Equational Theories of Tropical Semirings - brics

55 Pages·2001·0.35 MB·English

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BRICSReport Series RS-01-21
ISSN 0909-0878 June2001
Copyright(cid:13)c 2001, Luca Aceto &Zolta´n E´sik& Anna Ingo´lfsdo´ttir.
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This document insubdirectory RS/01/21/
Equational Theories of Tropical Semirings
Luca Aceto(cid:3) Zolta(cid:19)n E(cid:19)siky Anna Ingo(cid:19)lfsd(cid:19)ottir(cid:3)
Abstract
This paper studies the equational theory of various exotic semirings
presented in the literature. Exotic semirings are semirings whose un-
derlying carrier set is some subset of the set of real numbers equipped
with binary operations of minimum or maximum as sum, and addition
as product. Two prime examples of such structures are the (max;+)
semiring and the tropical semiring. It is shown that none of the exotic
semirings commonly considered in the literature has a (cid:12)nite basis for its
equations, and that similar results hold for the commutative idempotent
weak semirings that underliethem. Foreach ofthesecommutativeidem-
potentweaksemirings,thepapero(cid:11)erscharacterizationsoftheequations
that hold in them, decidability results for their equational theories, ex-
plicit descriptions of the free algebras in the varieties they generate, and
relative axiomatization results.
AMS Subject Classification (1991): 08A70, 08B05, 03C05, 68Q15,
68Q70.
CR Subject Classification (1991): D.3.1, F.1.1, F.4.1.
Keywords and Phrases: Equational logic, varieties, complete axioma-
tizations,relativeaxiomatizations, tropicalsemirings,commutativeidem-
potent weak semirings, convexity,exponential time complexity.
1 Introduction
Exotic semirings, i.e., semirings whose underlying carrier set is some subset
of the set of real numbers R equipped with binary operations of minimum or
maximum as sum, and addition as product, have been invented and reinvented
many times since the late 1950s in various (cid:12)elds of research. This family of
(cid:3)BRICS(BasicResearchinComputerScience),CentreoftheDanishNationalResearch
Foundation, Department of Computer Science, Aalborg University, Fr. Bajersvej 7E, 9220
Aalborg(cid:31),Denmark. Email: fluca,[email protected]. Fax: [phone].
yDepartment of Computer Science, Universityof Szeged, A(cid:19)rp(cid:19)adt(cid:19)er 2, 6720 Szeged, Hun-
gary. PartiallysupportedbyBRICSandresearchgrantno.T30511fromtheNational Foun-
dationofHungaryforScienti(cid:12)cResearch. Thisworkwaspartlycarriedoutwhilethisauthor
was visiting professor at the Department of Computer Science, Aalborg University. Email:
[email protected]. Fax: [phone].
1
structuresconsistsofsemiringswhosesumoperationis idempotent|twoprime
examples are the (max;+) semiring
(R[f−1g;max;+;−1;0)
(see [5, Chapter 3] for a general reference), and the tropical semiring
(N[f1g;min;+;1;0)
introducedin[34]. (Henceforth,weshallwrite_and^forthebinarymaximum
and minimum operations, respectively.) Interest in idempotent semirings arose
in the 1950s through the observation that some problems in discrete optimiza-
tioncouldbe linearizedoversuchstructures(see,e.g.,[10]forsomeofthe early
referencesand[38]fora survey). Since then, the study ofidempotentsemirings
has forgedproductive connections with such diverse (cid:12)elds as, e.g., performance
evaluationofmanufacturingsystems,discreteeventsystemtheory,graphtheory
(path algebra),Markovdecision processes,Hamilton-Jacobitheory, asymptotic
analysis (low temperature asymptotics in statistical physics, large deviations),
and automata and language theory (automata with multiplicities). The inter-
ested reader is referred to [14] for a survey of these more recent developments,
andto[12]forfurtherapplicationsofthe (max;+)semiring. Herewelimitour-
selves to mentioning some of the deep applications of variations on the tropical
semiring in automata theory and the study of formal power series.
The tropical semiring (N [ f1g;^;+;1;0) was originally introduced by
Simoninhissolution(see[34])toBrzozowski’scelebrated(cid:12)nitepowerproperty
problem|i.e., whether it is decidable if a regular language L has the property
that, for some m(cid:21)0,
L(cid:3) = 1+L+(cid:1)(cid:1)(cid:1)+Lm :
The basic idea in Simon’s argument was to use automata with multiplicities
in the tropical semiring to reformulate the (cid:12)nite power property as a Burnside
problem. (TheoriginalBurnsideproblemasksifa(cid:12)nitelygeneratedgroupmust
necessarily be (cid:12)nite if each element has (cid:12)nite order [7].) The tropical semiring
was also used by Hashiguchi in his independent solution to the aforementioned
problem [16], and in his study of the star height of regular languages (see, e.g.,
[17, 18, 19]). (For a tutorial introduction on how the tropical semiring is used
to solve the (cid:12)nite power property problem, we refer the reader to [31].) The
tropicalsemiring also plays a key role in Simon’s study of the nondeterministic
complexity of a standard (cid:12)nite automaton [36]. In his thesis [26], Leung intro-
duced topological ideas into the study of the limitedness problem for distance
automata (see also [27]). For an improved treatment of his solutions, and fur-
ther references, we refer the reader to [28]. Further examples of applications of
the tropical semiring may be found in, e.g., [24, 25, 35].
The study of automata and regular expressions with multiplicities in the
tropical semiring is by now classic, and has yielded many beautiful and deep
results|whose proofs have relied on the study of further exotic semirings. For
2
example,Krobhasshownthattheequalityproblemforregularexpressionswith
multiplicities in the tropical semiring is undecidable [22] by introducing the
equatorial semiring (Z[f1g;^;+;1;0), showing that the equality problem
for it is undecidable, and (cid:12)nally proving that the two decidability problems are
equivalent. Partial decidability results for certain kinds of equality problems
over the tropical and equatorial semirings are studied in [23].
Another classic question for the language of regular expressions, with or
without multiplicities, is the study of complete axiom systems for them (see,
e.g., [9, 21, 32]). Along this line of research, Bonnier-Rigny and Krob have
o(cid:11)ered a complete system of identities for one-letter regular expressions with
multiplicitiesinthetropicalsemiring[6]. However,tothebestofourknowledge,
there has not been a systematic investigation of the equational theory of the
di(cid:11)erentexoticsemiringsstudiedintheliterature. Thisistheaimofthispaper.
Our starting points are the results we obtained in [1, 2]. In [1] we studied
the equational theory of the max-plus algebra of the natural numbers N_ =
(N;_;+;0),andprovedthatnotonlyitsequationaltheoryisnot(cid:12)nitely based,
but, for every n, the equations in at most n variables that hold in it do not
formanequationalbasis. Another view ofthe non-existenceofa(cid:12)nite basisfor
the variety generatedby this algebrais o(cid:11)eredin[2], wherewe showedthat the
collection of equations in two variables that hold in it has no (cid:12)nite equational
axiomatization.
The algebraN_ is anexample ofastructure thatwe callin this paper com-
mutative idempotent weak semiring (abbreviated henceforth to ciw-semiring).
Since ciw-semirings underlie many of the exotic semirings studied in the liter-
ature, we begin our investigations in this paper by systematically generalizing
the results from [1] to the structures Z_ = (Z;_;+;0) and N^ = (N;^;+;0).
Our initial step in the study of the equational theory of these ciw-semirings is
the geometric characterizationof the (in)equations that hold in them (Proposi-
tions3.9and3.11). Thesecharacterizationspavethewaytoexplicitdescriptions
ofthefreealgebrasinthevarietiesV(Z_)andV(N^)generatedbyZ_ andN^,
respectively,andyield (cid:12)nite axiomatizationsof the varietiesV(N_) andV(N^)
relativetoV(Z_)(Theorem3.20). Wethenshowthat,likeV(N_),thevarieties
V(Z_) andV(N^) arenot (cid:12)nitely based. The non-(cid:12)nite axiomatizabilityof the
variety V(Z_) (Theorem 3.22) is a consequence of the similar result for V(N_)
and of its (cid:12)nite axiomatizability relative to V(Z_).
The proof of the non-existence of a (cid:12)nite basis for the variety V(N^) (The-
orem 3.23) is more challenging, and proceeds as follows. For each n (cid:21) 3, we
(cid:12)rstisolateanequatione^n inn variableswhichholdsinV(N^). We thenprove
that no (cid:12)nite collection of equations that hold in V(N^) can be used to de-
duce all of the equations of the form e^. The proof of this technical result is
n
model-theoreticinnature. Moreprecisely,foreveryintegern(cid:21)3,we construct
an algebra Bn satisfying all the equations in at most n−1 variables that hold
in V(N^), but in which e^n fails. Hence, as for V(N_), for every natural num-
ber n, the equations in at most n variables that hold in V(N^) do not form
an equational basis for this variety. A similar strengthening of the non-(cid:12)nite
axiomatizability result holds for the variety V(Z_).
3
We then move on to study the equational theory of the exotic semirings
presented in the literature that are obtained by adding bottom elements to the
above ciw-semirings. More speci(cid:12)cally, we examine the following semirings:
Z_;−1 = (Z[f−1g;_;+;−1;0) ;
N_;−1 = (N[f−1g;_;+;−1;0)and
N− = (N−[f−1g;_;+;−1;0) ;
_;−1
where N− stands for the set of nonpositive integers. Since Z_;−1 and N−_;−1
are easily seen to be isomorphic to the semirings
Z^;1 = (Z[f1g;^;+;1;0) and
N^;1 = (N[f1g;^;+;1;0) ;
respectively,theresultsthatweobtainapplyequallywelltothesealgebras. (The
semiringsZ^;1 andN^;1 areusuallyreferredtoastheequatorialsemiring[22]
and the tropical semiring [34], respectively. The semiring N_;−1 is called the
polar semiring in [24].)
Our study ofthe equationaltheory ofthese algebraswill proceedas follows.
First, we shall o(cid:11)er some general facts relating the equational theory of a ciw-
semiring A to the theory of the free commutative idempotent semiring A? it
generates. In particular, in Sect. 4.1 we shall relate the non-(cid:12)nite axiomatiz-
abilityofthe varietyV(A?)generatedbyA? tothe non-(cid:12)nite axiomatizability
of the variety V(A) generated by A. Then, in Sect. 4.2, we shall apply our
generalstudytoderivethefactsthatalltropicalsemiringsstudiedinthispaper
have exponential time decidable, but non-(cid:12)nitely based equational theory. Our
general results, together with those proven in Sect. 3, will also give geometric
characterizations of the valid equations in the tropical semirings Z_;−1 and
−
N_;−1, but not in N_;−1. The task of providing a geometric description of
the valid equations for the semiring N_;−1 will be accomplished in Sect. 4.3,
whereweshallalsoshowthatV(N_;−1)canbeaxiomatizedoverV(Z_;−1)by
a single equation.
Somesemiringsstudiedintheliteratureareobtainedbyaddingatopelement
> to a ciw-semiring A in lieu of a bottom element. We examine the general
relationships that exist between the equational theory of a ciw-semiring A and
that of the semiring A> so generated in Sect. 5. This general theory, together
with the previouslyobtained non-(cid:12)nite axiomatizability results, is then applied
to show that several semirings with > are not (cid:12)nitely based either.
We conclude our investigations by examining some variations on the afore-
mentioned semirings. These include, amongst others, structures whose carrier
sets are the (nonnegative)rationalor realnumbers (Sect. 6.1),semirings whose
product operation is standard multiplication (Sect. 6.2), the semirings studied
by Mascle and Leung in [30] and [26, 27], respectively, (Sect. 6.3), and a min-
plusalgebrabasedontheordinalsproposedbyMasclein[29](Sect.6.4). Forall
of these structures, we o(cid:11)er results to the e(cid:11)ect that their equational theory is
not(cid:12)nitelybased,andhasnoaxiomatizationinaboundednumberofvariables.
4
Throughout the paper, we shall use standard notions and notations from
universal algebra that can be found, e.g., in [8, 13].
This paper collects, and improves upon, all of the results (cid:12)rst announced
without proof in [3, 4]. In addition, the material in Sects. 5 and 6 is new.
2 Background De(cid:12)nitions
We begin by introducing some notions that will be used in the technical devel-
opments to follow.
A commutative idempotent weak semiring (henceforth abbreviated to ciw-
semiring) is an algebra A = (A;_;+;0) such that (A;_) is an idempotent
commutative semigroup, i.e., a semilattice, (A;+;0) is a commutative monoid,
and such that addition distributes over the _ operation. Thus, the following
equations hold in A:
x_(y_z) = (x_y)_z
x_y = y_x
x_x = x
x+(y+z) = (x+y)+z
x+y = y+x
x+0 = x
x+(y_z) = (x+y)_(x+z) :
A ciw-semiring A is positive if
x_0 = x
holds in A. It then follows that
x_(x+y) = x+y
alsoholdsinA. Ahomomorphismofciw-semiringsisafunctionwhichpreserves
the _ and + operations and the constant 0.
A commutative idempotent semiring, or ci-semiring for short, is an alge-
bra (A;_;+;?;0) such that (A;_;+;0) is a ciw-semiring which satis(cid:12)es the
equations
x_? = x
x+? = ? :
A homomorphism of ci-semirings also preserves ?.
SupposethatA=(A;_;+;0)isastructureequippedwithbinaryoperations
_and+andtheconstant0. Assumethat?62AandletA? =A[f?g. Extend
the operations _ and + given on A to A? by de(cid:12)ning
a_? = ?_a = a
a+? = ?+a = ? ;
5
foralla2A?. WeshallwriteA? fortheresultingalgebra(A?;_;+;?;0),and
A(?) for the algebra (A?;_;+;0) obtained by adding ? to the carrier set, but
not to the signature.
Lemma 2.1 For each ciw-semiring A, the algebra A? is a ci-semiring.
Remark 2.2 In fact, A? is the free ci-semiring generated by A.
Let Eciw denote the set of de(cid:12)ning axioms of ciw-semirings, Eci the set of
axioms of ci-semirings, and E+ the set of axioms of positive ciw-semirings.
ciw
Note thatEciw is includedin bothEci andEc+iw. Moreover,let Vciw denote the
variety axiomatized by Eciw, Vci the variety axiomatized by Eci and Vc+iw the
varietyaxiomatizedbyEc+iw. Thus,Vci isthevarietyofallci-semiringsandVc+iw
the variety of all positive ciw-semirings. Since Eciw (cid:18) Eci and Eciw (cid:18) Ec+iw,
it follows that Vciw includes both Vc+iw and the reduct of any algebra in Vci
obtained by forgetting about the constant?.
Remark 2.3 If an equation t= u can be derived from Eciw or Ec+iw, then the
set of variables occurring in t coincides with the set of variables occurring in u.
Inlight ofaxiomx+?=?, this does nothold true for the equations derivable
from Eci.
Notation 2.4 In the remainder of this paper, we shall use nx to denote the
n-fold addition of x with itself, and we take advantage of the associativity and
commutativity of the operations. By convention, nx stands for 0 when n = 0.
In the same way, the empty sum is de(cid:12)ned to be 0.
For eachinteger n(cid:21)0, we use [n] to stand for the set f1;:::;ng, so that [0]
is another name for the empty set.
Finally, we sometimes write t(x1;:::;xn) to emphasize that the variables
occurring in the term t are amongst x1;:::;xn.
Lemma 2.5 WithrespecttotheaxiomsystemEciw,everytermtinthelanguage
of ciw-semirings, in the variables x1;:::;xn, may be rewritten in the form
_
t = ti
i2[k]
where k >0, each ti is a \linear combination"
X
ti = cijxj ;
j2[n]
and each cij is in N.
W
Termsoftheform i2[k]ti,whereeachti (i2[k],k(cid:21)0)isalinearcombWination
of variables,will be referredto as simple terms. When k =0, the term i2[k]ti
is just ?. (Note that k =0 is only allowed for ci-semirings.)
6
Lemma 2.6 With respect to the axiom system Eci, every termt in the language
of ci-semirings, in the variables x1;:::;xn, may be rewritten in the form
_
t = ti
i2[k]
where k (cid:21)0, each ti is a \linear combination"
X
ti = cijxj ;
j2[n]
and each cij is in N.
For any commutative idempotent (weak) semiring A and a;b 2 A, we write
a (cid:20) b to mean a _b = b. In any such structure, the relation (cid:20) so de(cid:12)ned
is a partial order, and the + and _ operations are monotonic with respect to
it. Similarly, we say that an inequation t (cid:20) t0 between terms t and t0 holds in
A if the equation t_t0 = t0 holds. We shall write A j= t = t0 (respectively,
Aj= t(cid:20)t0) if the equation t=t0 (resp., the inequation t(cid:20)t0) holds in A. (In
that case, we say that A is a model of t = t0 or t (cid:20) t0, respectively.) If A is
a class of ciw-semirings, we write A j= t = t0 (respectively, A j= t (cid:20) t0) if the
equation t = t0 (resp., the inequation t(cid:20) t0) holds in every A 2 A. Note that,
if A is in the variety V+ , then the inequation 0(cid:20)x holds in A.
ciw
Definition 2.7 A simple inequation in the variables x1;:::;xn is of the form
_
t (cid:20) t ;
i
i2[k]
where k > 0, and t and the ti (i 2 [k]) are linear combinations of the vari-
ables x1;:::;xn. We say that the left-hand side of the above simple inequation
contains the variable xj, or that xj appears on the left-hand side of the simple
inequation, if the coe(cid:14)cient of xj in t is nonzero. Similarly, we say that right-
hand side of the above inequation contains the variable xj if for some i 2 [k],
the coe(cid:14)cient of xj in ti is nonzero.
Note that, for every linear combination t over variables x1;:::;xn, the inequa-
tion t(cid:20)? is not a simple inequation.
Corollary 2.8 With respect to the axiom system Eciw, any equation in the
language of ciw-semirings is equivalent to a (cid:12)nite set of simple inequations.
Similarly, with respect to Eci, any equation in the language of ci-semirings is
equivalent to a (cid:12)nite set of simple inequations or to an inequation of the form
x(cid:20)? (in which case the equation has only trivial models).
Proof: Let t = u be an equation in the language of ciw-semWirings. By
Lemma 2.5, using Eciw we can rewrite t and u to simple terms i2[k]ti and
7
W
j2[l]uj (k;l > 0), respectively. It is now immediate to see that the equation
t=u is equivalent to the family of simple inequations
_ _
fti (cid:20) uj; uj (cid:20) ti ji2[k];j 2[l]g :
j2[l] i2[k]
Assume now that t = u is an equation in the language of ci-semWirings. By
WLemma 2.6, using Eci we can rewrite t and u to simple terms i2[k]ti and
j2[l]uj (k;l (cid:21) 0), respectively. If k;l are both positive or both equal to zero,
thentheequationt=uisequivalenttothe(cid:12)nitesetofsimpleinequationsgiven
above. (Note that,if k andl areboth0,then this setis empty, andthe original
equation is equivalent to ? = ?.) If k = 0 and l > 0, then the equation is
equivalent to x(cid:20)?. (cid:3)
Notation 2.9 Henceforth in this study, we shall often abbreviate a simple
inequation
X _ X
d x (cid:20) c x
j j ij j
j2[n] i2[k]j2[n]
as d (cid:20) fc1;:::;ckg, where d = (d1;:::;dn) and ci = (ci1;:::;cin), for i 2 [k].
We shall sometimes refer to these inequations as simple _-inequations.
In the main body of the paper, we shall also study some ciw-semirings that,
like the structure N^ = (N;^;+;0), have the minimum operation in lieu of
maximumas their sum. The preliminary results that we have developedin this
section apply equally well to these structures. In particular, the axioms for
ciw-semirings dealing with _ become the standard ones describing the obvious
identities for the minimum operation, and its interplay with +, i.e.,
x^(y^z) = (x^y)^z
x^y = y^x
x^x = x
x+(y^z) = (x+y)^(x+z) :
Note that, for ciw-semirings of the form (A;^;+;0), the partial order (cid:21) is
de(cid:12)ned by b (cid:21) a i(cid:11) a ^ b = a. In Defn. 2.7, we introduced the notion of
simple _-inequation. Dually, we saythata simple ^-inequationin the variables
x=(x1;:::;xn) is an inequation of the form
^
d(cid:1)x (cid:21) c (cid:1)x ;
i
i2[k]
where d and the ci (i2[k]) are vectors in Nn. We shall often write
d (cid:21) fc ;:::;c g
1 k
as a shorthand for this inequation.
8