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2 Information and Computation Contents lists availale at SciVerse ScienceDirect Information and Computation A coalgeraic perspective on linear weighted automata Filippo Bonchi a,, Marcello Bonsangue,c, Michele Boreale d,janrutten c,e,alexandrasilva e,c,f a ENS Lyon, Université de Lyon, LIP UMR 5668 CNRS ENS Lyon UCBL INRIA, 46 Allée d Italie, Lyon, France Leiden Institute of Advanced Computer Science, Niels Bohrweg, 333 CA Leiden, The Netherlands c Centrum Wiskunde & Informatica, Science Park 3, 098 XG Amsterdam, The Netherlands d Dipartimento di Sistemi e Informatica, Università di Firenze, Viale Morgagni 65, I-5034 Firenze, Italy e Radoud Universiteit Nijmegen, Heyendaalseweg 35, 655 AJ Nijmegen, The Netherlands f HASLa/INESC TEC, Universidade do Minho, Braga, Portugal article info astract Article history: Received 3 March 0 Revised 9 Septemer 0 Availale online xxxx Weighted automata are a generalisation of non-deterministic automata where each transition, in addition to an input letter, has also a quantity expressing the weight e.g. cost or proaility of its execution. As for non-deterministic automata, their ehaviours can e expressed in terms of either weighted isimilarity or weighted language equivalence. Coalgeras provide a categorical framework for the uniform study of state-ased systems and their ehaviours. In this work, we show that coalgeras can suitaly model weighted automata in two different ways: coalgeras on Set the category of sets and functions characterise weighted isimilarity, while coalgeras on Vect the category of vector spaces and linear maps characterise weighted language equivalence. Relying on the second characterisation, we show three different procedures for computing weighted language equivalence. The first one consists in a generalisation of the usual partition refinement algorithm for ordinary automata. The second one is the ackward version of the first one. The third procedure relies on a syntactic representation of rational weighted languages. 0 Pulished y Elsevier Inc.. Introduction Weighted automata were introduced in Schützenerger s classical paper [38]. They are of great importance in computer science [0], arising in different areas of application, such as speech recognition [7], image compression [], control theory [0] and quantitative modelling [5,3]. They can e seen as a generalisation of non-deterministic automata, where each transition has a weight associated to it. This weight is an element of a semiring, representing, for example, the cost or proaility of taking the transition. The ehaviour of weighted automata is usually given in terms of weighted languages also called formal power series [37, 6], that are functions assigning a weight to each finite string w A over an input alphaet A. For computing the weight given to a certain word, the semiring structure plays a key role: the multiplication of the semiring is used to accumulate the weight of a path y multiplying the weights of each transition in the path, while the addition of the semiring computes the weight of a string w y summing up the weights of the paths laelled with w [4]. Alternatively, the ehaviour of weighted automata can e expressed in terms of weighted isimilarity [8], that is, an extension of isimilarity for non-deterministic * Corresponding author. addresses: filippo.onchi@ens-lyon.fr F. Bonchi, marcello@liacs.nl M. Bonsangue, oreale@dsi.unifi.it M. Boreale, jan.rutten@cwi.nl J. Rutten, alexandra@cs.ru.nl A. Silva /$ see front matter 0 Pulished y Elsevier Inc. doi:0.06/j.ic.0..00

3 78 F. Bonchi et al. / Information and Computation automata susuming several kinds of quantitative equivalences such as, for example, proailistic isimilarity []. As in the case of non-deterministic automata, weighted isimilarity implies strictly weighted language equivalence. In this paper, we study linear weighted automata, which are deterministic weighted automata where the set of states forms a vector space. A linear weighted automaton can e viewed as the result of determinizing an ordinary weighted automatonwithweightsinagenericfield, using some kind of weighted powerset construction. As such, linear weighted automata are typically infinite state. The key point is that the linear structure of the state space allows for finite representations of these automata and effective algorithms operating on them. To e more specific, the goal of the present paper is to undertake a systematic study of the ehavioural equivalences and minimisationalgorithms for linear weighted automata. To achieve this goal, we will enefit from a coalgeraic perspective on linear weighted automata. The theory of coalgeras offers a unifying mathematical framework for the study of many different types of state-ased systems and infinite data structures. Given a functor G : C C on a category C, ag-coalgera is a pair consisting of an oject X in C representing the state space of the system and a morphism f : X G X determining the dynamics of the system. Under mild conditions, functors G have a final coalgera unique up to isomorphism into which every G-coalgera can e mapped via a unique so-called G-homomorphism. The final coalgera can e viewed as the universe of all possile G-ehaviours: the unique homomorphism into the final coalgera maps every state of a coalgera to a canonical representative of its ehaviour. This provides a general notion of ehavioural equivalence G : two states are equivalent if and only if they are mapped to the same element of the final coalgera. Our first contriution in this paper is to recast oth weighted isimilarity and weighted language equivalence in the theory of coalgeras. We see weighted automata for a field K and alphaet A, as coalgeras of the functor W = K K A on Set the category of sets and functions. Concretely, a W-coalgera consists of a set of states X and a function o, t : X K K X A where, for each state x X, o : X K assigns an output weight in K and t : X K X A assigns a function in K X A. For each symol a A and state x X, txax is a weight k K representing the weight of a transition from x to x with lael a, in symols x a,k x.iftxax = 0, then there is no a-laelled transition from x to x. Note that there could exist several weighted transitions with the same lael outgoing from the same state: x a,k x, x a,k x,...,x a,k n x n. Adapting the aove reasoning, we model linear weighted automata as coalgeras of the functor L = K A on Vect the category of vector spaces and linear maps. A linear weighted automaton consists of a vector space V and a linear map o, t : V K V A where, as efore, o : V K defines the output and t : V V A the transition structure. More precisely, for each vector v V and a A, tva = v means that there is a transition from v to v with lael a, in symols v a v. Note that the transition structure is now deterministic, since for each vector v and input a A there is only one vector v V.WhenV = K X,eachvectorv V can e seen as a linear comination of states x,...,x n X, i.e., v = k x + +k n x n for some k,...,k n K. Therefore, the transitions x a,k x,...,x a,k n x n of a weighted automaton correspond to a single transition x a k x + +k n x n of a linear weighted automaton. We show that W i.e., the ehavioural equivalence induced y W coincides with weighted isimulation while L coincides with weighted language equivalence. Determinisation is the construction for moving from ordinary weighted automata and weighted isimilarity to linear weighted automata and weighted language equivalence. Similar to the powerset construction, determinisation comines all the states within one vector, ut unlike the determinisation of a nondeterministic automaton, the resulting state space will not e finite ut forming a vector space of finite dimension. On this respect, our determinisation differs from the construction descried y Mohri [7] for a suclass of weighted automata with weights on a semiring rather than a field, which associates states of the determinised weighted automaton with a set of states of the original weighted automaton. Once we have fixed the mathematical framework, we investigate three different types of algorithms for computing L. These algorithms work under the assumption that the underlying vector space has finite dimension. The first is a forward algorithm that generalises the usual partition-refinement algorithm for ordinary automata: one starts y decreeing as equivalent states with the same output values, then refines the otained relation y separating states for which outgoing transitions go to states that are not already equivalent. Linearity of the automata plays a crucial role to ensure termination of the algorithm. Indeed, the equivalences computed at each iteration can e represented as finite-dimensional suspaces in the given vector space. The resulting descending chain of suspaces must therefore converge in a finite numer of steps, despite the fact that the state space itself is infinite. We also show that each iteration of the algorithm coincides with the equivalence generated y each step of the standard construction of the final coalgera via the final sequence. The minimal linear representation of weighted automata over a field was first considered y Schützenerger [38]. This algorithm was reformulated in a more algeraic and slightly simplified fashion in the ook of Berstel and Reutenauer [6]. Their algorithm is different from our method, as it is related to the construction of a asis for a sugroup of a free group. Further, no evident connections can e traced etween their treatment and the notions of isimulation and coalgeras. The second algorithm proceeds in a similar way, ut uses a ackward procedure. It has een introduced y the third author together with linear weighted automata [7]. In this case, the algorithm starts from the complement inaprecise geometrical sense of the relation identifying vectors with equal weights. Then it incrementally computes the space of all states that are ackward reachale from this relation. The largest isimulation is otained y taking the complement of this space. The advantage of this algorithm over the previous one is that the size of the intermediate relations is typically much smaller. The presentation of this algorithm in [7] is somewhat more concrete, as there is no attempt at a coalgeraic

4 F. Bonchi et al. / Information and Computation treatment and the underlying field is fixed to R for example, this leads to using orthogonal complements rather than dual spaces and annihilators, which we consider in Section 4. No connection is made with rational series. Finally, the third algorithm is new and uses the fact that equivalent states are mapped y the unique homomorphism into the same element of the final coalgera. We characterise the final morphism in terms of so-called rational weighted languages which are also known as rational formal power series. This characterisation is useful for the computation of the kernel of the final homomorphism, which consists of weighted language equivalence. Taking again advantage of the linear structure of our automata, calculating the kernel of the aove homomorphism will correspond to solving a linear system of equations. The results in this paper are presented for weighted automata with weights taken from a field, as opposed to the more general and classical definition where weights from a semiring are considered. This restriction is convenient for presentation purposes and, as we will discuss in Section 6, many of the results ut not all can e extended to semirings. A coalgeraic perspective on weighted automata is y no means the only approach to understand their structure and properties, as is already clear from the various references to related work mentioned aove more will follow in Section 6. We have found the application of coalgera as a general framework for the study of dynamical systems and infinite ehaviour in the present setting useful for a numer of reasons, which we shall riefly discuss next. An important feature of the coalgeraic methodology is that once a class of systems is identified as the class of coalgeras of a certain type formally, a functor, then several things come for free, following from the general theory of universal coalgera [3]: i the semantics or ehaviour of each system is otained y a unique homomorphism into the final coalgera; ii with each coalgera type, a canonical notion of ehavioural equivalence is associated; iii the homomorphism into the final coalgera identifies all and only those states that are equivalent; iv consequently, the image of the system under this final coalgera homomorphism is its minimisation with respect to the canonical notion of ehavioural equivalence. Yet another advantage of the general perspective of coalgera is that it offers a framework in which it is possile to relate different types of systems in a rigorous manner. By identifying, in the present setting, the different types of weighted automata notaly, classical ranching weighted automata and linear weighted automata as different types of coalgeras, we otain immediately an appropriate notion of ehavioural equivalence for each of them. As a consequence, we have een ale to put the different existing notions of equivalence of weighted automata weighted isimilarity and weighted language equivalence into a coherent perspective. Using their coalgeraic characterisations, it was relatively straightforward to give a precise description of the transformation of ranching weighted automata into linear weighted automata, y means of a generalised version of the well-known powerset construction. Our coalgeraic characterisation has furthermore led to a canonical description of the minimisation of linear weighted automata, in Section 5. The details of this construction are very similar to the use of rational power series and linear systems of equations [6]. What is pleasant aout the coalgeraic approach is that the present description of the minimisation of linear weighted automata is an instance of the canonical and general insights from universal coalgera, mentioned aove. Structure of the paper. In Section we introduce weighted automata and coalgeras. We also show that W-coalgeras characterise weighted automata and weighted isimilarity. In Section 3., after recalling some preliminary notions of linear algeras, we show that each weighted automaton can e seen as a linear weighted automaton, i.e., an L-coalgera. This change of perspective allows us to coalgeraically capture weighted language equivalence. In Section 4, we show the forward and the ackward algorithm while, in Section 5, we first introduce a syntactic characterisation of rational weighted languages and then we show how to employ it in order to compute L. In Section 6, after summarising the main results of the paper, we discuss how to extend them to the case of automata with weights in a semiring. Sections.3 and 4.3 show some interesting minor results that could e safely skipped y the not interested reader. The presentation is self-contained and does not require any prior knowledge on the theory of coalgeras.. Weighted automata as coalgeras We will first introduce the fundamental definitions and facts aout weighted automata, weighted isimilarity and their characterisation as coalgeras over Set, the category of sets and functions. We will next introduce weighted language equivalence over weighted automata. In the final susection, we will discuss a further equivalence that naturally arises from the theory of coalgeras; this equivalence will play no role in the rest of the paper, though... Fundamental definitions First we fix some notation. We will denote sets y capital letters X, Y, Z,... and functions y lower case f, g, h,... Given a set X, id X is the identity function and, given two functions f : X Y and g : Y Z, g f is their composition. The product of two sets X, Y is X Y with the projection functions π : X Y X and π : X Y Y.Theproductoftwo functions f : X Y and f : X Y is f f defined for all x, x X X y f f x, x = f x, f x. The disjoint union of X, Y is X + Y with injections κ : X X + Y and κ : Y X + Y.Theunionof f : X Y and f : X Y is f + f defined for all z X +Y y f + f κ i z = κ i f i z for i {, }. The set of functions ϕ : Y X is denoted y X Y.For f : X X, the function f Y : X Y X Y is defined for all ϕ X Y y f Y ϕ = λy Y. f ϕy. The

5 80 F. Bonchi et al. / Information and Computation Fig.. The weighted automata X, o X, t X and Y, o Y, t Y from left to right. The dashed arrows denote the W-homomorphism h : X Y. This induces the equivalence relation R h = X X that equates all the states in X. collection of finite susets of X is denoted y P ω X and the empty set y. For a set of letters A, A denotes the set of all finite words over A; ɛ the empty word; and w w the concatenation of words w, w A. We fix a field K. Weusek, k,... to range over elements of K. ThesumofK is denoted y +, theproducty, the additive identity y 0 and the multiplicative identity y. The support of a function ϕ from a set X to a field K is the set {x X ϕx 0}. Weighted automata [38,0] are a generalisation of ordinary automata where transitions in addition to an input letter have also a weight in a field K and each state is not just accepting or rejecting ut has an associated output weight in K. Formally, a weighted automaton wa, for short with input alphaet A is a pair X, o, t, where X is a set of states, o : X K is an output function associating to each state its output weight and t : X K X A is the transition relation that associates a weight to each transition. We shall use the notation x a,k y meaning that txay = k. Weight 0 means no transition. If the set of states is finite, a wa can e conveniently represented in form of matrices. First of all, we have to fix an ordering x,...,x n of the set of states X. Then the transition relation t can e represented y a family of matrices {T a } a A where each T a K n n is a K-valued square matrix, with T a i, j specifying the value of the a-transition from x j to x i, i.e., tx j ax i. Note that we define the matrices T a to have the source state as column index and the target state as row index. Theoutputweightfunctiono can e represented as a K-valued row vector in K n that we will denote y the capital letter O. For a concrete example, let K = R the field of real numers and A ={a, } and consider the weighted automata X, o X, t X and Y, o Y, t Y in Fig.. Their representation as matrix is the following: O X =, T Xa = 3 0, T X = 0 0 0, O Y =, T Ya = 3, T Y = Weighted isimilarity generalises the astract semantics of several kind of proailistic and stochastic systems. This has een introduced y Buchholz in [8] for weighted automata with a finite state space. Here we extend that definition to possily infinite states automata with finite ranching, i.e., those X, o, t such that for all x X, a A, txax 0for finitely many x. This will e needed in the sequel, when we model weighted automata coalgeraically, to ensure that the final coalgera exists the final coalgera can e thought of the universe of possile ehaviours and will e used to provide semantics to each state of the automaton. Hereafter we will always implicitly refer to weighted automata with finite ranching. Moreover, given an x X and an equivalence relation R X X we will write [x] R to denote the equivalence class of x with respect to R. Definition. Let X, o, t e a weighted automaton. An equivalence relation R X X is a weighted isimulation if for all x, x R, it holds that:. ox = ox,. a A, x X, x [x ] R tx ax = x [x ] R tx ax. Weighted isimilarity in symols w is defined as the largest weighted isimulation. For instance, the relation R h in Fig. is a weighted isimulation. Now, we will show that weighted automata and weighted isimilarity can e suitaly characterised through coalgeras [3]. We first recall some asic definitions aout coalgeras. Given a functor G : C C on a category C, ag-coalgera is an oject X in C together with an arrow f : X G X. For many categories and functors, such a pair X, f represents a transition system, the type of which is determined y the functor G. Vice versa, many types of transition systems e.g., deterministic automata, laelled transition systems and proailistic transition systems can e captured y a functor.

6 F. Bonchi et al. / Information and Computation A G-homomorphism from a G-coalgera X, f to a G-coalgera Y, g is an arrow h : X Y preserving the transition structure, i.e., such that the following diagram commutes. X h Y f G X Gh g GY A G-coalgera Ω, ω is said to e final if for any G-coalgera X, f there exists a unique G-homomorphism G X : X Ω. Final coalgera can e viewed as the universe of all possile G-ehaviours: the unique homomorphism G X : X Ω maps every state of a coalgera X to a canonical representative of its ehaviour. This provides a general notion of ehavioural equivalence: two states x, x X are G-ehaviourally equivalent x G x iff x G X = x G X. The functors corresponding to many well-known types of systems are shown in [3]. In this section we will show a functor W : Set Set such that W coincides with weighted isimilarity. In order to do that, we need to introduce the field valuation functor. Definition Field valuation functor. Let K e a field. The field valuation functor K ω : Set Set is defined as follows. For each set X, Kω X is the set of functions from X to K with finite support. For each function h : X Y, Kh ω : KX ω KY ω is the function mapping each ϕ Kω X into ϕh Kω Y defined, for all y Y,y ϕ h y = x h y ϕ x. Note that the aove definition employs only the additive monoid of K, i.e., the element 0 and the + operator. For this reason, such functor is often defined in literature e.g., in [6] for commutative monoids instead of fields. We need two further ingredients. Given a set B, the functor B : Set Set maps every set X into B X and every function f : X Y into id B f : B X B Y.GivenafinitesetA, the functor A : Set Set maps X into X A and f : X Y into f A : X A Y A recall that f A is defined for all ϕ X A as f A ϕ = λa A. f ϕa. Now, the functor corresponding to weighted automata with input alphaet A over the field K is W = K K ω A : Set Set. Note that every function f : X WX consists of a pair of functions o, t with o : X K and t : X Kω X A. Therefore any W-coalgera X, f is a weighted automaton X, o, t and vice versa. Proposition. See [39]. The functor W has a final coalgera. Proof. By [7, Theorem 7.], the fact that W is ounded is enough to guarantee the existence of a final coalgera. Intuitively, a functor F is ounded y some cardinal numer c, ifforallf-coalgeras X, f and all states x X, thesetofstates reachale from x has cardinality smaller than or equal to c. For instance the powerset functor P is not ounded and does not have final coalgera [3], while the finite powerset functor P ω is ounded y ω. Also, the functor W is ounded y ω ecause of the finite ranching condition. In order to show that the equivalence induced y the final W-coalgera W coincides with weighted isimilarity w, it is instructive to spell out the notion of W-homomorphism. A function h : X Y is a W-homomorphism etween weighted automata X, o X, t X and Y, o Y, t Y if the following diagram commutes. X h Y o X,t X o Y,t Y K K X ω A id K h ω A K K Y ω A This means that for all x X, y Y, a A, o X x = o Y hx and t X xa x = t Y hx ay. x h y For every W-homomorphism h : X, o X, t X Y, o Y, t Y, the equivalence relation R h ={x, x hx = hx } is a weighted isimulation. Indeed, y the properties of W-homomorphisms and y definition of R h,forallx, x R h o X x = o Y hx = o Y hx = o X x Here we are implicitly assuming that C is a concrete category [], i.e., there exists a forgetful functor U : C Set. By writing x, x X, weformally mean that x, x UX and y x i G X,wemeanU G X x i.

7 8 F. Bonchi et al. / Information and Computation and for all a A, forally Y t X x a x = t Y hx ay = t Y hx ay = x h y Trivially, the latter implies that for all x X t X x a x = t X x a x. x [x ] Rh x [x ] Rh x h y t X x a x. For an example look at the function h depicted y the dashed arrows in Fig. : h is a W-homomorphism and R h is a weighted isimulation. Conversely, every isimulation R on X, o X, t X induces a coalgera X/R, o X/R, t X/R where X/R is the set of all equivalence classes of X w.r.t. R and o X/R : X/R K and t X/R : X/R Kω X/R A are defined for all x, x X, a A y o X/R [x ] R = o X x, t X/R [x ] R a [x ] R = t X x a x. x [x ] R Note that oth o X/R and t X/R are well-defined i.e., independent from the choice of the representative since R is a weighted isimulation. Most importantly, the function ε R : X X/R mapping x into [x] R is a W-homomorphism. W X X ε R X/R W X/R Ω o X,t X o X/R,t X/R ω WX Wε R WX/R W W X/R WΩ W W X Theorem. Let X, o, t e a weighted automaton and let x,x e two states in X. Then, x w x iff x W x, i.e., x W X = x W X. Proof. The proof follows almost trivially from the aove oservations. If x W x, i.e., x W X = x W X,thenx, x R W X and R W X is a weighted isimulation ecause W X is a W-homomorphism. Thus x w x. If x w x, then there exists a weighted isimulation R such that x, x R. LetX/R, o X/R, t X/R and ε R : X X/R e the W-coalgera and the W-homomorphism descried aove. Since there exists a unique W-homomorphism from X, o X, t X to the final coalgera, then W X = W X/R ε R. Sinceε R x = ε R x, then x W X = x W X, i.e., x W x... Weighted language equivalence The semantics of weighted automata can also e defined in terms of weighted languages. A weighted language over A and K is a function σ : A K assigning to each word in A a weight in K. Foreachwa X, o, t, the function l X : X K A assigns to each state x X its recognised weighted language. For all words a...a n A,itisdefinedy l X xa...a n = { k... k n k a x =,k a x n,k n xn and ox n = } k. We will often use the following characterisation: for all w A, { ox, if w = ɛ; l X xw = x X txax l X x w, if w = aw. Two states x, x X are said to e weighted language equivalent denoted y x l x ifl X x = l X x.in[8],itis shown that if two states are weighted isimilar then they are also weighted language equivalent. For completeness, we recall here the proof. Proposition. w l. Proof. We prove that if R is a weighted isimulation, then for all x, x R, l X x = l X x. We use induction on words w A.

8 F. Bonchi et al. / Information and Computation Fig.. The states x, y, z and u intheaoveautomatonrecognisethelanguagemappingaa into and the other words into 0. Although they are all language equivalent, they are not isimilar. If w = ɛ, thenl X x w = ox and l X x w = ox and ox = ox since R is a weighted isimulation. If w = aw,then l X x w = x X tx a x l X x w. By induction hypothesis for all x [x ] R, l X x w = l X x w. Thus in the aove summation we can group all the states x [x ] R as follows: l X x w = tx a x l X x w. [x ] R X/R x [x ] R Since x, x R and R is a weighted isimulation, the aove summation is equal to tx a x l X x w [x ] R X/R x [x ] R that, y the previous arguments, is equal to l X x w. The inverse inclusion does not hold: the states x, y, z and u in Fig. are language equivalent ut they are not equivalent according to weighted isimilarity..3. On the difference etween W-isimilarity and W-ehavioural equivalence We conclude this section with an example showing the difference etween W-ehavioural equivalence and hence weighted isimulation and another canonical equivalence notion from the theory of coalgera, namely W-isimulation. This result is not needed for understanding the next sections, and therefore this susection can e safely skipped. The theory of coalgeras provides an alternative definition of equivalence, namely G-isimilarity G, that coincides with G-ehavioural equivalence whenever the functor G preserves weak pullacks [3]. In the case of weighted automata, the functor W does not preserve weak pullacks and W is strictly included into W. Since weighted automata are one of the few interesting cases where this phenomenon arises, we now show an example of two states that are in W,ut not in W the paper [5] was of great inspiration for the construction of this example. First, let us instantiate the general coalgeraic definition of isimulation and isimilarity to the functor W. A W- isimulation etween two W-coalgeras X, o X, t X and Y, o Y, t Y is a relation R X Y such that there exists o R, t R : R WR making the following diagram commute. The largest W-isimulation is called W-isimilarity W. X π R π Y o X,t X o R,t R o Y,t Y W X WR Wπ Wπ WY Note that the actual definition of W relates the states of a single automaton. We can extend it in order to relate states of possily distinct automata: given X, o X, t X and Y, o Y, t Y, the states x X and y Y are equivalent w.r.t. W iff x W X = y W Y. Consider now the coalgeras in Fig. 3: x W y, ut x W y. For the former, it is enough to oserve that the functions h and h represented y the dashed arrows are W-homomorphisms, and y uniqueness of W : x W X = h x W Z = z W Z = h y W Z = y W Y.Forx W y, note that there exists no R X Y that is a W-isimulation and such that x, y R. Sincex and x 3 have different output values than y, then neither x, y nor x 3, y can elong to a isimulation. Thus, the only remaining non-empty relation on X Y is the one equating just x and y, i.e., R ={x, y }. But this is not a W-isimulation since there exists no o R, t R making the leftmost square of the aove diagram commute. In order to understand this fact, note that π x = and π x 3 =. Thus for all possile choices of

9 84 F. Bonchi et al. / Information and Computation Fig. 3. From left to right, three weighted automata over R: X, o X, t X, Z, o Z, t Z and Y, o Y, t Y. The dashed arrows denote the W-homomorphisms h : X Z and h : Y Z.Thestatesx and y are ehaviourally equivalent, ut they are not W-isimilar. o R, t R, the function Wπ o R, t R maps x, y into a pair k, ϕ where ϕax = 0 and ϕax 3 = 0. On the other side of the square, we have that o X, t X π x, y = o X x, t X x and t X x ax = and t X x ax 3 =. It is interesting to oserve that transitions with negative weight play an essential role for having x W y and x W y. Similar examples can e constructed y using commutative monoids which are not zero-sum free a monoid is said to e zero-sum free if k + k = 0impliesk = 0 = k. We refer the interested reader to [6], where the relationship etween zero-sum free monoids and weak-pullack preserving functors is discussed in detail. 3. Linear weighted automata as linear coalgeras In this section, we will introduce linear weighted automata as coalgeras for an endofunctor L : Vect Vect, where Vect is the category of vector spaces and linear maps over a field K. The goal of this approach is to characterise weighted language equivalence as the ehavioural equivalence induced y the final L-coalgera. 3.. Preliminaries First we fix some notations and recall some asic facts on vector spaces and linear maps. We use v, v,... to range over vectors and V, W,... to range over vector spaces on a field K. Given a vector space V, a vector v V and a k K, the scalar product is denoted y k v or kv for short. The space spanned y an I-indexed family of vectors B ={v i } i I is the space spanb of all v such that v = k v i + k v i + +k n v in where for all j, v i j B. In this case, we say that v is a linear comination of the vectors in B. Asetofvectorsislinearly independent if none of its elements can e expressed as the linear comination of the remaining ones. A asis for the space V is a linearly independent set of vectors that spans the whole V.AlltheasesofV have the same cardinality which is called the dimension of V denoted y dimv. If v,...,v n is a asis for V, then each vector v V is equal to k v + +k n v n for uniquely determined k,...,k n K. For this reason, each vector v can e represented as an n -column vector k v =... k n We use f, g,... to range over linear maps. Identity and composition of maps are denoted as usual. If B V = v,...,v n and B W = w,...,w m are, respectively, the ases of the vector spaces V and W, then every linear map f : V W can e represented as an m n-matrix. Indeed, for each v V, v = k v + +k n v n and f v = k f v + +k n f v n, y linearity of f.foreachv i, f v i can e represented as m -column vector y taking as asis B W. Thus the matrix corresponding to f w.r.t. B V and B W istheonehavingasi-th column the vector corresponding to f v i. In this paper we will use capital letters F, G,... to denote the matrices corresponding to linear maps f, g,... in lower case. By multiplying the matrix F with vector v in symols, F v we can compute f v. More generally, matrix multiplication corresponds to composition of linear maps, in symols: g f = G F. The product of two vector spaces V, W is written as V W, and the product of two linear maps f, f is f f,defined as for functions. It will e clear from the context whether refer to the multiplication of matrices or to the product of spaces or maps. Given a set X, and a vector space V,thesetV X i.e., the set of functions ϕ : X V carries a vector space structure where sum and scalar product are defined point-wise. Hereafter we will use V X to denote oth the vector space and the underlying carrier set. Given a linear map f : V V, the linear map f X : V X V X is defined as for functions. If A is a finite set we can conveniently think V A as the product of V with itself for A -times A is the cardinality of A. Alinearmap f : U V A can e decomposed in a family of maps indexed y A, f ={f a : U V } a A, such that for all u U, f a u = f ua. For a set X, thesetkω X i.e., the set of all finite support functions ϕ : X K carries a vector space structure where sum and scalar product are defined in the ovious way. This is called the free vector space generated y X and can e thought

10 F. Bonchi et al. / Information and Computation of as the space spanned y the elements of X: eachvectork x i + k x i + +k n x in corresponds to a function ϕ : X K such that ϕx i j = k j and for all x / {x ij j =,...,n}, ϕx = 0; conversely, each finite support function ϕ corresponds to a vector ϕx i x i + ϕx i x i + +ϕx in x in. A fundamental property holds in the free vector space generated y X: for all functions f from X to the carrier-set of a vector space V, there exists a linear map f : Kω X V that is called the linearisation of f. For allϕ KX ω, ϕ = k x i + k x i + +k n x in and f ϕ = k f x i + k f x i + +k n f x in. η X K X ω f X f V Note that f is the only linear map such that f = f η X, where η X x is the function assigning to x and 0 to all the other elements of X. The kernel ker f of a linear map f : V W is the suspace of V containing all the vectors v V such that f v = 0. The image im f of f is the suspace of W containing all the w W such that w = f v for some v V.IfV has finite dimension, the kernel and the image of f are related y the following equation: dimv = dim ker f + dim im f. Given two vector spaces V and V, their intersection V V is still a vector space, while their union V V might not e. Instead of union we consider the coproduct of vector spaces: we write V + V to denote the space spanv V note that in the category of vector spaces, product and coproduct coincide. 3.. From weighted automata to linear weighted automata We have now all the ingredients to introduce linear weighted automata and a coalgeraic characterisation of weighted language equivalence. Definition 3 lwa. A linear weighted automaton lwa, for short with input alphaet A over the field K is a coalgera for the functor L = K A : Vect Vect. More concretely [7], an lwa is a pair V, o, t, where V is a vector space representing the state space, o : V K is a linear map associating to each state its output weight and t : V V A is a linear map that for each input a A associates a next state i.e., a vector in V.Wewillwritev a v for tv a = v. The ehaviour of linear weighted automata is expressed in terms of weighted languages. The language recognised y a vector v V of an lwa V, o, t is defined for all words a...a n A as v L V a...a n = ov n where v n is the vector reached from v through a...a n, i.e., v a an v n. We will often use the following more compact characterisation: for all w A, v L V w = { ov, if w = ɛ; tva L V w, if w = aw. Here we use the notation L V ecause this is the unique L-homomorphism into the final L-coalgera. In Section 3.3, we will provide a proof for this fact and we will also discuss the exact correspondence with the function l X introduced in Section. Given a weighted automaton X, o, t, we can uild a linear weighted automaton Kω X, o, t, where Kω X is the free vector space generated y X and o and t are the linearisations of o and t. IfX is finite, we can represent t and o y the same matrices that we have introduced in the previous section for t and o. By fixing an ordering x,...,x n of the states in X, wehaveaasisforkω X, i.e., every vector v KX ω is equal to k x + +k n x n and it can e represented as an n -column vector. The values t va and o v can e computed via matrix multiplication as T a v and O v. For a concrete example, consider the weighted automaton X, o X, t X in Fig.. The corresponding linear weighted automaton Rω X, o X, t X has as state space the space of all the linear cominations of the states in X i.e., {k x + k x + k 3 x 3 k i R}. The function o X maps v = k x + k x + k 3 x 3 into k o X x + k o X x + k 3 o X x 3, i.e., k + k + k 3. By exploiting the correspondence etween functions and vectors in Kω X discussed in Section 3., we can write t X va = k t X x a + k t X x a + k 3 t X x 3 a that is k x + x + x 3 + k 3x + k 3 3x 3 and t X v = k 3x + k 3x + k 3 3x.This can e conveniently expressed in terms of matrix multiplication: k 0 0 k k o X v = k, t X va = 3 0 k, t X v = k. k k k 3

11 86 F. Bonchi et al. / Information and Computation Fig. 4. The weighted automata X, o X, t X left and Y, o Y, t Y right. The corresponding linear weighted automata R X ω, o X, t X and RY ω, o Y, t Y are isomorphic. Alinearmaph : V W is an L-homomorphism etween lwa V, o V, t V and W, o W, t W if the following diagram commutes. o V,t V V h W o W,t W K V A id h A K W A This means that for all v V, a A, o V v = o W hv and ht V va = t W hva. IfV and W have finite dimension, then we can represent all the morphisms of the aove diagram as matrices. In this case, the aove diagram commutes if and only if for all a A, O V = O W H and H T Va = T Wa H where T Va and T Wa are the matrix representations of t V and t W for any a A. For a function h : X Y, the function K h ω : KX ω KY ω formally introduced in Definition is a linear map. Note that if h is a W-homomorphism etween the wa X, o X, t X and Y, o Y, t Y, thenk h ω is an L-homomorphism etween the lwa Kω X, o X, t X and KY ω, o Y, t Y. For an example, look at the W-homomorphism h : X, o X,t X Y, o Y, t Y represented y the dashed arrows in Fig.. The linear map R h ω : RX ω RY ω is represented y the matrix H = and it is an L-homomorphism etween Rω X, o X, t X and RY ω, o Y, t Y. This can e easily checked y showing that O X = O Y H, H T Xa = T Ya H and H T X = T Y H. Note that two different weighted automata can represent the same up to isomorphism linear weighted automaton. As an example, look at the weighted automata X, o X, t X and Y, o Y, t Y in Fig. 4. They represent, respectively, the linear weighted automata Rω X, o X, t X and RY ω, o Y, t Y that are isomorphic. The transitions and the output functions for the two automata are descried y the following matrices: T Xa = , O X =, T Ya = , O Y =. Note that T Xa and T Ya are similar, i.e., they represent the same linear map. This can e immediately checked y showing that T Ya = j t Xa j, where j : R Y R X is the isomorphic map representing the change of ases from Y = x + x, x + x 3, x 3 + x to X = x, x, x 3 and j : R X R Y is its inverse. The matrix representation of j and j is the following: J = 0 0, J = 0. Also O X and O Y represent the same map in different ases. Indeed, O Y = O X J. At this point, it is easy to see that the linear isomorphism j : R X R Y is an L-homomorphism, ecause O X = O X J J = O Y J and J T Xa = J T Xa J J = T Ya J. Analogously for j : R Y R X Language equivalence and final L-coalgera We introduce the final L-coalgera and we show that the ehavioural equivalence L, induced y the functor L, coincides with weighted language equivalence.

12 F. Bonchi et al. / Information and Computation The set of all weighted languages K A carries a vector space structure: the sum of two languages σ, σ K A is the language σ + σ defined for each word w A as σ + σ w = σ w + σ w; the product of a language σ for a scalar k K is kσ defined as kσ w = k σ w; the element 0 of K A is the language mapping each word into the 0 of K. The empty function ɛ : K A K and the derivative function d : K A K A A are defined for all σ K A, a A as ɛσ = σ ɛ, dσ a = σ a where σ a : A K denotes the a-derivative of σ that is defined for all w A as σ a w = σ aw. Proposition 3. The maps ɛ : K A K and d : K A K A A are linear. Proof. We show the proof for d. The one for ɛ is analogous. Let σ, σ e two weighted languages in K A.Nowforalla A, w A, dσ + σ aw = σ + σ aw = σ aw + σ aw = dσ aw + dσ aw. Let k e a scalar in K and σ e a weighted language in K A.Nowforalla A, w A, k dσ aw = k σ aw = dkσ aw. Since K A is a vector space and since ɛ and d are linear maps, K A, ɛ, d is an L-coalgera. The following theorem shows that it is final. Theorem Finality. From every L-coalgera V, o, t there exists a unique L-homomorphism into K A, ɛ, d. o,t V L V K A ɛ,d LV L L V LK A Proof. The only function making the aove diagram commute is L V, i.e., the function mapping each vector v V into the weighted language that it recognises. Hereafter we show that L V is a linear map. By induction on w, we prove that for all v, v V, for all w A, v + v L V w = v L V w + v L V w. Suppose that w = ɛ. Then v + v L V ɛ = ov + v.sinceo is a linear map, this is equal to ov +ov = v L V ɛ+ v L V ɛ. Now suppose that w = aw.then v + v L V aw = tv + v a L V w.sincet is a linear map, this is equal to tv a + tv a L V w that y induction hypothesis is equal to tv a L V w + tv a L V w = v L V aw + v L V aw. We can proceed analogously for the scalar product. Thus, two vectors v, v V are L-ehaviourally equivalent v L v iff they recognise the same weighted language as defined in Section 3.. Proposition 4 elow shows that L K : ω X KX ω KA is the linearisation of the function l X : X K A defined in Section or, in other words, is the only linear map making the following diagram commute. K X ω η X L K X X l X K A Lemma. Let X, o, t e a wa and K X ω, o, t e the corresponding linear weighted automaton. Then for all x X, l X x = x L K X ω. Proof. We prove it y induction on w A. If w = ɛ, thenl X xw = o X x = o X x = x L K w. ω X

13 88 F. Bonchi et al. / Information and Computation If w = aw, then x L K X ω w = t xa L K X ω w. Note that y definition, t xa = x X txax x t xa L K X ω w is equal to and thus w x X txa x x 4 L K X ω which, y linearity of L, coincides with Kω X x X txa x x L K X ω w. By induction hypothesis x L K X ω w = l X x w and thus the aove coincides with txa x l X x w x X that is l X xw. Proposition 4. Let X, o, t e a wa and K X ω, o, t e the corresponding linear weighted automaton. Then, for all v = k x i + +k n x in, v L K X ω = k l X x i + +k n l X x in. Proof. It follows from Lemma and linearity of L K X ω Linear isimulations and suspaces We now introduce a convenient characterisation of language equivalence y means of linear weighted isimulations. Differently from ordinary weighted isimulations, these can e seen oth as relations and as suspaces. The latter characterisation will e exploited in the next section for defining an algorithm for checking language equivalence. First, we show how to represent relations over a vector space V as suspaces of V, following [40,7]. Definition 4 Linear relations. Let U e a suspace of V. The inary relation R U over V is defined y v R U v if and only if v v U. ArelationR is linear if there is a suspace U such that R equals R U. Note that a linear relation is a total equivalence relation on V.LetnowR e any inary relation over V.Thereisa canonical way of turning R into a linear relation, which we descrie in the following. The kernel of R in symols kerr is the set {v v v R v }. The linear extension of R, denoted R l, is defined y: v R l v if and only if v v spankerr. Lemma. Let U e a suspace of V, then kerr U = U. According to the aove lemma, a linear relation R is completely descried y its kernel, which is a suspace, that is v Rv if and only if v v kerr. Conversely, for any suspace U V there is a corresponding linear relation R U whose kernel is U. Hence, without loss of generality, we can identify linear relations on V with suspaces of V. For example, y slight ause of notation, we can write v Uv instead of v R U v ; and conversely, we will sometimes denote y R the suspace kerr. The context will e sufficient to tell whether we are actually referring to a linear relation or to the corresponding suspace kernel. Note that the suspace {0} corresponds to the identity relation on V, that is R {0} = Id V.Infact:v Id V v iff v = v iff v v = 0. Similarly, the space V itself corresponds to R V = V V. We are now ready to define linear weighted isimulation. This definition relies on the familiar step-y-step game played on transitions, plus an initial condition requiring that two related states have the same output weight. We call this form of isimulation linear to stress the difference with the one introduced in Definition. Definition 5 Linear weighted isimulation. Let V, o, t e a linear weighted automaton. A linear relation R V V is a linear weighted isimulation if for all v, v R, it holds that:

14 F. Bonchi et al. / Information and Computation ov = ov, a A, tv a Rtv a. For a concrete example, consider the automaton Rω X, o X, t X in Fig. 4. The relation R ={x, x 3 } is not linear, ecause U ={x x 3 } is not a suspace of Rω X.However,wecanturnR into a linear relation y applying its kernel kerr ={x x 3 }. The linear extension of R is R l ={k x + k x + k 3 x 3, k x + k x + k 3 x 3 k = k and k + k 3 = k + k 3 }.Itiseasytosee that R l is a linear weighted isimulation. The following lemma provides a characterisation of linear weighted isimulation as a suspace. Let us say that a suspace U is f -invariant if f U U. Bisimulations are transition-invariant relations that refine the kernel of the output map o. Lemma 3. Let V, o, t e an lwa and R e a linear relation over V. R is a linear weighted isimulation if and only if R kero, Rist a -invariant for each a A. This lemma will e fundamental in the next section for defining an algorithm computing the greatest linear weighted isimulation. In the remainder of this section, we show that the greatest linear weighted isimulation coincides with the kernel of the final map L V. More generally, the kernel of each L-homomorphism is a linear weighted isimulation R and, vice versa, for each linear weighted isimulation R there exists an L-homomorphism whose kernel is R. Proposition 5. Let V, o V, t V e an lwa. If f: V WisanL-homomorphism for some lwa W, o W, t W then ker f is a linear weighted isimulation. Conversely, if R is a linear weighted isimulation for V, o, t, then there exist an lwa W, o W, t W and an L-homomorphism f : V Wsuchthatker f = R. Proof. First, we suppose that f : V W is an L-homomorphism and we prove that ker f satisfies and of Lemma 3. Take a vector v ker f. Thus, f v = 0 and, since o W and t W are linear maps, o W f v = 0 and t W f va = 0forall a A. Since f is an L-homomorphism, we have that o V v = o W f v = 0, i.e., ker f kero V and f t V va = t W f va = 0 meaning that t V va ker f, i.e., ker f is t Va -invariant. In order to prove the second part, we need to recall quotient spaces and quotient maps from [8]. Given a suspace U of V, the equivalence class of v w.r.t. U is [v] U ={v + u u U}. Note that v [v ] U if and only if v R U v.the quotient space V /U is the space of all equivalence classes [v] U where scalar product k[v] U is defined as [kv] U and the sum [v ] U +[v ] U as [v + v ] U. It is easy to check that these operations are well-defined i.e., independent from the choice of the representative and turn V /U into a vector space where the element 0 is U. Most importantly, the quotient function ε U : V V /U mapping each vector v into [v] U is a linear map such that kerε U = U. Now, let U e the suspace corresponding to the linear weighted isimulation R. We can take W = V /U and we define o W as o W [v] U = o V v and t W as t W [v] U a =[tva] U. Note that oth o W and t W are well-defined: for all v [v] U ={v + u u U}, o W v = o W v since o V u = 0forallu U and t W v a [t W va] U since t V ua U for all u U. It is also easy to check that they are linear. Finally, we take f : V W as ε U and with the previous definition of o W and t W is trivial to check that ε U is an L-homomorphism. As said aove, its kernel is U. Theorem 3. Let V, o, t e an lwa and let L V : V KA e the unique L-morphism into the final coalgera. Then ker L V is the largest linear weighted isimulation on V. Proof. First of all, note that y the first part of Proposition 5, ker L V is a linear weighted isimulation. Now, suppose that R is a linear weighted isimulation. By the second part of Proposition 5, there exist an lwa W, o W, t W and an L-homomorphism f : V W such that R = ker f. Now note that, since W, o W, t W is an L-coalgera there exists an L-homomorphism L W : W KA to the final coalgera. Since the composition of two L- homomorphisms is still an L-homomorphism, also L W f : V KA is an L-homomorphism. Since L V is the unique L-homomorphism from V to K A,then L W f = L V. Finally, R = ker f ker L W f = ker L V. Corollary. L is the largest linear weighted isimulation. The characterisation of isimulations as suspaces seems to e possile in Vect and not in Set ecause the former category is aelian [3]: every map has a kernel that is a suspace and every suspace is the kernel of some map. We leave as future work to study at a more general level the categorical machinery allowing to characterise isimulations as suspaces. In Section, we have shown that the largest weighted isimulation w is strictly included in language equivalence, while here we have shown that the largest linear weighted isimulation coincides with language equivalence. However, it