ganzinger et al. [1998℄. Hen e, it is unlikely that there

1

is a simple redu tion, if any redu tion at all, from monadi SREU to monadi

SREU with ground rules, or else one would get a onsiderable simpli� ation

of Makanin's [1977℄ proof. The PSPACE-hardness of monadi SREU with

ground rules was shown by Goubault [1994℄.

To obtain the PSPACE result we use an extension of the interse tion

nonemptiness problem of a sequen e of �nite automata that we prove to be

in PSPACE. Moreover, using the same proof te hnique, we an show that

simultaneous rigid rea hability with ground rules remains in PSPACE, even

when just the rules are required to be monadi . Furthermore, in this ase

PSPACE-hardness holds already for a single onstraint with one variable,

ontrasting the fa t that SREU with one variable is solvable in polynomial

time [Degtyarev, Gurevi h, Narendran, Veanes & Voronkov 1998b℄.

Our se ond main result on erns (nonmonadi ) SRR with ground rules.

In se tion 4, we show that SRR with ground rules is EXPTIME- omplete for

\balan ed" systems of rea hability onstraints. Under balan ed systems fall

for example systems where all o urren es of ea h variable are at the same

depth. It is possible to obtain unde idability of (nonsimultaneous) rigid

rea hability with ground rules where all but one o urren e of all variables

o ur at the same depth [Ganzinger et al. 1998℄. Moreover, our de idability

result generalizes the de idability result by Degtyarev, Gurevi h, Narendran,

Veanes & Voronkov [1998a℄ of the largest known de idable fragment of SREU

with ground rules and implies EXPTIME- ompletess of the omplexity of

this fragment (whi h is left open in [Degtyarev et al. 1998a℄). We use �nite

tree automata te hniques over produ t languages, that have been used in

de ision pro edures for \automata with onstraints between brothers" [ f.

Comon, Dau het, Gilleron, Lugiez, Tison & Tommasi 1998℄.

2 Preliminaries

A signature � is a olle tion of fun tion symbols with �xed arities � 0 and,

unless otherwise stated, � is assumed to ontain at least one onstant, that

is, one fun tion symbol with arity 0. We use a; b; ; d; a

1

; : : : for onstants and

f; g; f

1

; : : : for fun tion symbols in general. A signature is alled monadi if

all fun tion symbols in it have arity � 1. A ground term is one that ontains

no variables. The set of all ground terms over a signature � is denoted by

T

�

.

We use s; t; l; r; s

1

; : : : for terms. The size ktk of a term t is de�ned

re ursively by: ktk = 1 if t is either a variable or a onstant and

kf(t

1

; : : : ; t

n

)k = kt

1

k+ : : :+ kt

n

k+ 1:

2

Positions in terms are sequen es of integers. We use p; p

1

et for positions,

� for the empty sequen e (root position) and pp

0

for the on atenation prod-

u t of two positions p and p

0

. We will also use the pre�x ordering � on

positions. Some positions p

1

; : : : ; p

n

are alled parallel if they are pairwise

un omparable with respe t to �.

We assume that the reader is familiar with the basi on epts in term

rewriting [e.g. Dershowitz & Jouannaud 1990, Baader & Nipkow 1998℄. We

write u[s℄ when s o urs as a subterm of u. In that ase u[t℄ denotes the re-

pla ement of the indi ated o urren e of s by t. An equation is an unordered

pair of terms, denoted by s � t. A rule is an ordered pair of terms, denoted

by s ! t. An equation or a rule is ground if the terms in it are ground. A

system is a �nite set. Let R be a system of ground rules, and s and t two

ground terms. Then s rewrites in R to t, denoted by s�!

R

t, if t is obtained

from s by repla ing an o urren e of a term l in s by a term r for some rule

l ! r in R. The term s redu es in R to t, denoted by s�!

�

R

t, if either s = t

or s rewrites to a term that redu es to t. R is alled symmetri if, with any

rule l ! r in R, R also ontains its onverse r ! l. Below we shall not

distinguish between systems of equations and symmetri systems of rewrite

rules. The size of a system R is the sum of the sizes of its omponents:

kRk =

P

l!r2R

(klk+ krk).

Rigid Rea hability. A rea hability onstraint, or simply a onstraint, in

a signature � is a triple (R; s; t) where R is a set of rules in �, and s and

t are �-terms. We refer to R, s and t as the rule set, the sour e term

and the target term, respe tively, of the onstraint. A substitution � in �,

solves (R; s; t) if � is grounding for R, s and t, and s��!

�

R�

t�: The problem

of solving onstraints is alled rigid rea hability. A system of onstraints is

solvable if there exists a substitution that solves all onstraints in that system.

Simultaneous rigid rea hability or SRR is the problem of solving systems

of onstraints. Monadi (simultaneous) rigid rea hability is (simultaneous)

rigid rea hability for monadi signatures.

Rigid E-uni� ation is rigid rea hability for onstraints (E; s; t) with sets

of equations E. Simultaneous Rigid E-uni� ation or SREU is de�ned a -

ordingly.

Finite tree automata. Finite bottom-up tree automata, or simply, tree

automata, from here on, are a generalization of lassi al automata [Doner

1970, That her & Wright 1968℄. Using a rewrite rule based de�nition [e.g.

Coquid�e, Dau het, Gilleron & V�agv�olgyi 1994, Dau het 1993℄, a tree au-

tomaton (or TA) A is a quadruple (Q;�; R; F ), where (i) Q is a �nite set

3

of onstants alled states, (ii) � is a �nite signature that is disjoint from Q,

(iii) R is a system of rules of the form f(q

1

; : : : ; q

n

) ! q, where f 2 � has

arity n � 0 and q; q

1

; : : : ; q

n

2 Q, and (iv) F � Q is the set of �nal states.

The size of a TA A is kAk = jQj+ j�j+ kRk.

We denote by L(A; q) the set ft 2 T

�

�

�

t�!

�

R

qg of ground terms a epted by

A in state q. The set of terms re ognized by the TA A is the set

S

q2F

L(A; q).

A set of terms is alled re ognizable or regular if it is re ognized by some TA.

A monadi TA is a TA with a monadi signature.

Finite string automata. For monadi signatures, we use the traditional,

equivalent on epts of alphabets, strings (or words), �nite automata, and

regular expressions. We will identify an NFA A with alphabet � with the set

of all rules a(q) ! p, also written as q�!

a

A

p, where there is a transition with

label a 2 � from state q to state p in A, and we denote this set of rules also

by A. A monadi term a

1

(a

2

(: : : a

n

(q))) is written, using the reversed Polish

notation, as the string qa

n

: : : a

1

.

Then A a epts a string a

1

a

2

� � �a

n

if and only if, for some �nal state q

and the initial state q

0

of A, a

n

(� � �a

2

(a

1

(q

0

)) � � � )�!

�

A

q, i.e.,

q

0

�!

a

1

A

q

1

�!

a

2

A

� � � �!

A

a

n

q:

The set of all strings a epted by A is denoted by L(A).

Produ t automata. Let � be a signature, m a positive integer, and ? a

new onstant. We write �

?

for � [ f?g and �

m

?

denotes the signature on-

sisting of, for all f

1

; f

2

; : : : ; f

m

2 �

?

, a unique fun tion symbol hf

1

f

2

� � � f

m

i

with arity equal to the maximum of the arities of the f

i

's.

Let t

i

2 T

�

[ ?, t

i

= f

i

(t

i1

; : : : ; t

ik

i

), where k

i

� 0, for 1 � i � m. Let

k be the maximum of all the k

i

and let t

ij

= ? for k

i

< j � k. The produ t

t

1

� � � t

m

of t

1

; : : : ; t

m

is de�ned by re ursion on the subterms:

t

1

� � � t

m

= hf

1

f

2

� � � f

m

i(t

11

� � � t

1k

; : : : ; t

m1

� � � t

mk

) (1)

For example:

f( ; g( )) f(g(d); f( ; g( ))) = hffi( g(d); g( ) f( ; g( )))

= hffi(h gi(? d); hgfi( ;? g( )))

= hffi(h gi(h?di; hgfi(h i; h?gi(? )))

= hffi(h gi(h?di; hgfi(h i; h?gi(h? i)))

4

We write T

m

�

for the set of all t in T

�

m

?

su h that t = t

1

� � � t

m

for some

t

1

; : : : ; t

m

2 T

�

[ ?. If s 2 T

m

�

and t 2 T

n

�

, where s = s

1

� � � s

m

and

t = t

1

� � � t

n

, then s t denotes the term s

1

� � � s

m

t

1

� � � t

n

in

T

m+n

�

. Given a sequen e

~

t = t

1

; : : : ; t

m

of terms in T

�

[?, we write

N

~

t for

the produ t term t

1

� � � t

m

Given two automata A

1

and A

2

over �

m

?

and �

n

?

, respe tively, the produ t

of A

1

and A

2

is an automaton A

1

A

2

over �

m+n

?

su h that

L(A

1

A

2

) = L(A

1

) L(A

2

) = ft

1

t

2

: t

1

2 L(A

1

); t

2

2 L(A

2

)g

The onstru tion of A

1

A

2

is straightforward, with a state q

(q

1

;q

2

)

for all

states q

1

in A

1

and q

2

in A

2

, [see e.g. Comon et al. 1998℄. In general,

N

n

i=1

A

i

is de�ned a ordingly.

We will use the following onstru tion of Dau het, Heuillard, Les anne &

Tison [1990℄ in our proofs.

Lemma 1 Let R be a ground rewrite system over a signature �. There is a

TA A su h that L(A) = fs t : s; t 2 T

�

; s�!

�

R

tg that an be onstru ted in

polynomial time from R and �.

3 Monadi SRR

We prove that monadi SRR with ground rules is PSPACE- omplete. Our

main tool is a de ision problem of NFAs that we de�ne next. In this se tion

we onsider only monadi signatures.

3.1 Constrained produ t nonemptiness of NFAs

Given a signature � and a positive integer m, we want to sele t only a

ertain subset from �

m

through sele tion onstraints (bounded by m). These

are unordered pairs of indi es written as i � j, where 1 � i; j � m, i 6= j.

Given a signature � and a set I of sele tion onstraints, we write �

m⇂I for

the following subset of �

m

:

�

m⇂I = fha

1

a

2

� � �a

m

i 2 �

m

: (8i � j 2 I) a

i

= a

j

g

For an automaton A, let A⇂I denote the redu tion of A to the alphabet �

m⇂I.

We write also L(A)⇂I for L(A⇂I). The automaton A⇂I has the same states

as A, and the transitions of A⇂I are pre isely all the transitions of A with

labels from �

m⇂I.

We onsider the following de ision problem, that is losely related to the

nonemptiness problem of the interse tion of a sequen e of NFAs. Consider

5

an alphabet �. Let (A

i

)

1�i�n

, n � 1, be a sequen e of (string produ t)

NFAs over the alphabets �

m

i

?

for 1 � i � n, respe tively. Let m be the

sum of all the m

i

and let I be a set of sele tion onstraints. The onstrained

produ t nonemptiness problem of NFAs is, given (A

i

)

1�i�n

, and I, to de ide if

(

N

n

i=1

L(A

i

))⇂I is nonempty. Our key lemma is the following one. Its proof is

a straightforward extension of the in lusion part of Kozen's [1977℄ PSPACE-

ompletess result of the interse tion nonemptiness problem of DFAs: given

a sequen e (A

i

)

1�i�n

of DFAs, is

T

n

i=1

L(A

i

) nonempty?

Lemma 2 Constrained produ t nonemptiness of NFAs (or monadi TAs) is

in PSPACE.

Proof. Let (A

i

)

1�i�n

, m

i

, m, �, and I be given as above. Assume also that

m

i

= 2 for 1 � i � n, i.e., m = 2n and ea h automaton has alphabet �

2

?

.

(Proof of the general ase is analogous.)

Furthermore, we an assume, without loss of generality, that none of the

automata a epts the empty string and that, for ea h string v that is a epted

by A

i

also h??iv is a epted by A

i

, e.g., we an assume that ea h automaton

has a transition with label h??i from the initial state to the initial state.

Consider the following nondeterministi de ision pro edure.

I (Initialize) Cal ulate the number of states in

N

n

i=1

A

i

and save it in

IterationLimit.

(This al ulation is easy, be ause ea h state of

N

n

i=1

A

i

orresponds to

a sequen e of states (q

i

)

1�i�n

, where q

i

is a state of A

i

.)

Save in State

i

the initial state of A

i

for 1 � i � n.

II (Guess the next letter from �

m

?

⇂I) Sele t (a1

; : : : ; a

m

) 2 �

m

?

⇂I and

store a

i

in Letter

i

.

III (Guess the next transition of (

N

n

i=1

A

i

)⇂I) For 1 � i � n, guess

nondeterministi ally a state q

i

from A

i

.

Che k that, for 1 � i � n, there is a hLetter

2i�1

Letter

2i

i-transition

in A

i

from State

i

to q

i

, and if so, save q

i

in State

i

. If there is no su h

transition then terminate and reje t.

IV (Che k a eptan e of (

N

n

i=1

A

i

)⇂I) If, for 1 � i � n, State

i

is an

a epting state of A

i

then terminate and a ept.

V (Iterate) If IterationLimit is 0 then terminate and reje t, else de-

rease IterationLimit by one and return to Step II.

6

The pro edure orresponds to walking through the graph of (

N

n

i=1

A

i

)⇂I, by

starting from the initial state, at ea h step just remembering the urrent state

and guessing a valid transition from that state to the next state. We only

need to he k if there exists a path of at most IterationLimit transitions

(as initialized in Step I) in L(

N

n

i=1

A

i

)⇂I from the initial state to a �nal state.

It is evident that the pro edure always terminates, and that it a epts if and

only if L(

N

n

i=1

A

i

)⇂I is nonempty.

It is obvious, through straightforward binary en odings, that no more

than polynomial spa e is required, in order to meet the spa e requirements

of the pro edure. Hen e, the pro edure runs in nondeterministi polynomial

spa e and thus in PSPACE, by using the result of Savit h [1970℄.

Finally, note that the only di�eren e between NFAs and monadi TAs is

that in the latter we may have several transitions of the form ! q, where

is a onstant and q a state. This orresponds roughly to allowing several

initial states in NFAs. ⊠

The proof of Lemma 2 an be extended in a straightforward manner to

�nite tree automata. The only di�eren e will be that the algorithm will

do \universal hoi es" when the arity of fun tion symbols (letters) in the

omponent automata is > 1. This leads to alternating PSPACE, and thus,

by the result of Chandra, Kozen & Sto kmeyer [1981℄, to EXPTIME upper

bound for the onstrained produ t nonemptiness problem of TAs.

Although we will not use this fa t, it is worth noting that the onstrained

produ t nonemptiness problem is also PSPACE-hard, and this so already for

DFAs (or monadi DTAs). It is easy to see that

T

n

i=1

L(A

i

) is nonempty if

and only if L(

N

n

i=1

A

i

)⇂fi � i + 1 : 1 � i < ng is nonempty.

3.2 Redu tion of monadi SRR with ground rules to

onstrained produ t nonemptiness of NFAs

We need the following notion of normal form of a system of rea hability

onstraints. We say that a system S of rea hability onstraints is at, if ea h

onstraint in S is either of the form

� (R; x; t), R is nonempty, x is a variable, and t is a ground term or a

variable distin t from x, or of the form

� (;; x; f(y)), where x and y are distin t variables and f is a unary fun -

tion symbol.

Note that solvability of a rea hability onstraint with empty rule set is simply

uni�ability of the sour e and the target. The following simple lemma is useful.

7

Lemma 3 Let S be a system of rea hability onstraints. There is a at

system that an be obtained in polynomial time from S, that is solvable if

and only if S is solvable.

Proof. Let S be a given system of rea hability onstraints and onsider the

following pro edure.

1. Repla e ea h onstraint (R; s; t), where s is not a variable, or when

s = t, by the onstraints (R; x; t) and (;; x; s), where x is a new variable.

2. Repla e ea h onstraint (R; x; t), where R is nonempty, x is a variable

and t is neither ground nor a variable, by the onstraints (R; x; y) and

(;; y; t), where y is a new variable.

3. Repla e ea h onstraint (;; x; f(s)), where s is not a variable and not

ground, by the onstraints (;; x; f(y)) and (;; y; s), where y is a new

variable.

4. Repeat the above steps until the system is at.

It is easy to he k that ea h step preserves solvability, and learly, the time

omplexity of this pro edure is polynomial in the size of S. ⊠

By using Lemma 2 and Lemma 3 we an now show the following theorem,

that is the main result of this se tion.

Theorem 1 Monadi SRR with ground rules is PSPACE- omplete.

Proof. The PSPACE-hardness has been proved already in the ase when

the rule sets are symmetri [Goubault 1994℄ and there is only one variable

[Gurevi h & Voronkov 1997℄. We prove in lusion in PSPACE by giving a

polynomial time redu tion to the onstrained produ t nonemptiness problem

of monadi TAs.

Let S be a system of rea hability onstraints with ground rules. Let

� be the signature of S. We may assume, by using Lemma 3, that S is

at. Enumerate all the onstraints in S as �

1

; : : : ; �

m

; �

m+1

; : : : ; �

n

, where

all the onstraints of the form (;; x; f(y)) are enumerated as �

m+1

; : : : ; �

n

.

Let �

i

= (R

i

; x

i

; t

i

) for 1 � i � m and �

i

= (;; x

i

; f

i

(y

i

)) for m < i � n.

For 1 � i � m, onstru t a TA A

i

su h that,

L(A

i

) = fx

i

� t

i

� : � solves �

i

g:

For m < i � n, onstru t a TA A

i

su h that,

L(A

i

) = fx

i

� y

i

� : � solves �

i

g:

8

q

0

q

g

q

f

q

h

h ?i

hg i hfgi

hgfi

h

g

h

i

h

h

g

i

h

f

h

i

h

h

f

i

h

h

i

hf i

hggi

h

f

f

i

hhhi

Figure 1: A DFA (or monadi DTA) A that re ognizes ff(s)s : s 2 T

�

g, where

� onsists of the unary fun tion symbols f , g, and h, and the onstant .

For example A re ognizes the string h ?ihg ihggihhgihfhi, i.e., the term

hfhi(hhgi(hggi(hg i(h ?i)))) that is the same as f(h(g(g( )))) h(g(g( ))).

(Su h an automaton is illustrated in Figure 1.) It follows from Lemma 1 that

all these TAs an be onstru ted in polynomial time.

Let I be the set of all the following sele tion onstraints (where 1 � i; j �

n and i 6= j):

1. If the sour e of a �

i

is a variable that o urs as the sour e of a �

j

, then

2i� 1 � 2j � 1 2 I.

2. If the sour e of a �

i

is a variable that o urs in the target of a �

j

, then

2i� 1 � 2j 2 I.

3. If the target of a �

i

is a variable that o urs in the target of a �

j

, then

2i � 2j 2 I.

It remains to be proved that L(

N

n

i=1

A

i

)⇂I is nonempty if and only if S is

solvable. (This proof is straightforward, and is illustrated in Example 1.)

The theorem follows then from Lemma 2. ⊠

The ru ial step in the proof of Theorem 1 is the onstru tion of an

automaton that re ognizes the language ff(s) s : s 2 T

�

g. (See Figure 1.)

The reason why the proof does not generalize to TAs is that the language

ff(s) s : s 2 T

�

g is not regular for nonmonadi signatures. The next

example illustrates how the redu tion in the proof of Theorem 1 works.

Example 1 Consider a at system S = f�

1

; �

2

; �

3

g with �

1

= (R; y; x), �

2

=

(;; y; f(z)) and �

3

= (;; z; g(x)), over a signature � = ff; g; g, where is a

onstant. (This system is solvable if and only if the onstraint (R; f(g(x)); x)

is solvable.)

9

The onstru tion in the proof of Theorem 1 gives us the monadi TAs

A

1

, A

2

and A

3

su h that

L(A

1

) = fs t : s�!

�

R

t; s; t 2 T

�

g;

L(A

2

) = ff(s) s : s 2 T

�

g;

L(A

3

) = fg(s) s : s 2 T

�

g;

and a set I = f1 � 3; 5 � 4; 6 � 2g of sele tion onstraints. So L(

N

3

i=1

A

i

)⇂I

is as follows.

L(A

1

A

2

A

3

)⇂I = fs t f(u) u g(v) v :

s; t; u; v 2 T

�

; s�!

�

R

tg⇂f1 � 3; 5 � 4; 6 � 2g

= fs t f(u) u g(v) v :

s; t; u; v 2 T

�

; s�!

�

R

t; s = f(u); g(v) = u; v = tg

= ff(g(t)) t f(g(t)) g(t) g(t) t :

t 2 T

�

; f(g(t))�!

�

R

tg

So, solvability of S is equivalent to nonemptiness of L(A

1

A

2

A

3

)⇂I.

3.3 Some de idable extensions of the monadi ase

Some restri tions imposed by only allowing monadi fun tion symbols an

be relaxed without losing de idability of SRR for the resulting lasses of

onstraints. One de idable fragment of SRR is obtained by requiring only

the rules to be ground and monadi . It an be shown that SRR for this

lass is still in PSPACE. Furthermore, an easy argument using the inter-

se tion nonemptiness problem of DFAs shows that PSPACE-hardness of this

fragment holds already for a single onstraint with one variable. This is in

ontrast with the fa t that SREU with one variable and a �xed number of

onstraints an be solved in polynomial time [Degtyarev et al. 1998b℄.

4 A de idable nonmonadi fragment

In this se tion, we onsider general signatures and give a riteria on the

sour e and target terms of a system of rea hability onstraints for the de id-

ability of SRR when the rules are ground. Moreover, we prove that SRR is

EXPTIME- omplete in this ase. Our de ision algorithm involves essentially

tree automata te hniques. Let � be a signature �xed for the rest of the

se tion.

10

4.1 Semi-linear sequen es of terms

We say that a sequen e of terms (t

1

; t

2

; : : : ; t

m

) of (possibly non ground)

�-terms or ? is semi-linear if one of the following onditions holds for ea h

t

i

:

1. t

i

is a variable, or

2. t

i

is a linear term and no variable in t

i

o urs in t

j

for i 6= j.

Note that if t

i

is ground then it satis�es the se ond ondition trivially.

Lemma 4 Let (s

1

; s

2

; : : : ; s

k

) be a semi-linear sequen e of �-terms. Then

the subset

�

s

1

� s

2

� � � � s

k

� : � is a grounding �-substitution

� T

m

�

is

re ognized by a TA the size of whi h is in O((ks

1

k+ k�k) : : : (ks

k

k+ k�k)).

Proof. Let � and ~s = s

1

; s

2

; : : : ; s

k

be given. Let A

i

be the TA that re og-

nizes fs

i

� : s

i

� 2 T

�

g for 1 � i � k. The desired TA is (

N

A

i

)⇂I, where I

is the set of all sele tion onstraints i � j su h that s

i

and s

j

are identi al

variables. ⊠

We shall also use the following lemma.

Lemma 5 Let A = (�; Q;R; F ) be a TA, s 2 T

�

, and p

1

; : : : ; p

k

parallel

positions in s. Then there is a TA A

0

, with kA

0

k 2 O

�

kAk

2k

�

, that re ognizes

the set

�

s

1

� � � s

k

: s

1

; : : : ; s

k

2 T

�

; s[p

1

s

1

; : : : ; p

k

s

k

℄ 2 L(A)

Proof. For all states q 2 Q, let A

q

be the automaton (�; Q;R; fqg). Let

f~q

i

g

1�i�m

be the olle tion of all sequen es ~q

i

= q

i1

; : : : ; q

ik

2 Q su h that,

for some q

f

2 F , s[p

1

q

i1

; : : : ; p

k

q

ik

℄�!

�

R

q

f

. For all su h sequen es ~q

i

,

1 � i � m, onstru t a TA A

i

that re ognizes

L(A

q

i1

) � � � L(A

q

ik

):

Here we an assume that ea h L(A

q

ij

) is nonempty, or else L(A

i

) is empty.

Assume that all the A

i

's have disjoint sets of states and let A

0

be the union of

all the A

i

's. It is easy to he k that A

0

re ognizes the given set of terms. Note

that m � jQj

k

. The size of A

0

is therefore kA

0

k �

P

m

i=1

kA

i

k �

P

m

i=1

kAk

k

�

jQj

k

� kAk

k ⊠

11

4.2 Parallel de omposition of sequen es of terms

For te hni al reasons, we generalize the notion of a produ t of m terms by

allowing nonground terms. The resulting term is in an extended signature

with as an additional variadi fun tion symbol. The de�nition is the same

as for ground terms (see (1)), with the additional ondition that if one of the

t

i

's is a variable then

t

1

� � � t

m

= (t

1

; : : : ; t

m

):

Consider a sequen e ~s = s

1

; : : : ; s

m

of terms and let ((

~

t

i

))

1�i�k

be the

sequen e of all the subterms of the produ t term

N

~s whi h have head symbol

. The parallel de omposition of ~s = s

1

; : : : ; s

m

or pd(~s) is the sequen e

(

~

t

i

)

1�i�k

, i.e., we forget the symbol . We need the following te hni al

notion in the proof of Lemma 6: pdp(~s) is the sequen e (p

i

)

1�i�k

, where p

i

is

the position of (

~

t

i

) in

N

~s.

The following example illustrates these new de�nitions and lemmas and

how they are used.

Example 2 Let s = f(g(z); g(x)) and t = f(y; f(x; y)) be two �-terms, and

let R be a ground rewrite system over �. We will show how to apture all

the solutions of the rea hability onstraint (R; s; t) as a ertain regular set of

�

2

?

-terms. First, onstru t the produ t s t.

s t = f(g(z); g(x)) f(y; f(x; y))

= hffi(g(z) y; g(x) f(x; y))

= hffi((g(z); y); hgfi(x x;? y))

= hffi((g(z); y); hgfi((x; x);(?; y)))

The preorder traversal of s t yields the sequen e (g(z); y), (x; x),

(?; y).

Finally, pd(s; t) is the semi-linear sequen e g(z); y; x; x;?; y. (Note

that pdp(s; t) is the sequen e 1; 21; 22.) It follows from Lemma 4 that

there is a TA A

1

su h that L(A

1

) =

�

g(z�) y� x� x� ? y� :

� is a grounding �-substitution

.

Now, onsider a TA A

R

that re ognizes the produ t of �!

�

R

, see Lemma 1,

i.e., L(A

R

) = fuv : u�!

�

R

v; u; v 2 T

�

g: From A

R

we an, by using Lemma 5,

onstru t a TA A

2

su h that

L(A

2

) =

�

s

1

s

21

s

22

: s

1

; s

21

; s

22

2 T

2

�

; hffi(s

1

; hgfi(s

21

; s

22

)) 2 L(A

R

)

12

Let A re ognize L(A

1

) \ L(A

2

). We get that

L(A) = L(A

1

) \ L(A

2

)

=

8

<

:

s

1

s

21

s

22

: (9x�; y�; z� 2 T

�

)

s

1

= g(z�) y�; s

21

= x� x�; s

22

= ? y�;

hffi(s

1

; hgfi(s

21

; s

22

)) 2 L(A

R

)

= fg(z�) � � � y� : hffi(g(z�) y�; hgfi(x� x�;? y�)) 2 L(A

R

)g

= fg(z�) � � � y� : � solves (R; s; t)g

Hen e L(A) 6= ; if and only if (R; s; t) is solvable.

The ru ial property that is needed in the example to prove the de id-

ability of the rigid rea hability problem is that the parallel de omposition of

the sequen e onsisting of its sour e and target terms is semi-linear. This

observation leads to the following de�nition.

4.3 Balan ed systems with ground rules

A system

�

(R

1

; s

1

; t

1

); : : : ; (R

n

; s

n

; t

n

)

�

of rea hability onstraints is alled

balan ed if the parallel de omposition pd(s

1

; t

1

; s

2

; t

2

; : : : ; s

n

; t

n

) is semi-

linear. The proof of Lemma 6 is a generalization of the onstru tion in

Example 2.

Lemma 6 From every balan ed system S of rea hability onstraints with

ground rules, we an onstru t in EXPTIME a TA A su h L(A) 6= ; i� S is

satis�able.

Proof. Let S =

�

(R

1

; s

1

; t

1

); : : : ; (R

n

; s

n

; t

n

)

�

be a given a balan ed system

of rea hability onstraints su h that R

1

, : : : ,R

n

are ground.

Using Lemma 1, we an asso iate a TA A

i

to ea h R

i

(i � n) su h that

L(A

i

) =

�

u v : u�!

�

R

i

v; u; v 2 T

�

The ground terms s

?

i

and t

?

i

are obtained from the sour e and target terms

by repla ement of every variable by the onstant ?.

Let U = s

?

1

t

?

1

: : : s

?

n

t

?

n

and (p

1

; : : : ; p

k

) = pdp(s

1

; t

1

; : : : ; s

n

; t

n

). We

an use Lemma 5 to onstru t a TA A

0

su h that

L(A

0

) =

n

v

1

: : : v

k

: v

1

; : : : ; v

k

2 T

n

�

;

U [p

1

v

1

; : : : ; p

k

v

k

℄ 2 L(

n

O

i=1

A

i

)

o

13

By hypothesis, the sequen e pd(s

1

; t

2

; : : : ; s

n

; t

n

), denoted (u

1

; : : : ; u

kn

), is

semi-linear. Therefore, it follows from Lemma 4 that there is a TA A

00

su h

that

L(A

00

) =

�

u

1

� : : : u

kn

� : � is a grounding �-substitution

Note that both L(A

0

) and L(A

00

) are subsets of T

kn

�

. Let A be a TA re og-

nizing L(A

0

) \ L(A

00

). We have that L(A) 6= ; if and only if S is satis�able.

Let t 2 T

kn

�

. Then t 2 L(A)

i� t = u

1

� : : :u

kn

� for some grounding �-substitution � (t 2 L(A

00

)), and

U [p

1

w

1

; : : : ; p

k

w

k

℄ 2 L(

N

n

i=1

A

i

), where w

i

=

N

i:n

j=(i�1)n+1

u

j

�

(t 2 L(A

00

)),

i� s

?

1

t

?

1

: : : s

?

n

t

?

n

[p

1

w

1

; : : : ; p

k

w

k

℄ 2 L(

N

n

i=1

A

i

),

i� s

1

� t

1

� : : : s

n

� t

n

� 2 L(

N

n

i=1

A

i

), be ause every variable of S

o urs in one of the u

1

; : : : ; u

k

, by de�nition of pd ,

i� s

1

��!

�

R

1

t

1

�; : : : ; s

n

��!

�

R

n

t

n

�.

Lets now �nally evaluate the size of A, the omplexity of its onstru tion

being linearly proportional to its size. For ea h i � n, the size of A

i

is

polynomial in kR

i

k, thus

N

n

i=1

A

i

� M

n

where M = maxfkR

i

k : i � ng

and is one onstant independent from the problem size Therefore, kA

0

k �

M

2 nk

, see Lemma 5. A ording to Lemma 4,

kA

00

k � ku

1

k � : : :� ku

kn

k � �

n

i=1

ks

i

k � �

n

i=1

kt

i

k � N

2n

;

where N = maxfks

i

k; kt

i

k : i � ng. Hen e,

kAk = kA

00

k � kA

0

k � N

2n

�M

2 nk

� kSk

2n( k+1)

:

⊠

Theorem 2 SRR is EXPTIME- omplete for balan ed systems with ground

rules.

Proof. The EXPTIME-hardness follows from [Ganzinger et al. 1998℄, where

we have proved that one an redu e the emptiness de ision for interse tion

of n tree automata to the satis�ability of a rigid rea hability onstraint

�

R; f(x; : : : ; x); f(q

1

; : : : ; q

n

)

�

, where R is ground and q

1

, : : : ,q

n

are on-

stants. ⊠

The balan ed ase an easily be used to show the de idability of the following

ase: for ea h variable x there exists an integer d

x

su h that x o urs only at

positions of length d

x

. For example with s

1

= f(x; g(y)), t

1

= f(f(y; y); x),

s

2

= g(x), and t

2

= g(f(a; y)), �rst \guess" a term a, g(x

1

), or f(x

1

; x

2

) for

x to obtain a system where all variables o ur at the same depth.

14

Referen es

Baader, F. & Nipkow, T. (1998), Term Rewriting and All That, Cambridge

University Press.

Chandra, A., Kozen, D. & Sto kmeyer, L. (1981), `Alternation', Journal of

the Asso iation for Computing Ma hinery 28(1), 114{133.

Comon, H., Dau het, M., Gilleron, R., Lugiez, D., Tison, S. & Tommasi,

M. (1998), Tree Automata Te hniques and Appli ations, unpublished

manus ript.

Coquid�e, J., Dau het, M., Gilleron, R. & V�agv�olgyi, S. (1994), `Bottom-

up tree pushdown automata: lassi� ation and onne tion with rewrite

systems', Theoreti al Computer S ien e 127, 69{98.

Dau het, M. (1993), Rewriting and tree automata, in H. Comon & J. Jouan-

naud, eds, `Term Rewriting (Fren h Spring S hool of Theoreti al Com-

puter S ien e)', Vol. 909 of Le ture Notes in Computer S ien e, Springer

Verlag, Font Romeux, Fran e, pp. 95{113.

Dau het, M., Heuillard, T., Les anne, P. & Tison, S. (1990), `De idability of

the on uen e of �nite ground term rewrite systems and of other related

term rewrite systems', Information and Computation 88, 187{201.

Degtyarev, A. & Voronkov, A. (1995), Simultaneous rigid E-uni� ation is

unde idable, UPMAIL Te hni al Report 105, Uppsala University, Com-

puting S ien e Department.

Degtyarev, A. & Voronkov, A. (1996), De idability problems for the prenex

fragment of intuitionisti logi , in `Eleventh Annual IEEE Symposium

on Logi in Computer S ien e (LICS'96)', IEEE Computer So iety

Press, New Brunswi k, NJ, pp. 503{512.

Degtyarev, A., Gurevi h, Y. & Voronkov, A. (1996), Herbrand's theorem

and equational reasoning: Problems and solutions, in `Bulletin of the

European Asso iation for Theoreti al Computer S ien e', Vol. 60. The

\Logi in Computer S ien e" olumn.

Degtyarev, A., Gurevi h, Y., Narendran, P., Veanes, M. & Voronkov,

A. (1998a), `De idability and omplexity of simultaneous rigid E-

uni� ation with one variable and related results', Theoreti al Computer

S ien e. To appear.

15

Degtyarev, A., Gurevi h, Y., Narendran, P., Veanes, M. & Voronkov,

A. (1998b), The de idability of simultaneous rigid E-uni� ation with

one variable, in T. Nipkow, ed., `Rewriting Te hniques and Appli a-

tions', Vol. 1379 of Le ture Notes in Computer S ien e, Springer Verlag,

pp. 181{195.

Degtyarev, A., Matiyasevi h, Y. & Voronkov, A. (1996), Simultaneous rigid

E-uni� ation and related algorithmi problems, in `Eleventh Annual

IEEE Symposium on Logi in Computer S ien e (LICS'96)', IEEE Com-

puter So iety Press, New Brunswi k, NJ, pp. 494{502.

Dershowitz, N. & Jouannaud, J.-P. (1990), Rewrite systems, in

J. Van Leeuwen, ed., `Handbook of Theoreti al Computer S ien e', Vol.

B: Formal Methods and Semanti s, North Holland, Amsterdam, hap-

ter 6, pp. 243{309.

Doner, J. (1970), `Tree a eptors and some of their appli ations', Journal of

Computer and System S ien es 4, 406{451.

Farmer, W. (1991), `Simple se ond-order languages for whi h uni� ation is

unde idable', Theoreti al Computer S ien e 87, 25{41.

Gallier, J., Narendran, P., Plaisted, D. & Snyder, W. (1988), Rigid E-

uni� ation is NP- omplete, in `Pro . IEEE Conferen e on Logi in Com-

puter S ien e (LICS)', IEEE Computer So iety Press, pp. 338{346.

Gallier, J., Raatz, S. & Snyder, W. (1987), Theorem proving using rigid E-

uni� ation: Equational matings, in `Pro . IEEE Conferen e on Logi in

Computer S ien e (LICS)', IEEE Computer So iety Press, pp. 338{346.

Ganzinger, H., Ja quemard, F. & Veanes, M. (1998), Rigid rea hability, in

J. Hsiang & A. Ohori, eds, `Advan es in Computing S ien e { ASIAN'98,

4th Asian Computing S ien e Conferen e, Manila, The Philippines, De-

ember 1998, Pro eedings', Vol. 1538 of Le ture Notes in Computer S i-

en e, Springer Verlag, pp. 4{21.

Goubault, J. (1994), Rigid

~

E-uni�ability is DEXPTIME- omplete, in `Pro .

IEEE Conferen e on Logi in Computer S ien e (LICS)', IEEE Com-

puter So iety Press.

Gurevi h, Y. & Voronkov, A. (1997), Monadi simultaneous rigid E-

uni� ation and related problems, in P. Degano, R. Corrieri &

16

A. Mar hetti-Spa amella, eds, `Automata, Languages and Program-

ming, 24th International Colloquium, ICALP'97', Vol. 1256 of Le ture

Notes in Computer S ien e, Springer Verlag, pp. 154{165.

Kozen, D. (1977), Lower bounds for natural proof systems, in `Pro .

18th IEEE Symposium on Foundations of Computer S ien e (FOCS)',

pp. 254{266.

Levy, J. (1998), De idable and unde idable se ond-order uni� ation prob-

lems, in T. Nipkow, ed., `Rewriting Te hniques and Appli ations, 9th

International Conferen e, RTA-98, Tsukuba, Japan, Mar h/April 1998,

Pro eedings', Vol. 1379 of Le ture Notes in Computer S ien e, Springer

Verlag, pp. 47{60.

Makanin, G. (1977), `The problem of solvability of equations in free semi-

groups',Mat. Sbornik (in Russian) 103(2), 147{236. English Translation

in Ameri an Mathemati al So . Translations (2), vol. 117, 1981.

Savit h, W. (1970), `Relationships between nondeterministi and determin-

isti tape omplexities', Journal of Computer and System S ien es

4(2), 177{192.

S hulz, K. (1990), Makanin's algorithm: Two improvements and a general-

ization, in K. S hulz, ed., `Pro eedings of the First International Work-

shop on Word Equations and Related Topi s, T�ubingen', number 572

in `Le ture Notes in Computer S ien e'.

That her, J. &Wright, J. (1968), `Generalized �nite automata theory with an

appli ation to a de ision problem of se ond-order logi ', Mathemati al

Systems Theory 2(1), 57{81.

Veanes, M. (1998), The relation between se ond-order uni� ation and simul-

taneous rigid E-uni� ation, in `Pro . Thirteenth Annual IEEE Sympo-

sium on Logi in Computer S ien e, June 21{24, 1998, Indianapolis,

Indiana (LICS'98)', IEEE Computer So iety Press, pp. 264{275.

Voronkov, A. (1998), `Simultaneous rigid E-uni� ation and other de ision

problems related to Herbrand's theorem', Theoreti al Computer S ien e.

Arti le after invited talk at LFCS'97.

17

���

�

��

k

I N F O R M A T I K

Below you �nd a list of the most re ent te hni al reports of the Max-Plan k-Institut f�ur Informatik. They

are available by anonymous ftp from ftp.mpi-sb.mpg.de under the dire tory pub/papers/reports. Most

of the reports are also a essible via WWW using the URL http://www.mpi-sb.mpg.de. If you have any

questions on erning ftp or WWW a ess, please onta t reports�mpi-sb.mpg.de. Paper opies (whi h

are not ne essarily free of harge) an be ordered either by regular mail or by e-mail at the address below.

Max-Plan k-Institut f�ur Informatik

Library

attn. Birgit Hofmann

Im Stadtwald

D-66123 Saarbr�u ken

GERMANY

e-mail: library�mpi-sb.mpg.de

MPI-I-1999-2-003 U. Waldmann Can ellative Superposition De ides the Theory of

Divisible Torsion-Free Abelian Groups

MPI-I-1999-2-001 W. Charatonik Automata on DAG Representations of Finite Trees

MPI-I-1999-1-001 A. Crauser, P. Ferragina A Theoreti al and Experimental Study on the

Constru tion of SuÆx Arrays in External Memory

MPI-I-98-2-018 F. Eisenbrand A Note on the Membership Problem for the First

Elementary Closure of a Polyhedron

MPI-I-98-2-017 M. Tzakova, P. Bla kburn Hybridizing Con ept Languages

MPI-I-98-2-014 Y. Gurevi h, M. Veanes Partisan Corroboration, and Shifted Pairing

MPI-I-98-2-013 H. Ganzinger, F. Ja quemard, M. Veanes Rigid Rea hability

MPI-I-98-2-012 G. Delzanno, A. Podelski Model Che king In�nite-state Systems in CLP

MPI-I-98-2-011 A. Degtyarev, A. Voronkov Equality Reasoning in Sequent-Based Cal uli

MPI-I-98-2-010 S. Ramangalahy Strategies for Conforman e Testing

MPI-I-98-2-009 S. Vorobyov The Unde idability of the First-Order Theories of One

Step Rewriting in Linear Canoni al Systems

MPI-I-98-2-008 S. Vorobyov AE-Equational theory of ontext uni� ation is

Co-RE-Hard

MPI-I-98-2-007 S. Vorobyov The Most Nonelementary Theory (A Dire t Lower

Bound Proof)

MPI-I-98-2-006 P. Bla kburn, M. Tzakova Hybrid Languages and Temporal Logi

MPI-I-98-2-005 M. Veanes The Relation Between Se ond-Order Uni� ation and

Simultaneous Rigid E-Uni� ation

MPI-I-98-2-004 S. Vorobyov Satis�ability of Fun tional+Re ord Subtype

Constraints is NP-Hard

MPI-I-98-2-003 R.A. S hmidt E-Uni� ation for Subsystems of S4

MPI-I-98-2-002 F. Jaquemard, C. Meyer, C. Weidenba h Uni� ation in Extensions of Shallow Equational

Theories

MPI-I-98-1-031 G.W. Klau, P. Mutzel Optimal Compa tion of Orthogonal Grid Drawings

MPI-I-98-1-030 H. Br�onniman, L. Kettner, S. S hirra,

R. Veltkamp

Appli ations of the Generi Programming Paradigm in

the Design of CGAL

MPI-I-98-1-029 P. Mutzel, R. Weiskir her Optimizing Over All Combinatorial Embeddings of a

Planar Graph

MPI-I-98-1-028 A. Crauser, K. Mehlhorn, E. Althaus,

K. Brengel, T. Bu hheit, J. Keller,

H. Krone, O. Lambert, R. S hulte,

S. Thiel, M. Westphal, R. Wirth

On the performan e of LEDA-SM

MPI-I-98-1-027 C. Burnikel Delaunay Graphs by Divide and Conquer

MPI-I-98-1-026 K. Jansen, L. Porkolab Improved Approximation S hemes for S heduling

Unrelated Parallel Ma hines

MPI-I-98-1-025 K. Jansen, L. Porkolab Linear-time Approximation S hemes for S heduling

Malleable Parallel Tasks

MPI-I-98-1-024 S. Burkhardt, A. Crauser, P. Ferragina,

H. Lenhof, E. Rivals, M. Vingron

q-gram Based Database Sear hing Using a SuÆx Array

(QUASAR)

MPI-I-98-1-023 C. Burnikel Rational Points on Cir les

MPI-I-98-1-022 C. Burnikel, J. Ziegler Fast Re ursive Division

MPI-I-98-1-021 S. Albers, G. S hmidt S heduling with Unexpe ted Ma hine Breakdowns

MPI-I-98-1-020 C. R�ub On Walla e's Method for the Generation of Normal

Variates

MPI-I-98-1-019 2nd Workshop on Algorithm Engineering WAE '98 -

Pro eedings

MPI-I-98-1-018 D. Dubhashi, D. Ranjan On Positive In uen e and Negative Dependen e

MPI-I-98-1-017 A. Crauser, P. Ferragina, K. Mehlhorn,

U. Meyer, E. Ramos

Randomized External-Memory Algorithms for Some

Geometri Problems

MPI-I-98-1-016 P. Krysta, K. Lory�s New Approximation Algorithms for the A hromati

Number

MPI-I-98-1-015 M.R. Henzinger, S. Leonardi S heduling Multi asts on Unit-Capa ity Trees and

Meshes

MPI-I-98-1-014 U. Meyer, J.F. Sibeyn Time-Independent Gossiping on Full-Port Tori

MPI-I-98-1-013 G.W. Klau, P. Mutzel Quasi-Orthogonal Drawing of Planar Graphs

MPI-I-98-1-012 S. Mahajan, E.A. Ramos,

K.V. Subrahmanyam

Solving some dis repan y problems in NC*

MPI-I-98-1-011 G.N. Frederi kson, R. Solis-Oba Robustness analysis in ombinatorial optimization

MPI-I-98-1-010 R. Solis-Oba 2-Approximation algorithm for �nding a spanning tree

with maximum number of leaves

MPI-I-98-1-009 D. Frigioni, A. Mar hetti-Spa amela,

U. Nanni

Fully dynami shortest paths and negative y le

dete tion on diagraphs with Arbitrary Ar Weights

MPI-I-98-1-008 M. J�unger, S. Leipert, P. Mutzel A Note on Computing a Maximal Planar Subgraph

using PQ-Trees

MPI-I-98-1-007 A. Fabri, G. Giezeman, L. Kettner,

S. S hirra, S. S h�onherr

On the Design of CGAL, the Computational Geometry

Algorithms Library

MPI-I-98-1-006 K. Jansen A new hara terization for parity graphs and a oloring

problem with osts

MPI-I-98-1-005 K. Jansen The mutual ex lusion s heduling problem for

permutation and omparability graphs

MPI-I-98-1-004 S. S hirra Robustness and Pre ision Issues in Geometri

Computation

MPI-I-98-1-003 S. S hirra Parameterized Implementations of Classi al Planar

Convex Hull Algorithms and Extreme Point

Compuations

MPI-I-98-1-002 G.S. Brodal, M.C. Pinotti Comparator Networks for Binary Heap Constru tion

MPI-I-98-1-001 T. Hagerup Simpler and Faster Stati AC

0

Di tionaries

MPI-I-97-2-012 L. Ba hmair, H. Ganzinger, A. Voronkov Elimination of Equality via Transformation with

Ordering Constraints

MPI-I-97-2-011 L. Ba hmair, H. Ganzinger Stri t Basi Superposition and Chaining

MPI-I-97-2-010 S. Vorobyov, A. Voronkov Complexity of Nonre ursive Logi Programs with

Complex Values

MPI-I-97-2-009 A. Bo kmayr, F. Eisenbrand On the Chv�atal Rank of Polytopes in the 0/1 Cube

MPI-I-97-2-008 A. Bo kmayr, T. Kasper A Unifying Framework for Integer and Finite Domain

Constraint Programming

MPI-I-97-2-007 P. Bla kburn, M. Tzakova Two Hybrid Logi s