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I N F O R M A T I K
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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
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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'.
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appli ation to a de ision problem of se ond-order logi ', Mathemati al
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taneous rigid E-uni� ation, in `Pro . Thirteenth Annual IEEE Sympo-
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problems related to Herbrand's theorem', Theoreti al Computer S ien e.
Arti le after invited talk at LFCS'97.
17
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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