Decidable and undecidable fragments of first-order branching ...

Decidable and undecidable fragments of first-order branching
temporal logics
Ian Hodkinson, Frank Wolter, and Michael Zakharyaschev
(e-mails:ÑÓºººÙ¸ÛÓÐØÖÒÓÖÑغÙÒ¹ÐÔÞº¸
Department of Computing, Imperial College
180 Queen’s
ÑÞ׺кºÙµ
Gate, London SW7 2BZ, U.K.;
Institut für Informatik, Universität Leipzig
Augustus-Platz 10-11, 04109 Leipzig, Germany;
Department of Computer Science, King’s College
ÓÒØØÙØÓÖÅÐÖÝ×Ú
Strand, London WC2R 2LS, U.K.
Ü´·µ¾¼¾½ ÑÞ׺кºÙ ÈÓÒ´·µ¾¼¾¾ ÔØÓÓÑÔÙØÖËÒ¸ÃÒ³×ÓÐиËØÖÒ¸ÄÓÒÓÒϾʾÄ˸ÍÃ
Abstract
In this paper we analyze the decision problem for fragments of first-order extensions of
branching time temporal logics such as computational tree logicsÌÄandÌÄ£or Prior’s Ockhamist
logic of historical necessity. On the one hand, we show that the one-variable fragments
of logics like first-order C T L£—such as the product of propositional C T L£with simple propositional
modal logicË, or even the one-variable bundled first-order temporal logic with sole
temporal operator ‘some time in the future’—are undecidable. On the other hand, it is proved
that by restricting applications of first-order quantifiers to state (i.e., path-independent) formulas,
and applications of temporal operators and path quantifiers to formulas with at most one free variable,
we can obtain decidable fragments. The same arguments show decidability of ‘non-local’
propositionalÌÄ£, in which truth values of propositional atoms depend on the history as well
as the current time. The positive decidability results can serve as a unifying framework for devising
expressive and effective time-dependent knowledge representation formalisms, e.g., temporal
description or spatio-temporal logics.
0

1 Introduction
Temporal logics of branching time originate in philosophy and computer science. In philosophy,
they formalize reasoning about indeterminate future; see e.g., [18, 19, 5, 20]. In computer science,
they are used to reason about state transition systems (computations of reactive systems, agents in an
unpredictable environment, etc.); see e.g., [6, 8, 16]. Families of different logics have been constructed
in both disciplines, and in both cases most of the logics are propositional (the only exceptions we know
of are [4, 22, 23]).
The main aim of this paper is to investigate the computational behavior—at least on the level of
decidability—of first-order branching temporal logics (FOBTLs).
The starting point of our investigation lies in recent work [14] on first-order linear temporal logic
(FOLTL), which showed that although very weak fragments (say, the monadic two-variable fragment)
of standard FOLTLs can be highly undecidable, by restricting applications of temporal operators
to formulas with at most one free variable, we obtain a fragment with much better computational
behavior. This ‘monodic’ fragment can be axiomatized in a natural way [25] and becomes decidable
if its ‘purely first-order part’ is restricted to a decidable fragment of first-order logic.
Of course, FOBTLs inherit bad computational properties of FOLTLs. For example, the guarded
L£
two-variable fragment of most FOBTLs is not even recursively enumerable; cf. [14]. Unfortunately
(and to our surprise) it turned out that the situation is much worse. As will be shown in Section 3,
the one-variable fragment of first-order CT L£is undecidable. This is a monodic fragment with very
simple decidable first-order part; it can be reformulated as the product of propositional CT L£with
propositionalË. Even the one-variable fragment of the bundled FOBTL [5] with sole temporal
operator ‘now or some time in the future’ is not decidable.
These ‘negative’ results become less surprising when we realize that first-order (bundled) CT
allows a form of quantification in three dimensions, something that is known to be associated with
undecidability [17, 13]. A natural way to limit the interaction between the three dimensions is to restrict
applications of first-order quantifiers to state (i.e., path-independent) formulas, and applications
of temporal operators and path quantifiers to formulas with at most one free variable. The resulting
fragment contains full propositional CT L£and full first-order logic. In this paper, we show in a
similar way to FOLTL that restricting its first-order part to a decidable first-order fragment yields a
decidable monodic fragment of first-order CT L£. A number of ‘positive’ results of this sort will be
proved in Section 4.
These results are interesting per se, and also because they identify a limit beyond which monodic
fragments of temporal logics are no longer decidable. More practically, the positive results serve as
a unifying framework for devising expressive and effective time-dependent knowledge representation
formalisms, say, temporal description or spatio-temporal logics; for details see [15].
2 Syntax and semantics
There are a number of approaches to constructing temporal logics based on the branching time
paradigm; see, e.g., [5, 7, 26, 10]. Many of the resulting languages are fragments of the language
we call here Q P CT L£, quantified CT L£with past operators. It is obtained in the standard way by
extending the language P L of classical predicate logic (without equality or function symbols) with
the binary temporal operatorsË(Since) andÍ(Until) and the path universal quantifier.
The intended models for Q P CT L£are based on ω-trees. Recall that a tree is a strict partial order
Wcontaining a unique-minimal point (the root of) and such that for all w¾W, the set
1

0
I´wµ
0c
v¾W : vwis linearly ordered by. A full branch ofis a maximal linearly-ordered subset of
W. An ω-tree is a tree whose full branches are all order-isomorphic toÆ.
Q P CT L£(as well as its sublanguages to be introduced below) is interpreted in structures of the
formÅÅHDI«, whereWis an ω-tree, H is a set (bundle) of full branches ofwith
ËHW, D is a non-empty I´wµ
I´wµ I´wµ
set called the domain ofÅ, and I is a function associating I´wµ
with I´vµ
every
moment of time w¾W a first-order P L-structure
I´wµDP the state ofÅat moment w. (Here, the Pi
are predicates on D interpreting the predicate symbols
P i of P L and the ci
are elements of D interpreting its constants.) We require that c c i i for
any wv¾W—i.e., that constants are ‘rigid’. The branches in the bundle H are called histories. If H
contains all full branches of, we say thatÅis a full tree model, or simply a tree model.
An assignment in D is a functionfrom the set of individual variables to D. (So assignments too
are ‘rigid’. We extendto constants via´cµcI´wµfor any w¾W . The truth relation´Åhwµϕ,
for w¾h¾H (or simply´hwµϕifÅis understood), is defined as follows:
¯´hwµP i´y 1yµiff I´wµP i´´y 1µ´yµµ, where the y i are variables or constants
(this is the so-called ‘local’ approach in that there is no dependence on h),
¯´hwµxψ iff´hwµψfor every assignmentin D that may differ fromonly on x,
¯´hwµχËψ iff there is vw such that´hvµψ and´huµχ for every u¾´vwµ,
where´vwµu¾W : vuw,
¯´hwµχÍψ iff there is v¾h such that vw,´hvµψand´huµχfor
ϕϕ¿Fϕ­ϕÍϕϕϕ
every u¾´wvµ, ¯´hwµψ iff´h¼wµψfor all h¼¾H such that w¾h¼,
plus the usual clauses for the Booleans. For a formula ϕ´¯xµand a tuple ā of elements of D, we write
´Åhwµϕ´āµif´Åhwµϕwhere´¯xµā. We will use the following standard abbreviations:
¿FϕÍϕ¾Fϕ¿Fϕ¾·F ϕϕ¾Fϕ¿·F
Thus,¿F can be read as ‘some time in the future’,¾·F as ‘from now on’,­as ‘at the next moment’
or ‘tomorrow’, andas ‘there exists a history in H ’.
We will also be considering the following sublanguages of Q P CT L£:
¯Q CT L£—that is, Q P CT L£without the past temporal operatorË;
¯Q CT L£F—that is, Q CT L£in which the binary operatorÍis replaced by¾·F ;
¯Q P CT L—the fragment of Q P CT L£in which temporal operators occur only in the form
´ψ 1Íψ 2µ,´ψ 1Ëψ 2µ,´ψ 1Íψ 2µ, or´ψ 1Ëψ 2µ.
To introduce one more fragment, Q P CT L s , we need the definition of state and path formulas:
¯all formulas without path quantifiers and temporal operators are state formulas;
¯the set of state formulas is closed under the Booleans, path and first-order quantifications;
¯every state formula is a path formula;
2

¯the set of path formulas is closed under the Booleans and the temporal operators;
¯if ψ is a path formula, thenψ is a state formula.
So for a state formula ψ, whether´hwµψdoes not depend on h. Now, Q P CT L s consists of
all state formulas in Q P CT L£. The main difference between full Q P CT L£and Q P CT L s is that
first-order quantifiers in Q P CT L s can be applied only to formulas which are history-independent.
(That Q P CT L s only contains state formulas is not important for decidability, since a path formula
ϕ is satisfiable
¾Fx´ÓÖÖØÑ´xµ­ÐÚÖØÑ´xµµ ¾Fx´ÓÖÖØÑ´xµ­ÐÚÖØÑ´xµµ x¾F´ÓÖÖØÑ´xµ­ÐÚÖØÑ´xµµ iff the state formulaϕ is satisfiable.) It should
be clear that
¢ Q P CT L
¢
s contains
propositional P CT L£and Q P CT L. As examples, we give three formulas in the various fragments,
trying to express that every ordered item is delivered in one day.
Q P CT L£Q P CT L s Q P CT L
For any of the languages L introduced above, denote byÄ(respectively,Ä) the set of all L-
formulas that are true at all points in all histories under every assignment in every bundled (respectively,
full) tree model. Such formulas are said to be valid, and a formula is satisfiable if its negation
is not valid. Thus,ÉÈÌÄ£is the set of Q P CT L£-formulas valid in bundled tree models, while
ÉÈÌÄ£is the set of Q P CT L£-formulas valid in all tree models. The logicÉÈÌÄ£is the firstorder
version of the bundled Ockhamist logic of historical necessity; cf. e.g. [10].
Q P CT L£-satisfiability in bundled tree models can be reduced to satisfiability in full tree models.
Indeed, given a Q P CT L£-formula ϕ, we take a propositional variable q not occurring in ϕ and denote
by ϕthe result of replacing each subformula of ϕ of the formψ by´¿F¾Fqψµ. Note that if ϕ
is in Q P CT L s then so is ϕ(however, ϕ¾QP CT L).
LEMMA 1. ϕ is satisfiable in a bundled tree model iff¿F¾Fqϕis satisfiable in a full tree model.
Proof. The implication´´µis easily seen. We prove´µµ. Using a Löwenheim–Skolem argument
(cf. [5]), we may assume ϕ to be satisfied in a modelÅwith a countable bundle H . We assume that
H is infinite, leaving the (easy) other case to the reader. Let h 0h 1be an enumeration of H . We
convertÅinto a full tree modelņ and define a truth-relation for q in it inductively as follows: Put
´h 0wµq for all w¾h 0 . Suppose we have already defined truth of q in´h iwµ, for all in. Consider
h n·1. There must be a w¾h n·1 such that the distance from w to each h i , in·1, is2 (the distance
is the length of the shortest path from w to a point in h i ). Then we put´h n·1w¼µq if w¼w and
w¼¾h n·1.
Say that a full branch h ofÅis marked if there is w¾h such that´hw¼µq for all w¼w,
w¼¾h. One can easily see that h is marked iff h¾H. In particular, if h¾H and for each n, t n is the
least element of hÒËmn h m , then´ht nµq andt 0t 1is infinite, so h is not marked. Now one
can prove by induction that for every subformula ψ of ϕ and every´hwµ, we have´Åhwµψ iff
´Å†hwµψ. It follows that¿F¾Fqϕis satisfied inņ .
3 Undecidable fragments
The following theorems indicate some limits beyond which one cannot hope to find decidable fragments
of first-order temporal logics.
Given a first-order temporal language L andω, we denote by Lthe-variable fragment of L
(i.e., every formula in Lcontains at mostdistinct individual variables). And by L mo we denote the
3

monadic fragment of L (i.e., the set of formulas which contain only unary predicates and propositional
variables). Both the two-variable and the monadic fragments of classical (non-temporal) first-order
logic are known to be decidable and have the finite model property; see [3] and references therein.
The computational behavior of the corresponding fragments of first-order temporal logics turns out to
be quite different. From linear time results (Theorem 2 of [14]) we easily obtain:
THEOREM 2. For any of the FOBTLsÄintroduced above,ÄL 2L mo is not recursively enumerable.
Another well-behaved fragment of classical predicate logic is the guarded fragment of [1]. The
corresponding fragment T GF of FOBTL is obtained by restricting the Q P CT L£quantification
formation rule to:
¯if xy are tuples of variables, G´xyµis an atomic formula, ϕ´xyµ¾TGF , and every free
variable occurring in ϕ´xyµoccurs in G´xyµas well, theny´G´xyµϕ´xyµµis in T GF .
The set T GF is called the guarded fragment of FOBTL. Again in contrast to the case of classical
predicate logic, we have the following consequence of Theorem 73 of [14]:
THEOREM 3. For any of the FOBTLsÄabove,ÄL 2T GF is not recursively enumerable.
The main result of this section is the following:
THEOREM 4. The one-variable fragments of the logicsÉÌÄ£F andÉÌÄ£F are undecidable. Hence,
so are the one-variable fragments ofÉÈÌÄ£,ÉÌÄ£,ÉÈÌÄ£, andÉÌÄ£.
Proof. We only considerÉÌÄ£F ; the other case follows from Lemma 1. The proof involves a
rather indirect reduction of the following tiling problem. An instance is a finite set T of square tiles,
the edges of each τ¾T being colored Left´τµRight´τµUp´τµDown´τµ(see Fig. 2 in Appendix). 1
T is a yes-instance of the problem iff for each τ¾T there is a T -tiling of¢that uses τ—a map
f :¢T with Right´f´ijµµLeft´f´i·1jµµ, Up´f´ijµµDown´f´ij·1µµfor all ij¾,
and (without loss of generality) f´00µτ. It is well known that this tiling problem is undecidable—
see, e.g., [2].
To code this problem intoÉÌÄ£F, we go via a construction originating in algebraic logic, which
has already been used to prove undecidability of 3-dimensional modal logics [13]. It was shown in
[12] that there is no algorithm to decide whether a finite Tarskian relation algebra representable. We
do not wish to assume any knowledge of algebraic logic here, so we will state some consequences of
this result without using terms from that field. Given a tiling instance T , we can effectively construct
a certain finite signature L´Tµof binary relation symbols, and a set satisfying´pqrµ¾
CL´Tµ3
Cµ´qrpµ¾C (and with further properties to be explained later). We call C the set of consistent
triples. Results in [12] imply that if T is a yes-instance then there exists a countable L´Tµ-structure
N satisfying:
Ïp¾L´Tµp´xyµÏq¾L´TµÒpq´xyµ¡Îp¾L´Tµxy ¯xy p´xyµ,
Ï´pqrµ¾C ¯xy p´xyµ¡°z q´yzµr´zxµ¡¡, for all qr¾L´Tµ.
Thus, each pair of points in N is labelled by a unique p¾L´Tµ; only consistently-labelled triangles
occur in N, but these must occur wherever possible. We will write a one-variable Q CT L£F-sentence
ϕ T , essentially the Yankov–Fine formula of a frame derived from C, that has a bundled model iff T
is a yes-instance. This will establish undecidability ofÉÌÄ£F.
1 Strictly, we also require that there is a ‘white’ tile in T whose sides are all colored white; no other tile has any white
edges. This is for technical reasons in [12] not relevant here.
4

In Q CT L£F, truth values of atomic formulas are history-independent (cf. Section 2). This is
inconvenient here, so we temporarily drop this ‘locality’ restriction. We introduce three unary relation
symbols p 0p 1p 2 for each p¾L´Tµ. Let ϕ T be the conjunction of
Ïp ¾·Fx p 0´xµÏqp q 0´xµ¡Îp¾L´Tµ¿·Fx p 0´xµand
pq¾L´Tµ qr¾L´Tµ pr¾L´Tµ
similarly for p 1p 2
¾·Fx ´Ï´pqrµ¾C p 0´xµµ°x´q 1´xµr 2´xµµ¡for all
¾·Fx ´Ï´pqrµ¾C q 1´xµµ°¿·F´r 2´xµp 0´xµµ¡for all
¾·Fx ´Ï´pqrµ¾C r 2´xµµ°´p 0´xµq 1´xµµ¡for all
If T is a yes-instance, take N as above. We identify (notationally) N with its domain. Choose an
ω-treeWwith a countably infinite set H of full branches such thatËHW. Let π 0 be the
identity map on N. Choose µ : ωN such that µ 1´aµis infinite for all a¾N, and define π 1 : WN
by π 1´wµµ´ht´wµµ, where ht´wµv¾W : vw. 1
Choose π 2 : HN such thatË´π
for all a¾N. Now we can define values for the p i j in a modelÆwith domain N, underlying tree,
and bundle H , by pulling back from N. For a¾N, h¾H,w¾h, and
1´wµµ 0´aµµ 2´hµµ
pqr¾L´Tµ, we let
´Æhwµp 0´aµiff Np´π 1´wµπ ´Æhwµq 1´aµiff Nq´π 2´hµπ ´Æhwµr 2´aµiff Nr´π 0´aµπ It can be checked that ϕ T is true inÆat the root.
We now show how to codeÆinto an official Q CT L£F-model with ‘local’ history-independent
atoms. We exploit the fact that the values of the atoms inÆdepend on only two coordinates. r 2
depends on aw, which is acceptable already. p 0 depends on wh (actually, on π 2´hµ) but not a; we
can express this by coding h as π 2´hµ, so reducing the dependence to w and a domain point. And q 1
depends on ha but not w; by regarding q 1 as true at ha iff it is in fact true at aw for cofinitely many
w¾h, we again reduce the dependence to wa.
In detail, we translate ϕ T into a Q CT L£F -sentenceϕ T of the signature of ϕ T plus a unary relation
symbol Q. For pqr¾L´Tµ, we define formulas
2´xµ 1´xµ Q´xµµ p 0´xµx´p 0´xµ¿·F¾·F
q 1´xµ¿·F¾·F q
r 2´xµr and letϕ T be the result of replacing each subformula p 0´xµof ϕ T byp 0´xµ, and similarly for q 1r 2 .
We form a Q CT L£F-structureÅwith the same tree, bundle, and domain asÆ. Enumerate H as
h 0h 1, and for nω let h nh nÒËmn h m , a cofinite subset of h n . For h¾H, w¾h, a¾N,
and pqr¾L´Tµ, define
iff aπ 2´hµ,
2´aµµW
Q´aµ ¯´ÅhwµQ´aµiff there is g¾H with w¾g and aπ 2´gµ—so´Åhwµ¿·F¾·F ¯´Åhwµp 0´aµiff there is g¾H with w¾g,´Ægwµp ¯´Åhwµq 1´aµiff there is g¾H with w¾g and´Ægwµq ¯´Åhwµr 2´aµiff´Æhwµr 2´aµ.
0´aµ, and π 2´gµa,
1´aµ,
These values do not depend on h, soÅis a bona fide Q CT L£F -structure. It can now be checked that
´Åhwµp 0´aµiff´Æhwµp 0´aµ, and similarly forq 1r . Hence,ϕ T is true inÅ.
5
2

For the converse,
1
we
2
will need more details about L´Tµand C from [12]. There is a ‘converse’
map p˘p defined on L´Tµ, with ˘ppfor all p. L´Tµcontains a relation symbol τ for each tile
τ¾T (we identify the two, so that TL´Tµ), the converse ˘τ of each tile τ, and also relation symbols
012·1 1 1·1 1 201100220 (and
iµ˘
others that do not concern us here), all distinct.
Each p¾L´Tµhas a start index st´pµand an end index end´pµ. The start and end indices ofi j are
ij, respectively; the start and end indices ofi·1 i , and 1 i are i, and the tiles have start index 1 and
end
¯´1001
index 2. We have g˘
i jg ji , ˘ii, and´·1 1 i . C satisfies:
¯If´pqrµ¾C then end´pµst´qµ, end´qµst´rµ, end´rµst´pµ, and´qrpµ´˘r˘q˘pµ¾C.
¯For all pq¾L´Tµ, p˘q iff´pqiµ¾C for some i3.
¯For all p¾L´Tµ,´2001pµ¾C iff p¾T .
1 1µ´2002·1 2µ¾C.
¯for τυ¾T ,´ῠ·1 1τµ¾C iff Right´υµLeft´τµ, and´τ 1 2ῠµ¾C iff Down´τµUp´υµ.
Now assume thatϕ T is true (at the root, say) in some Q CT L£F-modelÅwith domain D, tree
W, and bundle H . We will sketch a proof that T is a yes-instance. Let τ 00¾T be given; we
will read off fromÅatiling with τ 00 at the origin. As notation, for pqr¾L´Tµ, a¾D, h¾H,and
w¾h, we write´awhµpqr iff´Åhwµp 0´aµq 1´aµr 2´aµIt follows from the form ofp 0
and of ϕ T that
¯for all awh, there are unique pqr¾L´Tµwith´awhµpqr; moreover,´pqrµ¾C,
¯for all´pqrµ¾C there are ahw with´awhµpqr,
¯whenever´awhµpqr and´a¼whµp¼q¼r¼, we have pp¼,
0hµ
¯whenever´awhµpqr and´pq¼r¼µ¾C, there is a¼¾D with´a¼whµpq¼r¼.
Analogous facts hold for q 1r 2 . We use these facts below without explicit mention.
Asϕ T is true inÅ, and´2001τ00µ¾C, there are x 0¾D, h¾H, and y 0¾h with´x 0y 2001τ 00 . From this, and since´20022µ¾C, there must be u 0¾D with´u 0y 0hµ20022.
Similarly,´10011µ¾C, so there is v 0¾h with y 0v 0 and´x 0v 0hµ10011. In this way, by
induction, for each i¾we can find x iu i¾D and y 0y iv i¾h with
and´x i·1v ihµ1001 1 1 ,
¯´u iy ihµ20022 and´u iy i·1hµ2002·1 2 .
See Fig. 3 in Appendix.
Consider now´x iy jhµ(ij¾). We have´x iy jhµpqr for some unique´pqrµ¾C. Since
´u jy jhµ20022, we have p20. Since´x iv ihµ10011, we have q01. By the properties
of C, the only possibility for r is a tile, say τ i j .
It remains to show that´ijµτi j is a tiling of¢(with τ 00 at the origin of course). Take
ij¾; we check that Right´τi jµLeft´τi·1jµ. We will use the points x ix i·1v iy j , and one more
point t¾h. Let m be whichever of v iy j has greatest height in h. Since´x imhµp01r for some
pr, there must exist tm with t¾h and´x ithµ10011. Since´111µ¾C, there is g¾H
with t¾g and´x itgµ111. Note that v iy j¾g.
We consider some triples formed from the above with h and g (see Fig. 4 in Appendix). First,
we show that´x iv igµ111. Let´x iv igµpqr, say, for unique´pqrµ¾C. We know that
6
¯´x iv ihµ10011

jgµ
Also,´x iv ihµ10011, so r1 as well. So we must have p1
as required. In the same way we can successively show that´x i·1v igµ1·1 1 1 1 ,´x iy i j·1 1 τ i·1j . Therefore,´˘τ i j·1 1τi·1jµ¾C, and the cited properties of
C yield Left´τi·1jµRight´τi jµ, as required. That Up´τi jµDown´τij·1µis shown similarly.
We have shown that T is a yes-instance of the cited tiling problem iffϕ T has a bundled Q CT L£F -
model. The undecidability of the tiling problem now yields undecidability of satisfiability ofÉÌÄ£F -
sentences.
´x itgµ111, so q1.
˘τ i j1τi j , and´x i·1y jgµ˘τ
REMARK 5. A more general version of this result can be formulated in terms of products of propositional
modal logics; see [9] and references therein. Namely, there is no decidable set L of formulas
such thatâÌÄ£FLË¢ÌÄ£F . The same holds for the propositional bundled case as well.
4 Decidable fragments
The ‘negative’ results of Section 3 can be ‘explained’ by the fact that all the undecidable fragments
there are in a sense ‘three-dimensional’, which is often a cause of bad computational properties. The
three-variable fragment of classical first-order logic is undecidable even without equality [17], and
products of three propositional modal logics are usually undecidable [13]. In Theorem 4 we also have
quantification in three dimensions: temporal operators, path quantifiers and the domain quantification.
A natural way to reduce the interaction between the dimensions is to restrict first-order quantification
to state formulas—that is to work with the language Q P CT L s —and to limit the scope of the
path quantifiers (hence also of the temporal operators) to formulas with1 free variable; cf. [14, 24].
DEFINITION 6 (MONODIC FORMULAS). Let Q P CT L
s
1
be the set of all Q P CT L s -formulas ϕ such
that any subformula of ϕ of the formψ, ψ 1Íψ 2 , or ψ 1Ëψ 2 has at most one free variable. Such
formulas ϕ will be called monodic and Q P CT L
s
1
the monodic fragment of Q P CT L s .
It should be clear from the definition that Q P CT L
s
1
contains full propositional CT L£and the
full first-order (non-temporal) language. The latter means, in particular, that the monodic fragments
of the logics under consideration are still undecidable.
The main aim of this section is to prove a satisfiability criterion for the monodic formulas (Theorem
8) and then apply it in order to obtain various decidable fragments of FOBTLs. As in [14], the
idea is to encode models in structures called quasimodels and then express the statement ‘there exists
a quasimodel satisfying a given monodic sentence’ as a monadic second-order sentence.
In what follows we assume that ϕ¾Q P CT L s 1
. Denote by sub n´ϕµthe closure under negation
of the set of state subformulas of formulas in ϕ containingn free variables; con´ϕµis the set of
constants in ϕ. Without loss of generality, we may identify ψ andψ, so sub n´ϕµis finite. For every
formula ψ´xµχ´xµin sub 1´ϕµwith a free variable x, we reserve a unary predicate P ψ´xµ, and for
every sentence ψχ in sub 0´ϕµwe fix a propositional variable p ψ . P ψ´xµand p ψ are called the
surrogates of ψ´xµand ψ, respectively. Given a state formula ψ, denote by ψ the result of replacing
all its maximal subformulas of the formχ with their surrogates. Thus, ψ contains neither temporal
operators nor path quantifiers at all—it is a P L-formula.
Let x be a variable not occurring in ϕ and let sub x´ϕµψxy: ψ´yµ¾sub 1´ϕµ. Define a type
for ϕ as a subset t of sub x´ϕ) such that ψχ¾t iff ψ¾t and χ¾t, for every ψχ¾sub x ϕ, and
ψ¾t iff ψ¾t, for every ψ¾sub x ϕ. Given a type t for ϕ and a constant c¾con´ϕµ, the pairtcis
called an indexed type for ϕ. We will not be distinguishing between a type t and the conjunctionÎt
of formulas in it.
7

DP0c0
Suppose that T is a set of types for ϕ, and T contc:c¾con´ϕµis a set of indexed types
such thatt :tc¾T conT . Then the pairTT conis called a state candidate for ϕ. Consider
a first-order P L-structure
(1)
and suppose that a¾D. The set t´aµψ¾sub x´ϕµ:ψa℄is clearly a type for ϕ. Say that
realizes a state candidateTT conif Tt´aµ:a¾Dand T conÅt´cµc«:c¾con´ϕµ.
A state candidate is realizable if somerealizes it.
LetWbe an ω-tree. A state function for ϕ overis a map f associating with each w¾W
a realizable state candidate f´wµT wT wfor con ϕ.
For every subformula ψ´yµof ϕ, every full branch h in, every w¾h, and every function r
mapping each w¾W to a type for ϕ, define inductively a ‘formula’ cond´ψhrwµ:
¯cond´ψhrwµis ‘ψxy¾r´wµ’ if ψ´yµis a state formula;
¯if ψ 1ψ 2 is not a state formula, cond´ψ 1ψ 2hrwµis ‘cond´ψ 1hrwµcond´ψ 2hrwµ’;
¯ifψ is not a state formula, cond´ψhrwµis ‘cond´ψhrwµ’;
¯cond´ψ 1Íψ 2hrwµis ‘vw´v¾hcond´ψ 2hrvµu¾´wvµcond´ψ 1hruµµ’;
¯cond´ψ 1Ëψ 2hrwµis ‘vw´cond´ψ 2hrvµu¾´vwµcond´ψ 1hruµµ’.
Since every Q P CT L s -formula is built from state formulas using only the Booleans and the temporal
operators, this is well-defined. Let f be a state function for ϕ overW, with f´wµT wT con
for w¾W . By a run in f we mean a function r from W into the setËw¾W T w such that
¯r´wµ¾T w , for all w¾W ,
¯for allψ¾sub x ϕ and w¾W, we haveψ¾r´wµiffh¿wcond´ψhrwµ.
We call f a quasimodel for ϕ overif the following conditions hold:
¯for each c¾con´ϕµ, the function r c defined by r c´wµt, fortc¾T con
w , w¾W, is a run in f ,
¯for all w¾W and t¾T w , there exists a run r in f such that r´wµt.
Say that ϕ is satisfied in f if there are w¾W and t¾T w such that ϕ¾t.
THEOREM 7. A Q P CT L s 1
-sentence ϕ is satisfiable in a full tree model based onWiff it is
satisfied in a quasimodel for ϕ over.
Proof. Similar to the proof of Theorem 14 in [14]. For´µµ, we construct a quasimodel from a model
in the obvious way. For´´µ, given a quasimodel f for ϕ overW, we take a cardinal κ and
form a set R of κ copies of each run in f . Since the language P L is countable and does not contain
equality, it follows from classical model theory that if κ is large enough then each f´wµis realized by
a modelw with domain R , and with tw´rµr´wµfor all r¾R. LetÅR´wµw¾W. We
may now check by induction that for all state subformulas ψ´¯xµof ϕ, all ¯r in R , and all hw, we have
wψā℄iff´Åhwµψā℄for any h¿w. It follows that ϕ is satisfied inÅ.
In the proof of the next theorem we show that the existence of a quasimodel satisfying ϕ can be
expressed as a monadic second-order formula, which yields the following satisfiability criterion:
w
8

THEOREM 8. Let LQ P CT L
s 1
and suppose that there is an algorithm that decides for any L-
sentence ϕ whether an arbitrarily-given state candidate is realizable. Then the satisfiability problem
for L-sentences in both bundled and full tree models
ψ´xµ
is decidable.
Proof. We consider only full tree models, leaving the bundled case
ψ´xµ
to Lemma 1. For each ψ¾sub x´ϕµ,
let R ψ be a unary predicate variable, and for each type t for ϕ, let
R R
χ t´xµψ¾t
ψ¾sub x´ϕµÒt
The formula χ t´xµsays that the R ψ´xµdefine the type t at x. For any subformula ψ´yµof ϕ, an
individual variable x, and a set variable h, define inductively a formula γ´ψhxµof monadic secondorder
logic by taking γ´ψhxµR ψ´xµif ψ is a state formula, and for non-state formulas taking:
¯γ´ψ 1ψ 2hxµγ´ψ 1hxµγ´ψ 2hxµ;
¯γ´ψhxµγ´ψhxµ;
ρx
¯γ´ψ 1Íψ 2hxµy¾h´xyγ´ψ ¯γ´ψ 1Ëψ 2hxµy´yxγ´ψ
con´xµ 2hyµz´xzyγ´ψ 1hzµµµ;
2hyµz´yzxγ´ψ
1hzµµµ.
β´hxµγ´ψhxµ¡
PTT χ t´xµ¡x x´ϕµRψ´xµ°h TT con¾Σ
s´xµ
ψ¾sub
s¼´xµ ψρx
t´xµ¡
P
ss¼P ϕ¾ËTx x
PTT con´xµ
s¼¾Σ
R PTT con´xµχ t´xµµ tc¾T con
PTT R ψ´ρχ
Denote by β´hxµamonadic second-order formula saying that h is a full branch containing x. Let Σ
be the set of all realizable state candidates for ϕ (Σ can be constructed effectively), and let P s (s¾Σ)
be a unary predicate variable. Then
says that the R ψ´xµdefine a run through realizable state candidates in Σ defined with the help of the
P s . Finally, we define a monadic second-order sentence σ ϕ by:
P
s¾Σ
sxs¾Σ
t¾T
TT c¾con´ϕµ
con´xµ
x´ϕµ
con¾Σ
ψ¾sub TT con¾Σ
TT con¾Σ
t¾T
ψ¾sub x´ϕµ
It is not hard to check thatσ ϕ iff ϕ is satisfied in a quasimodel over. It remains to recall
that the monadic second-order theory of countably branching trees is decidable (this can easily be
shown by reduction to the monadic second-order theory of two successor functions, which is decidable
[21]). A Löwenheim–Skolem–Tarski argument (see Theorem 14 of the Appendix) will show that any
satisfiable Q P CT L 1 s -formula has a model based on a countably branching tree.
As a consequence we obtain
9

THEOREM 9. The following fragments are decidable:
¯the two-variable fragment ofÉÈÌÄ£Q P CT L s 1
;
¯the two-variable fragment ofÉÈÌÄ£Q P CT L s 1
;
¯the monadic fragment ofÉÈÌÄ£Q P CT L s 1
;
¯the monadic fragment ofÉÈÌÄ£Q P CT L s 1
;
¯the guarded fragment ofÉÈÌÄ£Q P CT L s 1
;
¯the guarded fragment ofÉÈÌÄ£Q P CT L p 1 .
A similar construction yields the following theorem answering a question that has puzzled many
temporal logicians since [11]:
THEOREM 10. It is decidable whether an arbitrarily-given propositional P CT L£-formula is satisfiable
in a ‘non-local’ full tree model—where truth values of atoms may depend on the branch of
evaluation. The same holds for non-local bundled tree models.
5 Conclusion
This paper may be regarded as a beginning of systematic research into the computational behavior
of first-order branching time logics. We have obtained decidability results of two kinds. The most
striking bad news is that the one-variable fragments of logics containingÉÌÄ£F andÉÌÄ£F are
undecidable, which contrasts with the situation in many other modal and temporal first-order logics
(cf. [14, 24]). The good news is that there are still ways of obtaining decidable fragments with nontrivial
interaction between first-order quantifiers, path quantifiers and temporal operators. In this
paper, we considered the case when the first-order quantifiers are applied only to state formulas, while
the path-quantifiers and temporal operators are applicable to formulas with1 free variable.
Another way to obtain decidable fragments is to restrict further the formulas that can have free
variables. For example, using quasimodels and a mosaic technique one can prove a result similar to
Theorem 8 for bundled models and the fragment of Q CT L£in which­may be applied to formulas
with at most one free variable,Í,andare applicable only to sentences, and there are no restrictions
on first-order quantification (for a precise formulation see [15]).
Two other promising ways are to allow arbitrary first-order quantification, but to restrict applications
of path quantifiers to closed formulas and those of linear temporal operators to formulas with
1 free variable, and vice versa. However, they remain open for investigation.
Acknowledgements
The work of the first and third authors was partially supported by UK EPSRC grant GR/R45369/01
“Analysis and mechanisation of decidable first-order temporal logics” (the third author was also partially
supported by grant 99-01-00968 from the Russian Foundation for Basic Research). The work
of the second author was supported by Deutsche Forschungsgemeinschaft (DFG) grant Wo583/3-1.
We are grateful to S. Bauer, A. Kurucz, A. Rabinovich, M. Reynolds, C. Stirling, and A. Zanardo for
stimulating discussions and comments.
10

References
[1] H. Andréka, J. van Benthem, and I. Németi. Modal languages and bounded fragments of predicate
logic. J. Philosophical Logic, 27:217–274, 1998.
[2] R. Berger. The undecidability of the domino problem, volume 66 of Memoirs. Amer. Math. Soc.,
Providence, RI, 1966.
[3] E. Börger, E. Grädel, and Yu. Gurevich. The Classical Decision Problem. Perspectives in
Mathematical Logic. Springer, 1997.
[4] J. Burgess. The unreal future. Theoria, 44:157–179, 1978.
[5] J. Burgess. Logic and time. J. Symbolic Logic, 44:566–582, 1979.
[6] E. Clarke, E. Emerson. Synthesis of synchronization skeletons for branching time temporal
logic. In Proc. IBM Workshop on Logic of Programs, pages 52–71, Springer-Verlag, 1981.
[7] E. Emerson. Temporal and modal logic. In J. van Leeuwen, editor, Handbook of Theoretical
Computer Science, pages 996–1076, 1990.
[8] E. Emerson and J. Halpern. ‘Sometimes’ and ‘not never’ revisited: on branching versus linear
time. J. ACM, 33:151–178, 1979.
[9] D. Gabbay, A. Kurucz, F. Wolter, and M. Zakharyaschev. Many-Dimensional Modal Logics:
Theory and Applications. Elsevier, 2002. SeeØØÔ»»ÛÛۺ׺кºÙ»×Ø»ÑÞ.
[10] D. Gabbay, M. Reynolds, and M. Finger. Temporal Logic: Mathematical Foundations and
Computational Aspects, Volume 2. Oxford University Press, 2000.
[11] Y. Gurevich and S. Shelah. The decision problem for branching time logic. J. Symbolic Logic,
50:668–681, 1985.
[12] R. Hirsch and I. Hodkinson. Representability is not decidable for finite relation algebras. Trans.
Amer. Math. Soc., 353:1403–1425, 2001.
[13] R. Hirsch, I. Hodkinson, and A. Kurucz. On modal logics betweenââÃandˢˢË.
J. Symbolic Logic, 2001. (In press.)
[14] I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal
logics. Annals of Pure and Applied Logic, 106:85–134, 2000.
[15] I. Hodkinson, F. Wolter, and M. Zakharyaschev. Monodic fragments of first-order temporal
logics: 2000–2001 A.D. In R. Nieuwenhuis and A. Voronkov, editors, Logic for Programming,
Artificial Intelligence and Reasoning, number 2250 of LNAI, Springer, 2001, pages 1–23.
[16] R. Ladner and J. Reif. The logic of distributed protocols. In J. Halpern, editor, Theoretical
Aspects of Reasoning about Knowledge, pp.207–222, Morgan Kaufmann, 1986.
[17] R. Maddux. The equational theory of CA 3 is undecidable. J. Symbolic Logic, 45:311–316, 1980.
[18] A. Prior. Past, Present and Future. Oxford University Press, 1967.
11

[19] A. Prior. Now. Nous, 2:101–119, 1968.
[20] O. Øhrstrøm and P.F.V. Hasle. Temporal Logic: From Ancient Ideas to Artificial Intelligence.
volume 57 of Studies in Logic and Philosophy. Kluwer Academic Publishers, 1996.
[21] M. O. Rabin. Decidability of second order theories and automata on infinite trees. American
Mathematical Society Transactions, 141:1–35, 1969.
[22] S. Woelf. Combinations of tense and modality for predicate logic. J. Philosophical Logic,
28:371–398, 1999.
[23] M. Wooldridge and M. Fisher. A first-order branching time logic of mutli-agent systems. In
B. Neumann, editor, Proc. 10th European Conference on Artificial
SeeØØÔ»»ÛÛۺ׺кºÙ»
Intelligence (ECAI92), pages
234 – 238, Chichester, 1992. John Wiley and Sons.
[24] F. Wolter and M. Zakharyaschev. Decidable fragments of first-order modal logics. J. Symbolic
Logic, 66:1415–1438, 2001.
[25] F. Wolter and M. Zakharyaschev. Axiomatizing the monodic fragment of first-order temporal
logic. Annals of Pure and Applied Logic (In print.)
×Ø»ÑÞ.
[26] A. Zanardo. Branching-time logic with quantification over branches: the point of view of modal
logic. J. Symbolic Logic, 61:1–39, 1996.
12

Appendix
1
2¾·FxP´xµ
A monodic Q C T L£F -sentence with only uncountable models
4xyz´´xxµ´xyyzxzµµ
P´xµ¿·F¾·Fy´P´yµyxµ¿·F¾·Fy´P´yµyxµ¡
P´yµ¿·F¾·Fz´P´zµzyµxy℄¿·F¾·Fz´P´zµzyµxy℄¡
1xy 3¾·Fx
We take a signature with relation symbols P (unary) and(binary). Let ϕ be the conjunction of the
following Q CT L£F-sentences:
ϕ is monodic and only uses the reflexive temporal operators¾·F¿·F . The idea is that ϕ forces an
embedding of 2 ω full branches into the domain.
LEMMA 11. ϕ has a full tree model.
Proof. Letω 2be
the tree withthe ordering of proper initial segment. We identify full
branches ofwith elements
Ð
of ω 2. Let Dω 2. Define a linear
ËË Ð¸
orderon D in the standard way.
First, the nodes ofare ordered ‘left–right’ by ‘projecting’ down onto a horizontal axis, as shown in
Figure 1.
000 001 010 011 100 101 110 111
00
01
10 11
0
1
¹
ε
Figure 1: Ordering the nodes ofω 2
The full branches h ofare then inserted in this ordering in their natural places: h goes to the left of
all t¾ω 2 such that tˆ0¾h, and to the right of all t¾ω 2 such that tˆ1¾h. This defineson D.
Now for each t¾ω 2 let M t be the structure with domain D, withinterpreted as above, and
with M tP´dµiff P´yµ¿·F¾·Fz´P´zµzyµxy℄¿·F¾·Fz´P´zµzyµxy℄¡
dt, for d¾D. We obtain a modelÅ´D´M t : t¾ω 2µµ.
Write ε for the root of. We check that´Åhεµϕ for any h. For sentence 1 above, let h be
any branch of. We require
´Åhεµxy ¿·F
We will take ‘x’ to be h¾D. Then for any d¾D with´Åhεµ¿·F P´dµ, we require´Åhεµ
¿·F¾·Fz´P´zµzdµhd. Such a d must be inω 2. Suppose that there is u¾h with
M vz´P´zµzdµfor all v¾h with vu. Choose v¾h higher than d. Since M vP´vµ, we
¸¸¸
13

´Åhtµ¿·F¾·Fy´P´yµytµ (2)
have vd. By definition ofwe have vd iff hd. So hd, as required. The other side (xy)
is similar.
Sentence 2,¾·FxP´xµ, is clearly true, since M tP´tµfor any t¾ω 2.
so´Ågtµ
For sentence 3,¾·Fx P´xµ¿·F¾·Fy´P´yµyxµ¿·F¾·Fy´P´yµyxµ¡, take
h, t¾h, d¾D. If´ÅhtµP´dµthen dt, so we can assume td. We require
and similarly on the other side with yt. So take g to be any branch containing tˆ0. By definition of
, every utˆ0 insatisfies ut in D. Thus,´Ågtˆ0µ¾·Fy´P´yµytµ, ¿·F¾·Fy´P´yµytµ, yielding (2). The second half of sentence 3 is handled similarly, using tˆ1.
Sentence 4 is trivially true.
LEMMA 12. Any full tree model of ϕ has uncountable domain.
Proof. Assume that for some ω-treeW,Å´D´M w : w¾Wµµis a full tree model of ϕ.
We can assume that ϕ is true at the root of. By sentence 2, for all w¾W we may pick d w¾D with
M wP´d wµ.
We define nodes w t¾W (for t¾ω 2) by induction ont. We let w ε be the root of, where ε
is the empty sequence inω 2. Let t¾ω 2 and assume that w t is defined. By sentence 3, there is a
full branch h ofwith w t¾h and´Åhw tµ¿·F¾·Fy´P´yµyd wtµ. So there is w tˆ0¾h
with w tˆ0w t and´Åhw tˆ0µ¾·Fy´P´yµyd wtµ. Similarly, there are g¿w tˆ1w t with
´Ågw tˆ1µ¾·Fy´P´yµyd wtµ.
For each full branch β ofω 2, pick a full branch ˆβ ofcontainingw t : t¾β. Sentence
βy℄¡
1 tells
us that there is d β¾D with
´Åˆβw εµy ¿·F P´yµ¿·F¾·Fz´P´zµzyµd βy℄¿·F¾·Fz´P´zµzyµd Let βγ be distinct branches ofω 2; we claim that d βd γ . Let t be the highest node in βγ. We
may assume without loss of generality that tˆ0¾β and tˆ1¾γ. Now,´Åˆβw wt
P´yµ
εµy ¿·F ¿·F¾·Fz´P´zµzyµd βy℄¡. We have´Åˆβw εµ¿·F P´d wtµsince M wtP´d wtµ. Hence,
´Åˆβw εµ¿·F¾·Fz´P´zµzd wtµd wtµ
βd By definition of w tˆ0 , we have´Åˆβw tˆ0µ¾·Fy´P´yµyd wtµ. So certainly,
´Åˆβw εµ¿·F¾·Fz´P´zµzd Hence,´Åˆβw εµd βd wt . That is, M wεd βd wt Similarly, M wεd γd wt . By sentence 4,
is transitive in M wε , so M wεd βd γ . Sinceis irreflexive, d βd , as claimed.
Thus,Dd β : β a branch ofω 22 , proving the lemma.
ω
γ
REMARK 13. We can make any model of ϕ have uncountable branching factor by adding a conjunct
¾·Fx­´Q´xµy´yxyxQ´yµµµplus the statement thatis rigid.
14

2 Q P C T L£-formulas with countable models
On the other hand, if we restrict quantifiers to apply only to state formulas, we get a downward
Löwenheim–Skolem–Tarski theorem:
THEOREM 14. Let ϕ be a Q P CT L s -sentence, and suppose that ϕ is satisfiable in a full tree model.
Then ϕ is satisfiable in a full tree model with countable tree and domain.
Proof. LetÅbe any full tree model. We may viewÅ´DIµas a three-sorted first-order structure,
with sorts for the domain, the tree, and the set of all full branches. Taking a countable elementary
substructure of this yields a bundled tree modelÆ´0D 0HI 0µwhose domain D 0 , tree0, and
bundle H are countable. It is easy to translate Q P CT L£-formulas to three-sorted first-order formulas
with the same meaning. It follows that for any ā in D 0 , h¾H , w¾h, and any Q P CT L£-formula
ψ´¯xµ, we have´Åhwµψ´āµiff´Æhwµψ´āµ.
Now letÆ´0D 0I 0µbe
´Ågwµϕ´āµ´µ´Ægwµϕ´āµ
the full tree model based onÆ. We claim that for all formulas ϕ´¯xµ
as in the formulation of the theorem, all full branches g of0 (g may not be in H ), all w¾g, and all
ā in D 0 , we have
The proof is by induction on ϕ. The atomic, boolean, and temporal cases are easy and we omit
them. Consider the caseϕ´¯xµand inductively assume the result for ϕ. If´Ægwµϕ´āµ, it is
easily seen that´Ågwµϕ´āµ. Conversely, if´Ågwµϕ´āµ, pick h¾H containing w.
Clearly,´Åhwµϕ´āµ, so´Æhwµϕ´āµ. So there is h¼¾H with´Æh¼wµϕ´āµ. Thus,
´Åh¼wµϕ´āµ. Inductively,´Æh¼wµϕ´āµ. So´Ægwµϕ´āµ, as required.
Finally consider the casexϕ´xȳµ, for a state formula ϕ for which we assume the result inductively.
If´Ægwµxϕ´xāµ, then´Ægwµϕ´bāµfor some b¾D 0 . Inductively,´Ågwµ
ϕ´bāµ, so´Ågwµxϕ´xāµ. Conversely, suppose that´Ågwµxϕ´xāµ. Pick h¾H containing
w. Then asxϕ´xȳµis a state formula,´Åhwµxϕ´xāµ. So´Æhwµxϕ´xāµ,
whence´Æhwµϕ´bāµfor some b¾D 0 . Then´Åhwµϕ´bāµ. Since ϕ is a state formula,
so´Ægwµxϕ´xāµas
´Ågwµϕ´bāµ. Inductively,´Ægwµϕ´bāµ,
3 Diagrams for Theorem 4
required.
Up´τµ
Down´τµ
Left´τµ Right´τµ
τ
Figure 2: a tile τ
15

Ú´yµ
Ô
Ú×
Ô
Ú ×
ÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔ Ô ÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔ Ú Ô× Ú×
× × ×
× × × ×
Ú × Ú
Ú×
× × ×
x 2 u 2 x 1
× × ×
Ú×
2001τ 02
2001τ 01
2002·1 2
2002·1 2
10011 1001 1 1
v 0
2002·1 2
0
2001τ
10 u 12001τ 00 y20022
2001τ 10
0
10011 1001 1
v 1
1
2002·1 2 2001τ 0 1
20022 y 1
10011 1001 1 1
v 2
y 2001τ
2
2
20022
1
y 2
v 1
y 1
Figure 3: The defined points
10011 1001 1 1
20022
x 0 u 0 x 1 u 1
x 2
¹ ´xµ
0
16

±±±±±±±±± ×
±±±±±±±±±
× × ±±±±±±±±±
±±±±±±±±± ±±±±±±±± ± × ×
±±±±
×
×
× ×
histories H
h
tree T
××
t
v i
10011
y jm
g
10011
2001τ i j
111
˘τ i j1τ i j
111
2001τ i·1j
1001 1 1
domain D
x i x i·1
˘τ i j·1 1 τ i·1j
1·1 1 1 1
Figure 4: Checking Right´τ i jµLeft´τ i·1jµ.
17