arXiv subjects

Lutz Schröder

Publications and source records attributed to Lutz Schröder.

At least 37 records · Page 2Linked to original sources

arXiv2024

Coalgebraic Satisfiability Checking for Arithmetic $μ$-Calculi

The coalgebraic $μ$-calculus provides a generic semantic framework for fixpoint logics over systems whose branching type goes beyond the standard relational setup, e.g. probabilistic, weighted, or game-based. Previous work on the coalgebraic $μ$-calculus includes an exponential-time upper bound on satisfiability checking, which however relies on the availability of tableau rules for the next-step modalities that are sufficiently well-behaved in a formally defined sense; in particular, rule matches need to be representable by polynomial-sized codes, and the sequent duals of the rules need to absorb cut. While such rule sets have been identified for some important cases, they are not known to exist in all cases of interest, in particular ones involving either integer weights as in the graded $μ$-calculus, or real-valued weights in combination with non-linear arithmetic. In the present work, we prove the same upper complexity bound under more general assumptions, specifically regarding the complexity of the (much simpler) satisfiability problem for the underlying one-step logic, roughly described as the nesting-free next-step fragment of the logic. The bound is realized by a generic global caching algorithm that supports on-the-fly satisfiability checking. Notably, our approach directly accommodates unguarded formulae, and thus avoids use of the guardedness transformation. Example applications include new exponential-time upper bounds for satisfiability checking in an extension of the graded $μ$-calculus with polynomial inequalities (including positive Presburger arithmetic), as well as an extension of the (two-valued) probabilistic $μ$-calculus with polynomial inequalities.

cs.LO↗

arXiv2024

Nominal Tree Automata With Name Allocation

Data trees serve as an abstraction of structured data, such as XML documents. A number of specification formalisms for languages of data trees have been developed, many of them adhering to the paradigm of register automata, which is based on storing data values encountered on the tree in registers for subsequent comparison with further data values. Already on word languages, the expressiveness of such automata models typically increases with the power of control (e.g. deterministic, non-deterministic, alternating). Language inclusion is typically undecidable for non-deterministic or alternating models unless the number of registers is radically restricted, and even then often remains non-elementary. We present an automaton model for data trees that retains a reasonable level of expressiveness, in particular allows non-determinism and any number of registers, while admitting language inclusion checking in elementary complexity, in fact in parametrized exponential time. We phrase the description of our automaton model in the language of nominal sets, building on the recently introduced paradigm of explicit name allocation in nominal automata.

cs.FL↗

arXiv2024

Graded Monads and Behavioural Equivalence Games

The framework of graded semantics uses graded monads to capture behavioural equivalences of varying granularity, for example as found on 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.

cs.LO↗

arXiv2024

Graded Semantics and Graded Logics for Eilenberg-Moore Coalgebras

Coalgebra, as the abstract study of state-based systems, comes naturally equipped with a notion of behavioural equivalence that identifies states exhibiting the same behaviour. In many cases, however, this equivalence is finer than the intended semantics. Particularly in automata theory, behavioural equivalence of nondeterministic automata is essentially bisimilarity, and thus does not coincide with language equivalence. Language equivalence can be captured as behavioural equivalence on the determinization, which is obtained via the standard powerset construction. This construction can be lifted to coalgebraic generality, assuming a so-called Eilenberg-Moore distributive law between the functor termining the type of accepted structure (e.g.\ word languages) and a monad capturing the branching type (e.g.\ nondeterministic, weighted, probabilistic). Eilenberg-Moore-style coalgebraic semantics in this sense has been shown to be essentially subsumed by the more general framework of graded semantics, which is centrally based on graded monads. Graded semantics comes with a range of generic results, in particular regarding invariance and, under suitable conditions, expressiveness of dedicated modal logics for a given semantics; notably, these logics are evaluated on the original state space. We show that the instantiation of such graded logics to the case of Eilenberg-Moore-style semantics works extremely smoothly, and yields expressive modal logics in essentially all cases of interest. We additionally parametrize the framework over a quantale of truth values, thus in particular covering both the two-valued notions of equivalence and quantitative ones, i.e. behavioural distances.

cs.LO↗

arXiv2024

Expressive Quantale-valued Logics for Coalgebras: an Adjunction-based Approach

We address the task of deriving fixpoint equations from modal logics characterizing behavioural equivalences and metrics (summarized under the term conformances). We rely on earlier work that obtains Hennessy-Milner theorems as corollaries to a fixpoint preservation property along Galois connections between suitable lattices. We instantiate this to the setting of coalgebras, in which we spell out the compatibility property ensuring that we can derive a behaviour function whose greatest fixpoint coincides with the logical conformance. We then concentrate on the linear-time case, for which we study coalgebras based on the machine functor living in Eilenberg-Moore categories, a scenario for which we obtain a particularly simple logic and fixpoint equation. The theory is instantiated to concrete examples, both in the branching-time case (bisimilarity and behavioural metrics) and in the linear-time case (trace equivalences and trace distances).

cs.LO↗

arXiv2024

Logical Predicates in Higher-Order Mathematical Operational Semantics

We present a systematic approach to logical predicates based on universal coalgebra and higher-order abstract GSOS, thus making a first step towards a unifying theory of logical relations. We first observe that logical predicates are special cases of coalgebraic invariants on mixed-variance functors. We then introduce the notion of a locally maximal logical refinement of a given predicate, with a view to enabling inductive reasoning, and identify sufficient conditions on the overall setup in which locally maximal logical refinements canonically exist. Finally, we develop induction-up-to techniques that simplify inductive proofs via logical predicates on systems encoded as (certain classes of) higher-order GSOS laws by identifying and abstracting away from their boiler-plate part.

cs.LO↗

arXiv2023

A Point-free Perspective on Lax extensions and Predicate liftings

Lax extensions of set functors play a key role in various areas including topology, concurrent systems, and modal logic, while predicate liftings provide a generic semantics of modal operators. We take a fresh look at the connection between lax extensions and predicate liftings from the point of view of quantale-enriched relations. Using this perspective, we show in particular that various fundamental concepts and results arise naturally and their proofs become very elementary. Ultimately, we prove that every lax extension is induced by a class of predicate liftings; we discuss several implications of this result.

math.CT↗

arXiv2023

Quasilinear-time Computation of Generic Modal Witnesses for Behavioural Inequivalence

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).

cs.LO↗

arXiv2023

Generic Model Checking for Modal Fixpoint Logics in COOL-MC

We report on COOL-MC, a model checking tool for fixpoint logics that is parametric in the branching type of models (nondeterministic, game-based, probabilistic etc.) and in the next-step modalities used in formulae. The tool implements generic model checking algorithms developed in coalgebraic logic that are easily adapted to concrete instance logics. Apart from the standard modal $μ$-calculus, COOL-MC currently supports alternating-time, graded, probabilistic and monotone variants of the $μ$-calculus, but is also effortlessly extensible with new instance logics. The model checking process is realized by polynomial reductions to parity game solving, or, alternatively, by a local model checking algorithm that directly computes the extensions of formulae in a lazy fashion, thereby potentially avoiding the construction of the full parity game. We evaluate COOL-MC on informative benchmark sets.

cs.LO↗

arXiv2023

Weak Similarity in Higher-Order Mathematical Operational Semantics

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 lambda-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.

cs.PL↗

arXiv2023

COOL 2 -- A Generic Reasoner for Modal Fixpoint Logics

There is a wide range of modal logics whose semantics goes beyond relational structures, and instead involves, e.g., probabilities, multi-player games, weights, or neighbourhood structures. Coalgebraic logic serves as a unifying semantic and algorithmic framework for such logics. It provides uniform reasoning algorithms that are easily instantiated to particular, concretely given logics. The COOL 2 reasoner provides an implementation of such generic algorithms for coalgebraic modal fixpoint logics. As concrete instances, we obtain in particular reasoners for the aconjunctive and alternation-free fragments of the graded $μ$-calculus and the alternating-time $μ$-calculus. We evaluate the tool on standard benchmark sets for fixpoint-free graded modal logic and alternating-time temporal logic (ATL), as well as on a dedicated set of benchmarks for the graded $μ$-calculus.

cs.LO↗

arXiv2023

The Alternating-Time μ-Calculus With Disjunctive Explicit Strategies

Alternating-time temporal logic (ATL) and its extensions, including the alternating-time $μ$-calculus (AMC), serve the specification of the strategic abilities of coalitions of agents in concurrent game structures. The key ingredient of the logic are path quantifiers specifying that some coalition of agents has a joint strategy to enforce a given goal. This basic setup has been extended to let some of the agents (revocably) commit to using certain named strategies, as in ATL with explicit strategies (ATLES). In the present work, we extend ATLES with fixpoint operators and strategy disjunction, arriving at the alternating-time $μ$-calculus with disjunctive explicit strategies (AMCDES), which allows for a more flexible formulation of temporal properties (e.g. fairness) and, through strategy disjunction, a form of controlled nondeterminism in commitments. Our main result is an ExpTime upper bound for satisfiability checking (which is thus ExpTime-complete). We also prove upper bounds QP (quasipolynomial time) and NP $\cap$ coNP for model checking under fixed interpretations of explicit strategies, and NP under open interpretation. Our key technical tool is a treatment of the AMCDES within the generic framework of coalgebraic logic, which in particular reduces the analysis of most reasoning tasks to the treatment of a very simple one-step logic featuring only propositional operators and next-step operators without nesting; we give a new model construction principle for this one-step logic that relies on a set-valued variant of first-order resolution.

cs.LO↗

arXiv2023

Kantorovich Functors and Characteristic Logics for Behavioural Distances

Behavioural distances measure the deviation between states in quantitative systems, such as probabilistic or weighted systems. There is growing interest in generic approaches to behavioural distances. In particular, coalgebraic methods capture variations in the system type (nondeterministic, probabilistic, game-based etc.), and the notion of quantale abstracts over the actual values distances take, thus covering, e.g., two-valued equivalences, (pseudo-)metrics, and probabilistic (pseudo-)metrics. Coalgebraic behavioural distances have been based either on liftings of SET-functors to categories of metric spaces, or on lax extensions of SET-functors to categories of quantitative relations. Every lax extension induces a functor lifting but not every lifting comes from a lax extension. It was shown recently that every lax extension is Kantorovich, i.e. induced by a suitable choice of monotone predicate liftings, implying via a quantitative coalgebraic Hennessy-Milner theorem that behavioural distances induced by lax extensions can be characterized by quantitative modal logics. Here, we essentially show the same in the more general setting of behavioural distances induced by functor liftings. In particular, we show that every functor lifting, and indeed every functor on (quantale-valued) metric spaces, that preserves isometries is Kantorovich, so that the induced behavioural distance (on systems of suitably restricted branching degree) can be characterized by a quantitative modal logic.

math.CT↗

arXiv2022

Common Knowledge of Abstract Groups

Epistemic logics typically talk about knowledge of individual agents or groups of explicitly listed agents. Often, however, one wishes to express knowledge of groups of agents specified by a given property, as in `it is common knowledge among economists'. We introduce such a logic of common knowledge, which we term abstract-group epistemic logic (AGEL). That is, AGEL features a common knowledge operator for groups of agents given by concepts in a separate agent logic that we keep generic, with one possible agent logic being ALC. We show that AGEL is EXPTIME-complete, with the lower bound established by reduction from standard group epistemic logic, and the upper bound by a satisfiability-preserving embedding into the full $μ$-calculus. Further main results include a finite model property (not enjoyed by the full $μ$-calculus) and a complete axiomatization.

cs.AI↗

arXiv2022

Quantitative Hennessy-Milner Theorems via Notions of Density

The classical Hennessy-Milner theorem is an important tool in the analysis of concurrent processes; it guarantees that any two non-bisimilar states in finitely branching labelled transition systems can be distinguished by a modal formula. Numerous variants of this theorem have since been established for a wide range of logics and system types, including quantitative versions where lower bounds on behavioural distance (e.g.~in weighted, metric, or probabilistic transition systems) are witnessed by quantitative modal formulas. Both the qualitative and the quantitative versions have been accommodated within the framework of coalgebraic logic, with distances taking values in quantales, subject to certain restrictions, such as being so-called value quantales. While previous quantitative coalgebraic Hennessy-Milner theorems apply only to liftings of set functors to (pseudo-)metric spaces, in the present work we provide a quantitative coalgebraic Hennessy-Milner theorem that applies more widely to functors native to metric spaces; notably, we thus cover, for the first time, the well-known Hennessy-Milner theorem for continuous probabilistic transition systems, where transitions are given by Borel measures on metric spaces, as an instance. In the process, we also relax the restrictions imposed on the quantale, and additionally parametrize the technical account over notions of closure and, hence, density, providing associated variants of the Stone-Weierstrass theorem; this allows us to cover, for instance, behavioural ultrametrics.

cs.LO↗

arXiv2022

Characteristic Logics for Behavioural Hemimetrics via Fuzzy Lax Extensions

In systems involving quantitative data, such as probabilistic, fuzzy, or metric systems, behavioural distances provide a more fine-grained comparison of states than two-valued notions of behavioural equivalence or behaviour inclusion. Like in the two-valued case, the wide variation found in system types creates a need for generic methods that apply to many system types at once. Approaches of this kind are emerging within the paradigm of universal coalgebra, based either on lifting pseudometrics along set functors or on lifting general real-valued (fuzzy) relations along functors by means of fuzzy lax extensions. An immediate benefit of the latter is that they allow bounding behavioural distance by means of fuzzy (bi-)simulations that need not themselves be hemi- or pseudometrics; this is analogous to classical simulations and bisimulations, which need not be preorders or equivalence relations, respectively. The known generic pseudometric liftings, specifically the generic Kantorovich and Wasserstein liftings, both can be extended to yield fuzzy lax extensions, using the fact that both are effectively given by a choice of quantitative modalities. Our central result then shows that in fact all fuzzy lax extensions are Kantorovich extensions for a suitable set of quantitative modalities, the so-called Moss modalities. For nonexpansive fuzzy lax extensions, this allows for the extraction of quantitative modal logics that characterize behavioural distance, i.e. satisfy a quantitative version of the Hennessy-Milner theorem; equivalently, we obtain expressiveness of a quantitative version of Moss' coalgebraic logic. All our results explicitly hold also for asymmetric distances (hemimetrics), i.e. notions of quantitative simulation.

cs.LO↗

arXiv2022

Stateful Structural Operational Semantics

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.

cs.LO↗

arXiv2022

Uniform Interpolation in Coalgebraic Modal Logic

A logic has uniform interpolation if its formulas can be projected down to given subsignatures, preserving all logical consequences that do not mention the removed symbols; the weaker property of (Craig) interpolation allows the projected formula -- the interpolant -- to be different for each logical consequence of the original formula. These properties are of importance, e.g., in the modularization of logical theories. We study interpolation in the context of coalgebraic modal logics, i.e. modal logics axiomatized in rank 1, restricting for clarity to the case with finitely many modalities. Examples of such logics include the modal logics K and KD, neighbourhood logic and its monotone variant, finite-monoid-weighted logics, and coalition logic. We introduce a notion of one-step (uniform) interpolation, which refers only to a restricted logic without nesting of modalities, and show that a coalgebraic modal logic has uniform interpolation if it has one-step interpolation. Moreover, we identify preservation of finite surjective weak pullbacks as a sufficient, and in the monotone case necessary, condition for one-step interpolation. We thus prove or reprove uniform interpolation for most of the examples listed above.

cs.LO↗