Presheaves and nominal sets provide alternative abstract models of sets of syntactic objects with free and bound variables, such as λ-terms. One distinguishing feature of the presheaf-based perspective is its elegant syntax-free characterization of substitution using a closed monoidal structure. In this paper, we introduce a corresponding closed monoidal structure on nominal sets, in the spirit of Fiore et al.’s substitution tensor for presheaves over finite sets. To this end, we present a general method to derive a closed monoidal structure on a category from a given action of a monoidal category on that category. We demonstrate that this method not only uniformly recovers known substitution tensors for various kinds of presheaf categories but also yields notions of substitution tensor for nominal sets and their relatives, such as renaming sets. In the process, we shed new light on different incarnations of nominal sets and (pre-)sheaf categories and establish a number of correspondences between them.

Codensity monads provide a universal method to generate complex monads from simple functors. Recently, a wide range of important monads in logic, denotational semantics, and probabilistic computation, such as several incarnations of the ultrafilter monad, the Vietoris monad, and the Giry monad, have been presented as codensity monads, using complex arguments. We propose a unifying categorical approach to codensity presentations of monads, based on the idea of relating the presenting functor to a dense functor via a suitable duality between categories. We prove a general presentation result applying to every such situation and demonstrate that most codensity presentations known in the literature emerge from this strikingly simple duality-based setup, drastically alleviating the complexity of their proofs and in many cases completely reducing them to standard duality results. Additionally, we derive a number of novel codensity presentations using our framework, including the first non-trivial codensity presentations for the filter monads on sets and topological spaces, the lower Vietoris monad on topological spaces, and the expectation monad on sets.

Compositionality proofs in higher-order languages are notoriously involved, and general semantic frameworks guaranteeing compositionality are hard to come by. In particular, Turi and Plotkin's bialgebraic abstract GSOS framework, which provides off-the-shelf compositionality results for first-order languages, so far does not apply to higher-order languages. In the present work, we develop a theory of abstract GSOS specifications for higher-order languages, in effect transferring the core principles of Turi and Plotkin's framework to a higher-order setting. In our theory, the operational semantics of higher-order languages is represented by certain dinatural transformations that we term (pointed) higher-order GSOS laws. We give a general compositionality result that applies to all systems specified in this way and discuss how compositionality of combinatory logics and the lambda-calculus w.r.t. a strong variant of Abramsky's applicative bisimilarity are obtained as instances. Extended and updated version of arXiv:2210.13387

Message Authentication Codes (MACs) represent a fundamental symmetric key primitive, serving to ensure the authenticity and integrity of transmitted data. As a building block in authenticated encryption and in numerous deployed standards, including TLS, IPsec, and SSH, MACs play a central role in practice. Due to their importance for practice, MACs have been subject to extensive research, leading to prominent schemes such as HMAC, CBCMAC, or LightMAC. Despite the existence of various MACs, there is still considerable interest in creating schemes that are more efficient, potentially parallelizable, or have specific non-cryptographic attributes, such as being patent-free. In this context, we introduce an automated method for analyzing and synthesizing MAC schemes. In order to achieve this goal, we have constructed a framework that restricts the class of MACs in such a way that it is sufficiently expressive to cover known constructions, yet also admits automated reasoning about the security guarantees of both known and new schemes. Our automated analysis has identified a novel category of MACs, termed “hybrid” MACs. These MACs operate by processing multiple blocks concurrently, with each block managed by a different, specified MAC scheme. A key finding is that in certain scenarios, the hybrid MAC marginally outperforms the simultaneous operation of the individual MACs. This improvement is attributed to the hybrid approach exploiting the strengths and compensating for the weaknesses of each distinct MAC scheme involved. Our implementation confirms that we have successfully identified new schemes that have comparable performance with state-of-the-art schemes and in some settings seem to be slightly more efficient.

Regular languages - the languages accepted by deterministic finite automata - are known to be precisely the languages recognized by finite monoids. This characterization is the origin of algebraic language theory. In this paper, we generalize the correspondence between automata and monoids to automata with generic computational effects given by a monad, providing the foundations of an effectful algebraic language theory. We show that, under suitable conditions on the monad, a language is computable by an effectful automaton precisely when it is recognizable by (1) an effectful monoid morphism into an effect-free finite monoid, and (2) a monoid morphism into a monad-monoid bialgebra whose carrier is a finitely generated algebra for the monad, the former mode of recognition being conceptually completely new. Our prime application is a novel algebraic approach to languages computed by probabilistic finite automata. Additionally, we derive new algebraic characterizations for nondeterministic probabilistic finite automata and for weighted finite automata over unrestricted semirings, generalizing previous results on weighted algebraic recognition over commutative rings.

Formal languages over infinite alphabets serve as abstractions of structures and processes carrying data. Automata models over infinite alphabets, such as classical register automata or, equivalently, nominal orbit-finite automata, tend to have computationally hard or even undecidable reasoning problems unless stringent restrictions are imposed on either the power of control or the number of registers. This has been shown to be ameliorated in automata models with name allocation such as regular nondeterministic nominal automata, which allow for deciding language inclusion in elementary complexity even with unboundedly many registers while retaining a reasonable level of expressiveness. In the present work, we demonstrate that elementary complexity survives under extending the power of control to alternation: We introduce regular alternating nominal automata (RANAs), and show that their non-emptiness and inclusion problems have elementary complexity even when the number of registers is unbounded. Moreover, we show that RANAs allow for nearly complete de-alternation, specifically de-alternation up to a single deadlocked universal state. As a corollary to our results, we improve the complexity of model checking for a flavour of Bar-µTL, a fixed-point logic with name allocation over finite data words, by one exponential level.

Extensions of Stone-type dualities have a long history in algebraic logic and have also been instrumental in proving results in algebraic language theory. We show how to extend abstract categorical dualities via monoidal adjunctions, subsuming various incarnations of classical extended Stone and Priestley duality as special cases, and providing the foundation for two new concrete dualities: First, we investigate residuation algebras, which are lattices with additional residual operators modeling language derivatives algebraically. We show that the subcategory of derivation algebras is dually equivalent to the category of profinite ordered monoids, restricting to a duality between Boolean residuation algebras and profinite monoids. We further refine this duality to capture relational morphisms of profinite ordered monoids, which dualize to natural morphisms of residuation algebras. Second, we apply the categorical extended duality to the discrete setting of sets and complete atomic Boolean algebras to obtain a concrete description for the dual of the category of all small categories.

Reasoning about program equivalence in imperative languages is notoriously challenging, as the presence of states (in the form of variable stores) fundamentally increases the observational power of program terms. The key desideratum for any notion of equivalence is compositionality , guaranteeing that subprograms can be safely replaced by equivalent subprograms regardless of the context. To facilitate compositionality proofs and avoid boilerplate work, one would hope to employ the abstract bialgebraic methods provided by Turi and Plotkin's powerful theory of mathematical operational semantics (a.k.a. abstract GSOS ) or its recent extension by Goncharov et al. to higher-order languages. However, multiple attempts to apply abstract GSOS to stateful languages have thus failed. We propose a novel approach to the operational semantics of stateful languages based on the formal distinction between readers (terms that expect an initial input store before being executed), and writers (running terms that have already been provided with a store). In contrast to earlier work, this style of semantics is fully compatible with abstract GSOS, and we can thus leverage the existing theory to obtain coinductive reasoning techniques. We demonstrate that our approach generates non-trivial compositionality results for stateful languages with first-order and higher-order store and that it flexibly applies to program equivalences at different levels of granularity, such as trace, cost, and natural equivalence.

Abstract Extensions of Stone-type dualities have a long history in algebraic logic and have also been instrumental for proving results in algebraic language theory. We show how to extend abstract categorical dualities via monoidal adjunctions, subsuming various incarnations of classical extended Stone and Priestley duality as a special case. Guided by these categorical foundations, we investigate residuation algebras, which are algebraic models of language derivatives, and show the subcategory of derivation algebras to be dually equivalent to the category of profinite ordered monoids, restricting to a duality between boolean residuation algebras and profinite monoids. We further extend this duality to capture relational morphisms of profinite ordered monoids, which dualize to natural morphisms of residuation algebras.

Logical relations constitute a key method for reasoning about contextual equivalence of programs in higher-order languages. They are usually developed on a per-case basis, with a new theory required for each variation of the language or of the desired notion of equivalence. In the present paper we introduce a general construction of (step-indexed) logical relations at the level of Higher-Order Mathematical Operational Semantics, a highly parametric categorical framework for modeling the operational semantics of higherorder languages. Our main result states that for languages whose weak operational model forms a lax bialgebra, the logical relation is automatically sound for contextual equivalence. Our abstract theory is shown to instantiate to combinatory logics and λ-calculi with recursive types, and to different flavours of contextual equivalence.

The initial algebra for an endofunctor F provides a recursion and induction scheme for data structures whose constructors are described by F. The initial-algebra construction by Adámek (1974) starts with the initial object (e.g. the empty set) and successively applies the functor until a fixed point is reached, an idea inspired by Kleene's fixed point theorem. Depending on the functor of interest, this may require transfinitely many steps indexed by ordinal numbers until termination.

Compositionality proofs in higher-order languages are notoriously involved, and general semantic frameworks guaranteeing compositionality are hard to come by. In particular, Turi and Plotkin's bialgebraic abstract GSOS framework, which has been successfully applied to obtain off-the-shelf compositionality results for first-order languages, so far does not apply to higher-order languages. In the present work, we develop a theory of abstract GSOS specifications for higher-order languages, in effect transferring the core principles of Turi and Plotkin's framework to a higher-order setting. In our theory, the operational semantics of higher-order languages is represented by certain dinatural transformations that we term pointed higher-order GSOS laws. We give a general compositionality result that applies to all systems specified in this way and discuss how compositionality of the SKI calculus and the $λ$-calculus w.r.t. a strong variant of Abramsky's applicative bisimilarity are obtained as instances.

The Vietoris space of compact subsets of a given Hausdorff space yields an endofunctor V on the category of Hausdorff spaces. Vietoris polynomial endofunctors on that category are built from V, the identity and constant functors by forming products, coproducts and compositions. These functors are known to have terminal coalgebras and we deduce that they also have initial algebras. We present an analogous class of endofunctors on the category of extended metric spaces, using in lieu of V the Hausdorff functor ℋ. We prove that the ensuing Hausdorff polynomial functors have terminal coalgebras and initial algebras. Whereas the canonical constructions of terminal coalgebras for Vietoris polynomial functors takes ω steps, one needs ω + ω steps in general for Hausdorff ones. We also give a new proof that the closed set functor on metric spaces has no fixed points.

We propose a novel topological perspective on data languages recognizable by orbit-finite nominal monoids. For this purpose, we introduce pro-orbit-finite nominal topological spaces. Assuming globally bounded support sizes, they coincide with nominal Stone spaces and are shown to be dually equivalent to a subcategory of nominal boolean algebras. Recognizable data languages are characterized as topologically clopen sets of pro-orbit-finite words. In addition, we explore the expressive power of pro-orbit-finite equations by establishing a nominal version of Reiterman's pseudovariety theorem.

Higher-order abstract GSOS is a recent extension of Turi and Plotkin’s framework of Mathematical Operational Semantics to higher-order languages. The fundamental well-behavedness property of all specifications within the framework is that coalgebraic strong (bi)similarity on their operational model is a congruence. In the present work, we establish a corresponding congruence theorem for weak similarity, which is shown to instantiate to well-known concepts such as Abramsky’s applicative similarity for the λ-calculus. On the way, we develop several techniques of independent interest at the level of abstract categories, including relation liftings of mixed-variance bifunctors and higher-order GSOS laws, as well as Howe’s method.

Positive data languages are languages over an infinite alphabet closed under possibly non-injective renamings of data values. Informally, they model properties of data words expressible by assertions about equality, but not inequality, of data values occurring in the word. We investigate the class of positive data languages recognizable by nondeterministic orbit-finite nominal automata, an abstract form of register automata introduced by Bojańczyk, Klin, and Lasota. As our main contribution we provide a number of equivalent characterizations of that class in terms of positive register automata, monadic second-order logic with positive equality tests, and finitely presentable nondeterministic automata in the categories of nominal renaming sets and of presheaves over finite sets.

Compositionality proofs in higher-order languages are notoriously involved, and general semantic frameworks guaranteeing compositionality are hard to come by. In particular, Turi and Plotkin’s bialgebraic abstract GSOS framework, which has been successfully applied to obtain off-the-shelf compositionality results for first-order languages, so far does not apply to higher-order languages. In the present work, we develop a theory of abstract GSOS specifications for higher-order languages, in effect transferring the core principles of Turi and Plotkin’s framework to a higher-order setting. In our theory, the operational semantics of higher-order languages is represented by certain dinatural transformations that we term pointed higher-order GSOS laws . We give a general compositionality result that applies to all systems specified in this way and discuss how compositionality of the SKI calculus and the λ-calculus w.r.t. a strong variant of Abramsky’s applicative bisimilarity are obtained as instances.

Compositionality of denotational semantics is an important concern in programming semantics. Mathematical operational semantics in the sense of Turi and Plotkin guarantees compositionality, but seen from the point of view of stateful computation it applies only to very fine-grained equivalences that essentially assume unrestricted interference by the environment between any two statements. We introduce the more restrictive stateful SOS rule format for stateful languages. We show that compositionality of two more coarse-grained semantics, respectively given by assuming read-only interference or no interference between steps, remains an undecidable property even for stateful SOS. However, further restricting the rule format in a manner inspired by the cool GSOS formats of Bloom and van Glabbeek, we obtain the streamlined and cool stateful SOS formats, which respectively guarantee compositionality of the two more abstract equivalences.

The framework of graded semantics uses graded monads to capture behavioural equivalences of varying granularity, for example as found in the linear-time / branching-time spectrum, over general system types. We describe a generic Spoiler-Duplicator game for graded semantics that is extracted from the given graded monad, and may be seen as playing out an equational proof; instances include standard pebble games for simulation and bisimulation as well as games for trace-like equivalences and coalgebraic behavioural equivalence. Considerations on an infinite variant of such games lead to a novel notion of infinite-depth graded semantics. Under reasonable restrictions, the infinite-depth graded semantics associated to a given graded equivalence can be characterized in terms of a determinization construction for coalgebras under the equivalence at hand.

Abstract Partition refinement is a method for minimizing automata and transition systems of various types. Recently we have developed a partition refinement algorithm and the tool that is generic in the transition type of the input system and matches the theoretical run time of the best known algorithms for many concrete system types. Genericity is achieved by modelling transition types as functors on sets and systems as coalgebras. Experimentation has shown that memory consumption is a bottleneck for handling systems with a large state space, while running times are fast. We have therefore extended an algorithm due to Blom and Orzan, which is suitable for a distributed implementation to the coalgebraic level of genericity, and implemented it in . Experiments show that this allows to handle much larger state spaces. Running times are low in most experiments, but there is a significant penalty for some.

We provide a generic algorithm for constructing formulae that distinguish behaviourally inequivalent states in systems of various transition types such as nondeterministic, probabilistic or weighted; genericity over the transition type is achieved by working with coalgebras for a set functor in the paradigm of universal coalgebra. For every behavioural equivalence class in a given system, we construct a formula which holds precisely at the states in that class. The algorithm instantiates to deterministic finite automata, transition systems, labelled Markov chains, and systems of many other types. The ambient logic is a modal logic featuring modalities that are generically extracted from the functor; these modalities can be systematically translated into custom sets of modalities in a postprocessing step. The new algorithm builds on an existing coalgebraic partition refinement algorithm. It runs in time O((m+n) log n) on systems with n states and m transitions, and the same asymptotic bound applies to the dag size of the formulae it constructs. This improves the bounds on run time and formula size compared to previous algorithms even for previously known specific instances, viz. transition systems and Markov chains; in particular, the best previous bound for transition systems was O(mn).

An old theorem of Adámek constructs initial algebras for sufficiently cocontinuous endofunctors via transfinite iteration over ordinals in classical set theory. We prove a new version that works in constructive logic, using "inflationary" iteration over a notion of size that abstracts from limit ordinals just their transitive, directed and well-founded properties. Borrowing from Taylor's constructive treatment of ordinals, we show that sizes exist with upper bounds for any given signature of indexes. From this it follows that there is a rich class of endofunctors to which the new theorem applies, provided one admits a weak form of choice (WISC) due to Streicher, Moerdijk, van den Berg and Palmgren, and which is known to hold in the internal constructive logic of many kinds of topos.

We introduce a framework for universal algebra in categories of relational structures given by finitary relational signatures and finitary or infinitary Horn theories, with the arity $λ$ of a Horn theory understood as a strict upper bound on the number of premisses in its axioms; key examples include partial orders ($λ=ω$) or metric spaces ($λ=ω_1$). We establish a bijective correspondence between $λ$-accessible enriched monads on the given category of relational structures and a notion of $λ$-ary algebraic theories (i.e. with operations of arity $

Infinite words over infinite alphabets serve as models of the temporal development of the allocation and (re-)use of resources over linear time. We approach ω-languages over infinite alphabets in the setting of nominal sets, and study languages of infinite bar strings, i.e. infinite sequences of names that feature binding of fresh names; binding corresponds roughly to reading letters from input words in automata models with registers. We introduce regular nominal nondeterministic Büchi automata (Büchi RNNAs), an automata model for languages of infinite bar strings, repurposing the previously introduced RNNAs over finite bar strings. Our machines feature explicit binding (i.e. resource-allocating) transitions and process their input via a Büchi-type acceptance condition. They emerge from the abstract perspective on name binding given by the theory of nominal sets. As our main result we prove that, in contrast to most other nondeterministic automata models over infinite alphabets, language inclusion of Büchi RNNAs is decidable and in fact elementary. This makes Büchi RNNAs a suitable tool for applications in model checking.

We provide a generic algorithm for constructing formulae that distinguish behaviourally inequivalent states in systems of various transition types such as nondeterministic, probabilistic or weighted; genericity over the transition type is achieved by working with coalgebras for a set functor in the paradigm of universal coalgebra. For every behavioural equivalence class in a given system, we construct a formula which holds precisely at the states in that class. The algorithm instantiates to deterministic finite automata, transition systems, labelled Markov chains, and systems of many other types. The ambient logic is a modal logic featuring modalities that are generically extracted from the functor; these modalities can be systematically translated into custom sets of modalities in a postprocessing step. The new algorithm builds on an existing coalgebraic partition refinement algorithm. It runs in time $\mathcal{O}((m+n) \log n)$ on systems with $n$ states and $m$ transitions, and the same asymptotic bound applies to the dag size of the formulae it constructs. This improves the bounds on run time and formula size compared to previous algorithms even for previously known specific instances, viz. transition systems and Markov chains; in particular, the best previous bound for transition systems was $\mathcal{O}(m n)$.

Abstract We introduce a new measure on regular languages: theirnondeterministic syntactic complexity. It is the least degree of any extension of the ‘canonical boolean representation’ of the syntactic monoid. Equivalently, it is the least number of states of anysubatomicnondeterministic acceptor. It turns out that essentially all previous structural work on nondeterministic state-minimality computes this measure. Our approach rests on an algebraic interpretation of nondeterministic finite automata as deterministic finite automata endowed with semilattice structure. Crucially, the latter form a self-dual category.

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Like notions of process equivalence, behavioural preorders on processes come in many flavours, ranging from fine-grained comparisons such as ready simulation to coarse-grained ones such as trace inclusion. Often, such behavioural preorders are characterized in terms of theory inclusion in dedicated characteristic logics; e.g. simulation is characterized by theory inclusion in the positive fragment of Hennessy-Milner logic. We introduce a unified semantic framework for behavioural preorders and their characteristic logics in which we parametrize the system type as a functor on the category Pos of partially ordered sets following the paradigm of universal coalgebra, while behavioural preorders are captured as graded monads on Pos, in generalization of a previous approach to notions of process equivalence. We show that graded monads on Pos are induced by a form of graded inequational theories that we introduce here. Moreover, we provide a general notion of modal logic compatible with a given graded behavioural preorder, along with a criterion for expressiveness, in the indicated sense of characterization of the behavioural preorder by theory inclusion. We illustrate our main result on various behavioural preorders on labelled transition systems and probabilistic transition systems.

Logics and automata models for languages over infinite alphabets, such as Freeze LTL and register automata, serve the verification of processes or documents with data. They relate tightly to formalisms over nominal sets, such as nondetermininistic orbit-finite automata (NOFAs), where names play the role of data. Reasoning problems in such formalisms tend to be computationally hard. Name-binding nominal automata models such as {regular nondeterministic nominal automata (RNNAs)} have been shown to be computationally more tractable. In the present paper, we introduce a linear-time fixpoint logic Bar-μTL} for finite words over an infinite alphabet, which features full negation and freeze quantification via name binding. We show by a nontrivial reduction to extended regular nondeterministic nominal automata that even though Bar-μTL} allows unrestricted nondeterminism and unboundedly many registers, model checking Bar-μTL} over RNNAs and satisfiability checking both have elementary complexity. For example, model checking is in 2ExpSpace, more precisely in parametrized ExpSpace, effectively with the number of registers as the parameter.

Abstract Partition refinement is a method for minimizing automata and transition systems of various types. Recently, we have developed a partition refinement algorithm that is generic in the transition type of the given system and matches the run time of the best known algorithms for many concrete types of systems, e.g. deterministic automata as well as ordinary, weighted, and probabilistic (labelled) transition systems. Genericity is achieved by modelling transition types as functors on sets, and systems as coalgebras. In the present work, we refine the run time analysis of our algorithm to cover additional instances, notably weighted automata and, more generally, weighted tree automata. For weights in a cancellative monoid we match, and for non-cancellative monoids such as (the additive monoid of) the tropical semiring even substantially improve, the asymptotic run time of the best known algorithms. We have implemented our algorithm in a generic tool that is easily instantiated to concrete system types by implementing a simple refinement interface. Moreover, the algorithm and the tool are modular, and partition refiners for new types of systems are obtained easily by composing pre-implemented basic functors. Experiments show that even for complex system types, the tool is able to handle systems with millions of transitions.

Abstract For every finitary monad $T$ on sets and every endofunctor $F$ on the category of $T$-algebras, we introduce the concept of an ffg-Elgot algebra for $F$, i.e. an algebra admitting coherent solutions for finite systems of recursive equations with effects represented by the monad $T$. The goal is to study the existence and construction of free ffg-Elgot algebras. To this end, we investigate the locally ffg fixed point $\varphi F$, i.e. the colimit of all $F$-coalgebras with free finitely generated carrier, which is shown to be the initial ffg-Elgot algebra. This is the technical foundation for our main result: the category of ffg-Elgot algebras is monadic over the category of $T$-algebras.

Abstract Finitary monads on Pos are characterized as precisely the free-algebra monads of varieties of algebras. These are classes of ordered algebras specified by inequations in context. Analogously, finitary enriched monads on Pos are characterized: here we work with varieties of coherent algebras which means that their operations are monotone.

Profinite equations are an indispensable tool for the algebraic classification of formal languages. Reiterman’s theorem states that they precisely specify pseudovarieties, i.e., classes of finite algebras closed under finite products, subalgebras and quotients. In this article, Reiterman’s theorem is generalized to finite Eilenberg-Moore algebras for a monad T on a category D: we prove that a class of finite T -algebras is a pseudovariety iff it is presentable by profinite equations. As a key technical tool, we introduce the concept of a profinite monad T ^ associated to the monad T , which gives a categorical view of the construction of the space of profinite terms.

Abstract This paper studies fundamental questions concerning category-theoretic models of induction and recursion. We are concerned with the relationship between well-founded and recursive coalgebras for an endofunctor. For monomorphism preserving endofunctors on complete and well-powered categories every coalgebra has a well-founded part, and we provide a new, shorter proof that this is the coreflection in the category of all well-founded coalgebras. We present a new more general proof of Taylor’s General Recursion Theorem that every well-founded coalgebra is recursive, and we study conditions which imply the converse. In addition, we present a new equivalent characterization of well-foundedness: a coalgebra is well-founded iff it admits a coalgebra-to-algebra morphism to the initial algebra.

We present a generic partition refinement algorithm that quotients coalgebraic systems by behavioural equivalence, an important task in system analysis and verification. Coalgebraic generality allows us to cover not only classical relational systems but also, e.g. various forms of weighted systems and furthermore to flexibly combine existing system types. Under assumptions on the type functor that allow representing its finite coalgebras in terms of nodes and edges, our algorithm runs in time $\mathcal{O}(m\cdot \log n)$ where $n$ and $m$ are the numbers of nodes and edges, respectively. The generic complexity result and the possibility of combining system types yields a toolbox for efficient partition refinement algorithms. Instances of our generic algorithm match the run-time of the best known algorithms for unlabelled transition systems, Markov chains, deterministic automata (with fixed alphabets), Segala systems, and for color refinement.

Using recent developments in coalgebraic and monad-based semantics, we present a uniform study of various notions of machines, e.g., finite state machines, multi-stack machines, Turing machines, valence automata, and weighted automata. They are instances of Jacobs’s notion of a T - automaton , where T is a monad. We show that the generic language semantics for T -automata correctly instantiates the usual language semantics for a number of known classes of machines/languages, including regular, context-free, recursively-enumerable, and various subclasses of context free languages (e.g., deterministic and real-time ones). Moreover, our approach provides new generic techniques for studying the expressivity power of various machine-based models.

This extended abstract first presents a new category theoretic approach to equationally axiomatizable classes of algebras. This approach is well-suited for the treatment of algebras equipped with additional computationally relevant structure, such as ordered algebras, continuous algebras, quantitative algebras, nominal algebras, or profinite algebras. We present a generic HSP theorem and a sound and complete equational logic, which encompass numerous flavors of equational axiomizations studied in the literature. In addition, we use the generic HSP theorem as a key ingredient to obtain Eilenberg-type correspondences yielding algebraic characterizations of properties of regular machine behaviours. When instantiated for orbit-finite nominal monoids, the generic HSP theorem yields a crucial step for the proof of the first Eilenberg-type variety theorem for data languages.

State-based models of concurrent systems are traditionally considered under a variety of notions of process equivalence. In the case of labelled transition systems, these equivalences range from trace equivalence to (strong) bisimilarity, and are organized in what is known as the linear time - branching time spectrum. A combination of universal coalgebra and graded monads provides a generic framework in which the semantics of concurrency can be parametrized both over the branching type of the underlying transition systems and over the granularity of process equivalence. We show in the present paper that this framework of graded semantics does subsume the most important equivalences from the linear time - branching time spectrum. An important feature of graded semantics is that it allows for the principled extraction of characteristic modal logics. We have established invariance of these graded logics under the given graded semantics in earlier work; in the present paper, we extend the logical framework with an explicit propositional layer and provide a generic expressiveness criterion that generalizes the classical Hennessy-Milner theorem to coarser notions of process equivalence. We extract graded logics for a range of graded semantics on labelled transition systems and probabilistic systems, and give exemplary proofs of their expressiveness based on our generic criterion.

Abstract This paper proposes a new category theoretic account of equationally axiomatizable classes of algebras. Our approach is well-suited for the treatment of algebras equipped with additional computationally relevant structure, such as ordered algebras, continuous algebras, quantitative algebras, nominal algebras, or profinite algebras. Our main contributions are a generic HSP theorem and a sound and complete equational logic, which are shown to encompass numerous flavors of equational axiomizations studied in the literature.

For finite automata as coalgebras in a category C , we study languages they accept and varieties of such languages. This generalizes Eilenberg’s concept of a variety of languages, which corresponds to choosing as C the category of Boolean algebras. Eilenberg established a bijective correspondence between pseudovarieties of monoids and varieties of regular languages. In our generalization, we work with a pair C / D of locally finite varieties of algebras that are predual, i.e., dualize on the level of finite algebras, and we prove that pseudovarieties of D -monoids bijectively correspond to varieties of regular languages in C . As one instance, Eilenberg’s result is recovered by choosing D = sets and C = Boolean algebras. Another instance, Pin’s result on pseudovarieties of ordered monoids, is covered by taking D = posets and C = distributive lattices. By choosing as C amp;equals; D the self-predual category of join-semilattices, we obtain Polák’s result on pseudovarieties of idempotent semirings. Similarly, using the self-preduality of vector spaces over a finite field K , our result covers that of Reutenauer on pseudovarieties of K -algebras. Several new variants of Eilenberg’s theorem arise by taking other predualities, e.g., between the categories of non-unital Boolean rings and of pointed sets. In each of these cases, we also prove a local variant of the bijection, where a fixed alphabet is assumed and one considers local varieties of regular languages over that alphabet in the category C .

The syntactic monoid of a language is generalized to the level of a symmetric monoidal closed category $\mathcal D$. This allows for a uniform treatment of several notions of syntactic algebras known in the literature, including the syntactic monoids of Rabin and Scott ($\mathcal D=$ sets), the syntactic ordered monoids of Pin ($\mathcal D =$ posets), the syntactic semirings of Pol\'ak ($\mathcal D=$ semilattices), and the syntactic associative algebras of Reutenauer ($\mathcal D$ = vector spaces). Assuming that $\mathcal D$ is a commutative variety of algebras or ordered algebras, we prove that the syntactic $\mathcal D$-monoid of a language $L$ can be constructed as a quotient of a free $\mathcal D$-monoid modulo the syntactic congruence of $L$, and that it is isomorphic to the transition $\mathcal D$-monoid of the minimal automaton for $L$ in $\mathcal D$. Furthermore, in the case where the variety $\mathcal D$ is locally finite, we characterize the regular languages as precisely the languages with finite syntactic $\mathcal D$-monoids.

The rational fixed point of a set functor is well-known to capture the behaviour of finite coalgebras. In this paper we consider functors on algebraic categories. For them the rational fixed point may no longer be fully abstract, i.e. a subcoalgebra of the final coalgebra. Inspired by \'Esik and Maletti's notion of a proper semiring, we introduce the notion of a proper functor. We show that for proper functors the rational fixed point is determined as the colimit of all coalgebras with a free finitely generated algebra as carrier and it is a subcoalgebra of the final coalgebra. Moreover, we prove that a functor is proper if and only if that colimit is a subcoalgebra of the final coalgebra. These results serve as technical tools for soundness and completeness proofs for coalgebraic regular expression calculi, e.g. for weighted automata.

For an endofunctor H on a hyper-extensive category preserving countable coproducts we describe the free corecursive algebra on Y as the coproduct of the terminal coalgebra for H and the free H-algebra on Y. As a consequence, we derive that H is a cia functor, i.e., its corecursive algebras are precisely the cias (completely iterative algebras). Also all functors H(-) + Y are then cia functors. For finitary set functors we prove that, conversely, if H is a cia functor, then it has the form H = W \times (-) + Y for some sets W and Y.

The rational fixed point of a set functor is well-known to capture the behaviour of finite coalgebras. In this paper we consider functors on algebraic categories. For them the rational fixed point may no longer be a subcoalgebra of the final coalgebra. Inspired by Ésik and Maletti's notion of proper semiring, we introduce the notion of a proper functor. We show that for proper functors the rational fixed point is determined as the colimit of all coalgebras with a free finitely generated algebra as carrier and it is a subcoalgebra of the final coalgebra. Moreover, we prove that a functor is proper if and only if that colimit is a subcoalgebra of the final coalgebra. These results serve as technical tools for soundness and completeness proofs for coalgebraic regular expression calculi, e.g. for weighted automata.

We present a generic partition refinement algorithm that quotients coalgebraic systems by behavioural equivalence, an important task in reactive verification; coalgebraic generality implies in particular that we cover not only classical relational systems but also various forms of weighted systems. Under assumptions on the type functor that allow representing its finite coalgebras in terms of nodes and edges, our algorithm runs in time O(m log n) where n and m are the numbers of nodes and edges, respectively. Instances of our generic algorithm thus match the runtime of the best known algorithms for unlabelled transition systems, Markov chains, and deterministic automata (with fixed alphabets), and improve the best known algorithms for Segala systems.

The OSEK industrial standard governs the design of embedded real-time operating systems in the automotive domain. We report on efforts to develop verification methods for OSEK-conformant compilers, specifically of a code generator that weaves system calls and application code using a static configuration file, producing a stand-alone application that incorporates the relevant parts of the kernel. Our methodology involves two verification steps: On the one hand, we extract an OS-application interaction graph during the compilation phase and verify that it conforms to the standard, in particular regarding prioritized scheduling and interrupt handling. To this end, we generate from the configuration file a temporal specification of standard-conformant behaviour and model check the arising formulas on a labelled transition system extracted from the interaction graph. On the other hand, we verify that the actual generated code conforms to the interaction graph; this is done by graph isomorphism checking of the interaction graph against a dynamically-explored state-transition graph of the generated system.

Eilenberg-type correspondences, relating varieties of languages (e.g., of finite words, infinite words, or trees) to pseudovarieties of finite algebras, form the backbone of algebraic language theory. We show that they all arise from the same recipe: one models languages and the algebras recognizing them by monads on an algebraic category, and applies a Stone-type duality. Our main contribution is a variety theorem that covers e.g. Wilke's and Pin's work on infinity-languages, the variety theorem for cost functions of Daviaud, Kuperberg, and Pin, and unifies the two categorical approaches of Bojanczyk and of Adamek et al. In addition we derive new results, such as an extension of the local variety theorem of Gehrke, Grigorieff, and Pin from finite to infinite words.

Motivated by the recent interest in models of guarded (co-)recursion, we study their equational properties. We formulate axioms for guarded fixpoint operators generalizing the axioms of iteration theories of Bloom and Ésik. Models of these axioms include both standard (e.g., cpo-based) models of it eration theories and models of guarded recursion such as complete metric spaces or the topos of trees studied by Birkedal et al. We show that the standard result on the satisfaction of all Conway axioms by a unique dagger operation generalizes to the guarded setting. We also introduce the notion of guarded trace operator on a category, and we prove that guarded trace and guarded fixpoint operators are in one-to-one correspondence. Our results are intended as first steps leading, hopefully, towards future description of classifying theories for guarded recursion.

The syntactic monoid of a language is generalized to the level of a symmetric monoidal closed category D. This allows for a uniform treatment of several notions of syntactic algebras known in the literature, including the syntactic monoids of Rabin and Scott (D = sets), the syntactic semirings of Polak (D = semilattices), and the syntactic associative algebras of Reutenauer (D = vector spaces). Assuming that D is an entropic variety of algebras, we prove that the syntactic D-monoid of a language L can be constructed as a quotient of a free D-monoid modulo the syntactic congruence of L, and that it is isomorphic to the transition D-monoid of the minimal automaton for L in D. Furthermore, in case the variety D is locally finite, we characterize the regular languages as precisely the languages with finite syntactic D-monoids.

Models of concurrent systems employ a wide variety of semantics inducing various notions of process equivalence, ranging from linear-time semantics such as trace equivalence to branching-time semantics such as strong bisimilarity. Many of these generalize to system types beyond standard transition systems, featuring, for example, weighted, probabilistic, or game-based transitions; this motivates the search for suitable coalgebraic abstractions of process equivalence that cover these orthogonal dimensions of generality, i.e. are generic both in the system type and in the notion of system equivalence. In recent joint work with Kurz, we have proposed a parametrization of system equivalence over an embedding of the coalgebraic type functor into a monad. In the present paper, we refine this abstraction to use graded monads, which come with a notion of depth that corresponds, e.g., to trace length or bisimulation depth. We introduce a notion of graded algebras and show how they play the role of formulas in trace logics.

Kurz et al. have recently shown that infinite lambda-trees with finitely many free variables modulo alpha-equivalence form a final coalgebra for a functor on the category of nominal sets. Here we investigate the rational fixpoint of that functor. We prove that it is formed by all rational lambda-trees, i.e. those lambda-trees which have only finitely many subtrees (up to isomorphism). This yields a corecursion principle that allows the definition of operations such as substitution on rational lambda-trees.

Eilenberg's variety theorem, a centerpiece of algebraic automata theory, establishes a bijective correspondence between varieties of languages and pseudovarieties of monoids. In the present paper this result is generalized to an abstract pair of algebraic categories: we introduce varieties of languages in a category C, and prove that they correspond to pseudovarieties of monoids in a closed monoidal category D, provided that C and D are dual on the level of finite objects. By suitable choices of these categories our result uniformly covers Eilenberg's theorem and three variants due to Pin, Polák and Reutenauer, respectively, and yields new Eilenberg-type correspondences.

Coalgebras provide a uniform framework for studying dynamical systems, including several types of automata. In this article, we make use of the coalgebraic view on systems to investigate, in a uniform way, under which conditions calculi that are sound and complete with respect to behavioral equivalence can be extended to a coarser coalgebraic language equivalence, which arises from a generalized powerset construction that determinizes coalgebras. We show that soundness and completeness are established by proving that expressions modulo axioms of a calculus form the rational fixpoint of the given type functor. Our main result is that the rational fixpoint of the functor FT , where T is a monad describing the branching of the systems (e.g., non-determinism, weights, probability, etc.), has as a quotient the rational fixpoint of the determinized type functor F , a lifting of F to the category of T -algebras. We apply our framework to the concrete example of weighted automata, for which we present a new sound and complete calculus for weighted language equivalence. As a special case, we obtain nondeterministic automata in which we recover Rabinovich’s sound and complete calculus for language equivalence.

Coproducts of monads on $\Set$ have arisen in both the study of computational effects and universal algebra. We describe coproducts of consistent monads on $\Set$ by an initial algebra formula, and prove also the converse: if the coproduct exists, so do the required initial algebras. That formula was, in the case of ideal monads, also used by Ghani and Uustalu. We deduce that coproduct embeddings of consistent monads are injective; and that a coproduct of injective monad morphisms is injective. Two consistent monads have a coproduct iff either they have arbitrarily large common fixpoints, or one is an exception monad, possibly modified to preserve the empty set. Hence a consistent monad has a coproduct with every monad iff it is an exception monad, possibly modified to preserve the empty set. We also show other fixpoint results, including that a functor (not constant on nonempty sets) is finitary iff every sufficiently large cardinal is a fixpoint.

Bloom and Ésik's concept of iteration theory summarises all equational properties that iteration has in common applications, for example, in domain theory, where to every system of recursive equations, the least solution is assigned. This paper shows that in the coalgebraic approach to iteration, the more appropriate concept is that of a functorial iteration theory (called Elgot theory). These theories have a particularly simple axiomatisation, and all well-known examples of iteration theories are functorial. Elgot theories are proved to be monadic over the category of sets in context (or, more generally, the category of finitary endofunctors of a locally finitely presentable category). This demonstrates that functoriality is an equational property from the perspective of sets in context. In contrast, Bloom and Ésik worked in the base category of signatures rather than sets in context, and there iteration theories are monadic but Elgot theories are not. This explains why functoriality was not included in the definition of iteration theories.

Stream circuits are a convenient graphical way to represent streams (or stream functions) computed by finite dimensional linear systems. We present a sound and complete expression calculus that allows us to reason about the semantic equivalence of finite closed stream circuits. For our proof of the soundness and completeness we build on recent ideas of Bonsangue, Rutten and Silva. They have provided a "Kleene theorem'' and a sound and complete expression calculus for coalgebras for endofunctors of the category of sets. The key ingredient of the soundness and completeness proof is a syntactic characterization of the final locally finite coalgebra. In the present paper we extend this approach to the category of real vector spaces. We also prove that a final locally finite (dimensional) coalgebra is, equivalently, an initial iterative algebra. This makes the connection to existing work on the semantics of recursive specifications.

Iterative monads were introduced by Calvin Elgot in the 1970's and are those ideal monads in which every guarded system of recursive equations has a unique solution. We prove that every ideal monad has an iterative reflection, that is, an embedding into an iterative monad with the expected universal property. We also introduce the concept of iterativity for algebras for the monad , following in the footsteps of Evelyn Nelson and Jerzy Tiuryn, and prove that is iterative if and only if all free algebras for are iterative algebras.

Several concepts of algebras with solutions of recursive equation systems are compared: CPO-enrichable algebras are proved to be iteration algebras of Z. Ésik, and iteration algebras are a special case of the recently introduced Elgot algebras (which are the monadic algebras for the free iterative monad). Another special case of iteration algebras are the iterative algebras of E. Nelson and J. Tiuryn, which are algebras with unique solutions of all guarded systems. For each of the above classes of algebras an example is provided showing that the inclusion in a wider class is proper.

Denotational semantics can be based on algebras with additional structure (order, metric, etc.) which makes it possible to interpret recursive specifications. It was the idea of Elgot to base denotational semantics on iterative theories instead, i.e., theories in which abstract recursive specifications are required to have unique solutions. Later Bloom and Esik studied iteration theories and iteration algebras in which a specified solution has to obey certain axioms. We propose so-called Elgot algebras as a convenient structure for semantics in the present paper. An Elgot algebra is an algebra with a specified solution for every system of flat recursive equations. That specification satisfies two simple and well motivated axioms: functoriality (stating that solutions are stable under renaming of recursion variables) and compositionality (stating how to perform simultaneous recursion). These two axioms stem canonically from Elgot's iterative theories: We prove that the category of Elgot algebras is the Eilenberg-Moore category of the monad given by a free iterative theory.

Iterative theories, which were introduced by Calvin Elgot, formalise potentially infinite computations as unique solutions of recursive equations. One of the main results of Elgot and his coauthors is a description of a free iterative theory as the theory of all rational trees. Their algebraic proof of this fact is extremely complicated. In our paper we show that by starting with ‘iterative algebras’, that is, algebras admitting a unique solution of all systems of flat recursive equations, a free iterative theory is obtained as the theory of free iterative algebras. The (coalgebraic) proof we present is dramatically simpler than the original algebraic one. Despite this, our result is much more general: we describe a free iterative theory on any finitary endofunctor of every locally presentable category .Reportedly, a blow from the welterweight boxer Norman Selby, also known as Kid McCoy, left one victim proclaiming,‘It's the real McCoy!’.

By the Final Coalgebra Theorem of Aczel and Mendler, every endofunctor of the category of sets has a final coalgebra, which, however, may be a proper class. We generalise this to all ‘well-behaved’ categories .

Every finitary endofunctor of $\Set$ is proved to generate a free iterative theory in the sense of Elgot. This work is based on coalgebras, specifically on parametric corecursion, and the proof is presented for categories more general than just $\Set$ .