arXiv subjects

Lutz Schröder

Publications and source records attributed to Lutz Schröder.

At least 55 records · Page 3Linked to original sources

arXiv2021

Coalgebraic Reasoning with Global Assumptions in Arithmetic Modal Logics

We establish a generic upper bound ExpTime for reasoning with global assumptions (also known as TBoxes) in coalgebraic modal logics. Unlike earlier results of this kind, our bound does not require a tractable set of tableau rules for the instance logics, so that the result applies to wider classes of logics. Examples are Presburger modal logic, which extends graded modal logic with linear inequalities over numbers of successors, and probabilistic modal logic with polynomial inequalities over probabilities. We establish the theoretical upper bound using a type elimination algorithm. We also provide a global caching algorithm that potentially avoids building the entire exponential-sized space of candidate states, and thus offers a basis for practical reasoning. This algorithm still involves frequent fixpoint computations; we show how these can be handled efficiently in a concrete algorithm modelled on Liu and Smolka's linear-time fixpoint algorithm. Finally, we show that the upper complexity bound is preserved under adding nominals to the logic, i.e. in coalgebraic hybrid logic.

cs.LO↗

arXiv2021

Explaining Behavioural Inequivalence Generically in Quasilinear Time

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 $\mathcal{O}(m n)$.

cs.LO↗

arXiv2021

A Linear-Time Nominal $μ$-Calculus with Name Allocation

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-muTL 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-muTL allows unrestricted nondeterminism and unboundedly many registers, model checking Bar-muTL 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.

cs.FL↗

arXiv2021

Nominal Büchi Automata with Name Allocation

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 omega-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.

cs.FL↗

arXiv2021

Monads on Categories of Relational Structures

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 $<λ$), with the syntax of algebraic theories induced by the relational signature (e.g. inequations or equations-up-to-$ε$). We provide a generic sound and complete derivation system for such relational algebraic theories, thus in particular recovering (extensions of) recent systems of this type for monads on partial orders and metric spaces by instantiation. In particular, we present an $ω_1$-ary algebraic theory of metric completion. The theory-to-monad direction of our correspondence remains true for the case of $κ$-ary algebraic theories and $κ$-accessible monads for $κ<λ$, e.g. for finitary theories over metric spaces.

math.CT↗

arXiv2021

A Metalanguage for Guarded Iteration

Notions of guardedness serve to delineate admissible recursive definitions in various settings in a compositional manner. In recent work, we have introduced an axiomatic notion of guardedness in symmetric monoidal categories, which serves as a unifying framework for various examples from program semantics, process algebra, and beyond. In the present paper, we propose a generic metalanguage for guarded iteration based on combining this notion with the fine-grain call-by-value paradigm, which we intend as a unifying programming language for guarded and unguarded iteration in the presence of computational effects. We give a generic (categorical) semantics of this language over a suitable class of strong monads supporting guarded iteration, and show it to be in touch with the standard operational behaviour of iteration by giving a concrete big-step operational semantics for a certain specific instance of the metalanguage and establishing soundness and (computational) adequacy for this case.

cs.LO↗

arXiv2021

Behavioural Preorders via Graded Monads

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 $\mathsf{Pos}$ of partially ordered sets following the paradigm of universal coalgebra, while behavioural preorders are captured as graded monads on $\mathsf{Pos}$, in generalization of a previous approach to notions of process equivalence. We show that graded monads on $\mathsf{Pos}$ 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.

cs.LO↗

arXiv2021

Quasipolynomial Computation of Nested Fixpoints

It is well-known that the winning region of a parity game with $n$ nodes and $k$ priorities can be computed as a $k$-nested fixpoint of a suitable function; straightforward computation of this nested fixpoint requires $\mathcal{O}(n^{\frac{k}{2}})$ iterations of the function. Calude et al.'s recent quasipolynomial-time parity game solving algorithm essentially shows how to compute the same fixpoint in only quasipolynomially many iterations by reducing parity games to quasipolynomially sized safety games. Universal graphs have been used to modularize this transformation of parity games to equivalent safety games that are obtained by combining the original game with a universal graph. We show that this approach naturally generalizes to the computation of solutions of systems of \emph{any} fixpoint equations over finite lattices; hence, the solution of fixpoint equation systems can be computed by quasipolynomially many iterations of the equations. We present applications to modal fixpoint logics and games beyond relational semantics. For instance, the model checking problems for the energy $μ$-calculus, finite latticed $μ$-calculi, and the graded and the (two-valued) probabilistic $μ$-calculus -- with numbers coded in binary -- can be solved via nested fixpoints of functions that differ substantially from the function for parity games but still can be computed in quasipolynomial time; our result hence implies that model checking for these $μ$-calculi is in QP. Moreover, we improve the exponent in known exponential bounds on satisfiability checking.

cs.CC↗

arXiv2021

Nominal Automata with Name Binding

Automata models for data languages (i.e. languages over infinite alphabets) often feature either global or local freshness operators. We show that Bollig et al.'s session automata, which focus on global freshness, are equivalent to regular nondeterministic nominal automata (RNNA), a natural nominal automaton model with explicit name binding that has appeared implicitly in the semantics of nominal Kleene algebra (NKA), an extension of Kleene algebra with name binding. The expected Kleene theorem for NKA is known to fail in one direction, i.e. there are nominal languages that can be accepted by an RNNA but are not definable in NKA; via session automata, we obtain a full Kleene theorem for RNNAs for an expression language that extends NKA with unscoped name binding. Based on the equivalence with RNNAs, we then slightly rephrase the known equivalence checking algorithm for session automata. Reinterpreting the data language semantics of name binding by unrestricted instead of clean alpha-equivalence, we obtain a local freshness semantics as a quotient of the global freshness semantics. Under local freshness semantics, RNNAs turn out to be equivalent to a natural subclass of Bojanczyk et al.'s nondeterministic orbit-finite automata. We establish decidability of inclusion under local freshness by modifying the RNNA-based algorithm; in summary, we obtain a formalism for local freshness in data languages that is reasonably expressive and has a decidable inclusion problem.

cs.FL↗

arXiv2020

Finitary Monads on the Category of Posets

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

math.CT↗

arXiv2020

From Generic Partition Refinement to Weighted Tree Automata Minimization

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.

cs.DS↗

arXiv2020

Graded Monads and Graded Logics for the Linear Time -- Branching Time Spectrum

State-based models of concurrent systems are traditionally considered under a variety of notions of process equivalence. In the particular 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 exemplaric proofs of their expressiveness based on our generic criterion.

cs.LO↗

arXiv2020

Explaining Non-Bisimilarity in a Coalgebraic Approach: Games and Distinguishing Formulas

Behavioural equivalences can be characterized via bisimulation, modal logics, and spoiler-duplicator games. In this paper we work in the general setting of coalgebra and focus on generic algorithms for computing the winning strategies of both players in a bisimulation game. The winning strategy of the spoiler (if it exists) is then transformed into a modal formula that distinguishes the given non-bisimilar states. The modalities required for the formula are also synthesized on-the-fly, and we present a recipe for re-coding the formula with different modalities, given by a separating set of predicate liftings. Both the game and the generation of the distinguishing formulas have been implemented in a tool called T-BEG.

cs.LO↗

arXiv2020

Automata Learning: An Algebraic Approach

We propose a generic categorical framework for learning unknown formal languages of various types (e.g. finite or infinite words, weighted and nominal languages). Our approach is parametric in a monad T that represents the given type of languages and their recognizing algebraic structures. Using the concept of anautomata presentation of T-algebras, we demonstrate that the task of learning a T-recognizable language can be reduced to learning an abstract form of algebraic automaton whose transitions are modeled by a functor. For the important case of adjoint automata, we devise a learning algorithm generalizing Angluin's L*. The algorithm is phrased in terms of categorically described extension steps; we provide for a termination and complexity analysis based on a dedicated notion of finiteness. Our framework applies to structures like omega-regular languages that were not within the scope of existing categorical accounts of automata learning. In addition, it yields new learning algorithms for several types of languages for which no such algorithms were previously known at all, including sorted languages, nominal languages with name binding, and cost functions.

cs.FL↗

arXiv2020

Non-iterative Modal Logics are Coalgebraic

A modal logic is \emph{non-iterative} if it can be defined by axioms that do not nest modal operators, and \emph{rank-1} if additionally all propositional variables in axioms are in scope of a modal operator. It is known that every syntactically defined rank-1 modal logic can be equipped with a canonical coalgebraic semantics, ensuring soundness and strong completeness. In the present work, we extend this result to non-iterative modal logics, showing that every non-iterative modal logic can be equipped with a canonical coalgebraic semantics defined in terms of a copointed functor, again ensuring soundness and strong completeness via a canonical model construction. Like in the rank-1 case, the canonical coalgebraic semantics is equivalent to a neighbourhood semantics with suitable frame conditions, so the known strong completeness of non-iterative modal logics over neighbourhood semantics is implied. As an illustration of these results, we discuss deontic logics with factual detachment, which is captured by axioms that are non-iterative but not rank~1.

cs.LO↗

arXiv2020

NP Reasoning in the Monotone $μ$-Calculus

Satisfiability checking for monotone modal logic is known to be (only) NP-complete. We show that this remains true when the logic is extended with aconjunctive and alternation-free fixpoint operators as well as the universal modality; the resulting logic -- the aconjunctive alternation-free monotone $μ$-calculus with the universal modality -- contains both concurrent propositional dynamic logic (CPDL) and the alternation-free fragment of game logic as fragments. We obtain our result from a characterization of satisfiability by means of Büchi games with polynomially many Eloise nodes.

cs.LO↗

arXiv2020

Efficient and Modular Coalgebraic Partition Refinement

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.

cs.DS↗

arXiv2019

Generic Partition Refinement and Weighted Tree Automata

cs.DS↗