A New Foundation for Finitary Corecursion

4.1 Generalized Powerset Construction

The determinization of a non-deterministic automaton using the powerset construction is an instance of a more general framework, described by Silva, Bonchi, Bonsangue, and Rutten [41] based on an observation by Bartels [10] (see also Jacobs [26]). In that generalized powerset construction, an automaton with side-effects is turned into an ordinary automaton by internalizing the side-effects in the states. The LFF interacts well with this construction, because it precisely captures the behaviours of finite-state automata with side effects. The notion of side-effect is formalized by a monad, which induces the category, in which the LFF is considered.

In the following we assume that readers are familiar with monads and Eilenberg-Moore algebras (see e.g. [29] for an introduction). For a monad T on \(\mathcal {C} \) we denote by \(\mathcal {C} ^T\) the category of Eilenberg-Moore algebras. Recall from [8, Corollary 2.75] that if \(\mathcal {C} \) is lfp (in most of our examples \(\mathcal {C} \) is \(\mathsf {Set}\)) and T is finitary then \(\mathcal {C} ^T\) is lfp, too, and for every fp object X the free Eilenberg-Moore algebra TX is fp in \(\mathcal {C} ^T\). In all the examples we consider below, the classes of fp and fg objects either provably differ or it is still unknown whether these classes coincide.

Now an automaton with side-effects is modelled as an HT-coalgebra, where T is a finitary monad on \(\mathcal {C} \) providing the type of side-effect. For example, for \(HX = 2 \times X^\varSigma \), where \(\varSigma \) is an input alphabet, \(2 = \{0,1\}\) and T the finite powerset monad on \(\mathsf {Set}\), HT-coalgebras are non-deterministic automata. However, the coalgebraic semantics using the final HT-coalgebra does not yield the usual language semantics of non-deterministic automata. To obtain this one considers the final coalgebra of a lifting of H to \(\mathcal {C} ^T\). Denote by \(U: \mathcal {C} ^T \rightarrow \mathcal {C} \) the canonical forgetful functor.

If such a (not necessarily unique) lifting exists, the generalized powerset construction transforms an HT-coalgebra into a \(H^T\)-coalgebra on \(\mathcal {C} ^T\): For a coalgebra \(x: X\rightarrow HTX\), HTX carries an Eilenberg-Moore algebra, and one uses freeness of the Eilenberg-Moore algebra TX to obtain a canonical T-algebra homomorphism \(x^\sharp : (TX,\mu ^T) \rightarrow H^T(TX,\mu ^T)\). The coalgebraic language semantics of (X, x) is then given by Open image in new window , i.e. by composing the unique coalgebra morphism induced by \(x^\sharp \) with \(\eta _X\). This construction yields a functor \( T': \mathsf {Coalg} (HT) \rightarrow \mathsf {Coalg} H^T \) mapping coalgebras \(X\xrightarrow {x} HTX\) to \(x^\sharp \) and homomorphisms f to Tf (see e.g. [12, ProofofLemma 3.27] for a proof).

Now our aim is to show that the LFF of \(H^T\) characterizes precisely the coalgebraic language semantics of all fp-carried HT-coalgebras. As the right adjoint U preserves monos and is faithful, we know that \(H^T\) preserves monos, and as T is finitary, filtered colimits in \(\mathcal {C} ^T\) are created by the forgetful functor to \(\mathcal {C} \), and we therefore see that \(H^T\) is finitary. Thus, by Theorem 3.8, \(\vartheta H^T\) exists and is a subcoalgebra of \(\nu H^T\). By [37] and [10, Corollary 3.4.19], we know that \(\nu H^T\) is carried by \(\nu H\) equipped with a canonical algebra structure.

Now let us turn to the desired characterization of \(\vartheta H^T\). Formally, the coalgebraic language semantics of all fp-carried HT-coalgebras is collected by forming the colimit \(k: K \rightarrow HK\) of the diagram \( \mathsf {Coalg}_{\mathsf {fg}} HT \xrightarrow {T'} \mathsf {Coalg} H^T \xrightarrow {U} \mathsf {Coalg} H. \) This coalgebra K is not yet a subcoalgebra of \(\nu H\) (for \(\mathcal {C} = \mathsf {Set} \) that means, not all behaviourally equivalent states are identified in K), but taking its image in \(\nu H\) we obtain the LFF:

One can also directly take the union of all desired behaviours, for \(\mathcal {C} = \mathsf {Set} \):

Theorem 4.4

The locally finite fixpoint of the lifting \(H^T\) comprises precisely the images of determinized HT-coalgebras:

Open image in new window

(1)

This result suggests that the locally finite fixpoint is the right object to consider in order to represent finite behaviour. We now instantiate the general theory with examples from the literature to characterize several well-known notions as LFF.

4.2 The Languages of Non-deterministic Automata

Let us start with a simple standard example. We already mentioned that non-deterministic automata are coalgebras for the functor \(X \mapsto 2 \times {\mathcal {P}_ \mathsf f } (X)^\varSigma \). Hence they are HT-coalgebras for \(H = 2 \times (-)^\varSigma \) and \(T = {\mathcal {P}_ \mathsf f } \) the finite powerset monad on \(\mathsf {Set} \). The above generalized powerset construction then instantiates as the usual powerset construction that assigns to a given non-deterministic automaton its determinization.

Now note that the final coalgebra for H is carried by the set \(\mathcal L = {\mathcal {P}} (\varSigma ^*)\) of all formal languages over \(\varSigma \) with the coalgebra structure given by \(o: \mathcal L \rightarrow 2\) with \(o(L) = 1\) iff L contains the empty word and \(t: \mathcal L \rightarrow \mathcal {L}^\varSigma \) with \(t(L)(s) = \{ w \mid sw \in L\}\) the left language derivative. The functor H has a canonical lifting \(H^T\) on the Eilenberg-Moore category of \({\mathcal {P}_ \mathsf f } \), viz. the category of join semi-lattices. The final coalgebra \(\nu H^T\) is carried by all formal languages with the join semi-lattice structure given by union and \(\emptyset \) and with the above coalgebra structure. Furthermore, the coalgebraic language semantics of \(x: X \rightarrow HTX\) assigns to every state of the non-deterministic automaton X the language it accepts. Observe that join semi-lattices form a so-called locally finite variety, i.e. the finitely presentable algebras are precisely the finite ones. Hence, Theorem 4.4 states that the LFF of \(H^T\) is precisely the subcoalgebra of \(\nu H^T\) formed by all languages accepted by finite NFA, i.e. regular languages.

Note that in this example the LFF and the rational fixpoint coincide since both fp and fg join semi-lattices are simply the finite ones. Similar characterizations of the coalgebraic language semantics of finite coalgebras follow from Theorem 4.4 in other instances of the generalized powerset construction from [41] (cf. e.g. the treatment of the behaviour of finite weighted automata in [12]).

We now turn to examples that, to the best of our knowledge, cannot be treated using the rational fixpoint.

4.3 The Behaviour of Stack Machines

Push-down automata are finite state machines with infinitely many configurations. It is well-known that deterministic and non-deterministic pushdown automata recognize different classes of context-free languages. We will characterize both as instances of the locally finite fixpoint, using the results from [23] on stack machines, which can push or read multiple elements to or from the stack in a single transition, respectively.

That is, a transition of a stack machine in a certain state consists of reading an input character, going to a successor state based on the stack’s topmost elements and of modifying the topmost elements of the stack. These stack operations are captured by the stack monad.

Note that the parameter k gives a bound on how may of the topmost stack cells the machine can access in one step.

Using the stack monad, stack machines are HT-coalgebras, where \(H=B\times (-)^\varSigma \) is the Moore automata functor for the finite input alphabet \(\varSigma \) and the set B of all predicates mapping (initial) stack configurations to output values from 2, taking only the topmost k elements into account: \( B = \{ p \in 2^{\varGamma ^*} \mid \exists k\in \mathbb {N}_0: \forall w,u\in \varGamma ^*, |w|\ge k: p(wu) = p(w) \} \subseteq 2^{\varGamma ^*} \).

The final coalgebra \(\nu H\) is carried by \(B^{\varSigma ^*}\) which is (modulo power laws) a set of predicates, mapping stack configurations to formal languages. Goncharov et al. [23] show that H lifts to \(\mathsf {Set} ^T\) and conclude that finite-state HT-coalgebras match the intuition of deterministic pushdown automata without spontaneous transitions. The languages accepted by those automata are precisely the real-time deterministic context-free languages; this notion goes back to Harrison and Havel [24]. We obtain the following, with \(\gamma _0\) playing the role of an initial symbol on the stack:

Just as for pushdown automata, the expressiveness of stack machines increases when equipping them with non-determinism. Technically, this is done by considering the non-deterministic stack monad\(T'\), i.e. \(T'\) denotes a submonad of the non-deterministic store monad \({\mathcal {P}_ \mathsf f } (-\times \varGamma ^*)^{\varGamma ^*}\), as described in [23, Sect. 6]. In the non-deterministic setting, a similar property holds, namely that the determinized \(HT'\)-coalgebras with finite carrier describe precisely the context-free languages [23, Theorem 6.5]. Combine this with (1):

4.4 Context-Free Languages and Constructively S-Algebraic Power Series

One generalizes from formal (resp. context-free) languages to weighted formal (resp. context-free) languages by assigning to each word a weight from a fixed semiring. More formally, a weighted language – a.k.a. formal power series – over an input alphabet X is defined as a map \(X^* \rightarrow S\), where S is a semiring. The set of all formal power series is denoted by \(S\langle \!\langle X\rangle \!\rangle \). Ordinary formal languages are formal power series over the boolean semiring \(\mathbb {B}= \{0,1\}\), i.e. maps \(X^*\rightarrow \{0,1\}\).

An important class of formal power series is that of constructively S-algebraic formal power series. We show that this class arises precisely as the LFF of the standard functor for deterministic Moore automata \(H = S\times (-)^\varSigma \), but on an Eilenberg-Moore category of a \(\mathsf {Set}\) monad. As a special case, constructively \(\mathbb {B}\)-algebraic series are the context-free weighted languages and are precisely the LFF of the automata functor in the category of idempotent semirings.

The original definition of constructively S-algebraic formal power series goes back to Fliess [19], see also [17]. Here, we use the equivalent coalgebraic characterization by Winter et al. [44].

Let \(S\langle X\rangle \subseteq S\langle \!\langle X\rangle \!\rangle \) the subset of those maps, that are 0 for all but finitely many \(w\in X^*\). If S is commutative, then \(S\langle -\rangle \) yields a finitary monad and thus also \(T=S\langle -+\varSigma \rangle \) by Example 4.1. The algebras for \(S\langle -\rangle \) are associative S-algebras (over the commutative semiring S), i.e. S-modules together with a monoid structure that is a module morphism in both arguments. The algebras for T are \(\varSigma \)-pointed S-algebras. The following notions are special instances of S-algebras.

Winter et al. [44, Proposition 4] show that the final H-coalgebra is carried by \(S\langle \!\langle \varSigma \rangle \!\rangle \) and that constructively S-algebraic series are precisely those elements of \(S\langle \!\langle \varSigma \rangle \!\rangle \) that arise as the behaviours of those coalgebra \(c: X\rightarrow HS\langle X\rangle \) with finite X, after determinizing them to some \(c^\sharp : S\langle X\rangle \rightarrow HS\langle X\rangle \) (see [44, Theorem 23]).

It follows that \(\hat{c}^\dagger = c^{\sharp \dagger }\cdot S\langle \mathsf {inl}\rangle \) and thus their images in \(\nu H\) are identical. Hence, a formal power series is constructively S-algebraic iff it is in the image of some \(c^{\sharp \dagger }\cdot S\langle \mathsf {inl}\rangle \), and by (1), iff it is in the locally finite fixpoint of \(H^T\).

Open image in new window

4.5 Courcelle’s Algebraic Trees

For a fixed signature \(\varSigma \) of so called givens, a recursive program scheme (or rps, for short) contains mutually recursive definitions of new operations \(\varphi _1,\ldots ,\varphi _k\) (with respective arities \(n_1,\ldots ,n_k\)). The recursive definition of \(\varphi _i\) may involve symbols from \(\varSigma \), operations \(\varphi _1,\ldots ,\varphi _k\) and \(n_i\) variables \(x_1,\ldots ,x_{n_i}\). The (uninterpreted) solution of an rps is obtained by unravelling these recursive definitions, producing a possibly infinite \(\varSigma \)-tree over \(x_1,\ldots ,x_{n_i}\) for each operation \(\varphi _i\). Figure 1 shows an rps over the signature Open image in new window and its solution. In general, an algebraic \(\varSigma \)-tree is a \(\varSigma \)-tree which is definable by an rps over \(\varSigma \) (see Courcelle [16]). Generalizing from a signature to a finitary endofunctor \(H:\mathcal {C} \rightarrow \mathcal {C} \) on an lfp category, Adámek et al. [7] describe an rps as a coalgebra for a functor \(\mathcal {H}_ \mathsf f \) on \(H/\mathsf {Mnd}_ \mathsf f (\mathcal {C})\), in which objects are finitary H-pointed monads on \(\mathcal {C} \), i.e. finitary monads M together with a natural transformation \(H\rightarrow M\). They introduce the context-free monad \(C^H\) of H, which is an H-pointed monad that is a subcoalgebra of the final coalgebra for \(\mathcal {H}_ \mathsf f \) and which is the monad of Courcelle’s algebraic \(\varSigma \)-trees in the special case where \(\mathcal {C} = \mathsf {Set} \) and H is a polynomial functor associated to a signature \(\varSigma \). We will prove that this monad is the LFF of \(\mathcal {H}_ \mathsf f \), and thereby we characterize it by a universal property – solving the open problem in [7].

The setting is again an instance of the generalized powerset construction, but this time with \(\mathsf {Fun}_ \mathsf f (\mathcal {C})\) as the base category in lieu of \(\mathsf {Set} \). Let \(\mathcal {C} \) be an lfp category in which the coproduct injections are monic and consider a finitary, mono-preserving endofunctor \(H: \mathcal {C} \rightarrow \mathcal {C} \). Denote by \(\mathsf {Fun}_ \mathsf f (\mathcal {C})\) the category of finitary endofunctors on \(\mathcal {C} \). Then H induces an endofunctor \(H\cdot (-)+ Id \) on \(\mathsf {Fun}_ \mathsf f (\mathcal {C})\), denoted \(\dot{H}\) and mapping an endofunctor V to the functor \(X\mapsto HVX+X\). This functor \(\dot{H}\) gets precomposed with a monad on \(\mathsf {Fun}_ \mathsf f (\mathcal {C})\) as we now explain.

For example, if H is a polynomial functor associated to a signature \(\varSigma \), then \(F^{H}X\) is the usual term algebra that contains all finite \(\varSigma \)-trees over the set of generators X. Proposition 4.11 implies that \(H \mapsto F^H\) is the object assignment of a monad on \(\mathsf {Fun}_ \mathsf f (\mathcal {C})\). The Eilenberg-Moore category of \(F^{(-)}\) is easily seen to be \(\mathsf {Mnd}_ \mathsf f (\mathcal {C})\), the category of finitary monads on \(\mathcal {C} \). Here, fp and fg objects differ, see [45, Sect. 5.4.1] for a proof.

Similarly as in the case of context-free languages, we will work with the monad \(E^{(-)} = F^{H+(-)}\), so we get H-pointed finitary monads as the \(E^{(-)}\)-algebras. This category is equivalent to a slice category: the universal property induced by \(F^{(-)}\) states, that for any finitary monad B the natural transformations \(H\rightarrow B\) are in one-to-one correspondence with monad morphisms \(F^H \rightarrow B\); so the category \(H/\mathsf {Mnd}_ \mathsf f (\mathcal {C})\) of finitary H-pointed monads on \(\mathcal {C} \) is isomorphic to the slice category \(F^H/\mathsf {Mnd}_ \mathsf f (\mathcal {C})\). This finishes the description of the base category and we now lift the functor \(\dot{H}\) to this category.

Consider an H-pointed monad \((B,\beta : H\rightarrow \mathcal {C}) \in H/\mathsf {Mnd}_ \mathsf f (\mathcal {C})\). By [21], the endofunctor \(H\cdot B+ Id \) carries a canonical monad structure. Furthermore, we have an obvious pointing \(\mathsf {inl}\cdot H\eta ^B: H\rightarrow H\cdot B + Id \). By [32], this defines an endofunctor on H-pointed monads, \( \mathcal {H}_ \mathsf f : H/\mathsf {Mnd}_ \mathsf f (\mathcal {C})\rightarrow H/\mathsf {Mnd}_ \mathsf f (\mathcal {C}), \) which is a lifting of \(\dot{H}\). In order to verify that \(\mathcal {H}_ \mathsf f \) is finitary, we first need to know how filtered colimits look in \(H/\mathsf {Mnd}_ \mathsf f (\mathcal {C})\).

Clearly, the canonical projection functor \(H/\mathsf {Mnd}_ \mathsf f (\mathcal {C}) \rightarrow \mathsf {Mnd}_ \mathsf f (\mathcal {C})\) creates filtered colimits, too. Therefore, filtered colimits in the slice category \(H/\mathsf {Mnd}_ \mathsf f (\mathcal {C})\) are formed on the level of \(\mathsf {Fun}_ \mathsf f (\mathcal {C})\), i.e. object-wise. The functor \(\dot{H}\) is finitary on \(\mathsf {Fun}_ \mathsf f (\mathcal {C})\) and thus also its lifting \(\mathcal {H}_ \mathsf f \) is finitary. So all requirements from Assumption 3.1 are met: we have a finitary endofunctor \(\mathcal {H}_ \mathsf f \) on the lfp category \(H/\mathsf {Mnd}_ \mathsf f (\mathcal {C})\), and by [7, Corollary 2.20] \(\mathcal {H}_ \mathsf f \) preserves monos since H does. By Theorem 3.8, \(\mathcal {H}_ \mathsf f \) has a locally finite fixpoint.

Adámek et al. [7] characterize a (guarded) recursive program scheme as a natural transformation \(V \rightarrow H \cdot E^{V} + Id \) with V fp (in \(\mathsf {Fun}_ \mathsf f (\mathcal {C})\)), or equivalently, via the generalized powerset construction w.r.t. the monad \(E^{(-)}\) as an \(\mathcal {H}_ \mathsf f \)-coalgebra on the carrier \(E^V\) (in \(\mathsf {Mnd}_ \mathsf f (\mathcal {C})\)). These \(\mathcal {H}_ \mathsf f \)-coalgebras on carriers \(E^{V}\) where \(V\in \mathsf {Fun}_ \mathsf f (\mathcal {C})\) is fp form the full subcategory \(\mathsf {EQ}_\text {} \subseteq \mathsf {Coalg} \mathcal {H}_ \mathsf f \). They show two equivalent ways of constructing the monad of Courcelle’s algebraic trees for the case \(\mathcal {C} =\mathsf {Set} \): as the image of \({\mathsf {EQ}_\text {colim}}\) in the final coalgebra T of Remark 4.13, and as the colimit of \(\mathsf {EQ}_\text {2} \), where \(\mathsf {EQ}_\text {2} \) is the closure of \(\mathsf {EQ}_\text {} \) under strong quotients. We now provide a third characterization, and show that the monad of Courcelle’s algebraic trees is the locally finite fixpoint of \(\mathcal {H}_ \mathsf f \).

To this end it suffices to show that \(\mathsf {EQ}_\text {2} \) is precisely the diagram of \(\mathcal {H}_ \mathsf f \)-coalgebras with an fg carrier. This is established with the help of the following two technical lemmas. We now assume that \(\mathcal {C} = \mathsf {Set} \).

We have the following variation of Proposition 3.17:

Note that the last property holds because \(H/\mathsf {Mnd}_ \mathsf f (\mathsf {Set})\) is an Eilenberg-Moore category and \(E^V\) is the free Eilenberg-Moore algebra on the fp object V. It follows from Lemma 4.15 that \(\mathsf {Coalg}_{\mathsf {fg}} \mathcal {H}_ \mathsf f \) is the same category as \(\mathsf {EQ}_\text {2} \); thus their colimits in \(\mathsf {Coalg} \mathcal {H}_ \mathsf f \) are isomorphic and we conclude: