In rendez-vous protocols an arbitrarily large number of indistinguishable finite-state agents interact in pairs. The cut-off problem asks if there exists a number $B$ such that all initial configurations of the protocol with at least $B$ agents in a given initial state can reach a final configuration with all agents in a given final state. In a recent paper (Horn and Sangnier, CONCUR 2020), Horn and Sangnier proved that the cut-off problem is decidable (and at least as hard as the Petri net reachability problem) for protocols with a leader, and in EXPSPACE for leaderless protocols. Further, for the special class of symmetric protocols they reduce these bounds to PSPACE and NP, respectively. The problem of lowering these upper bounds or finding matching lower bounds was left open. We show that the cut-off problem is P-complete for leaderless protocols and in NC for leaderless symmetric protocols. Further, we also consider a variant of the cut-off problem suggested in (Horn and Sangnier, CONCUR 2020), which we call the bounded-loss cut-off problem and prove that this problem is P-complete for leaderless protocols and NL-complete for leaderless symmetric protocols. Finally, by reusing some of the techniques applied for the analysis of leaderless protocols, we show that the cut-off problem for symmetric protocols with a leader is NP-complete, thereby improving upon all the elementary upper bounds of (Horn and Sangnier, CONCUR 2020).

We study encodings of the lambda-calculus into the pi-calculus in the unexplored case of calculi with non-determinism and failures. On the sequential side, we consider lambdafail, a new non-deterministic calculus in which intersection types control resources (terms); on the concurrent side, we consider spi, a pi-calculus in which non-determinism and failure rest upon a Curry-Howard correspondence between linear logic and session types. We present a typed encoding of lambdafail into spi and establish its correctness. Our encoding precisely explains the interplay of non-deterministic and fail-prone evaluation in lambdafail via typed processes in spi. In particular, it shows how failures in sequential evaluation (absence/excess of resources) can be neatly codified as interaction protocols.

We introduce monoidal width as a measure of complexity for morphisms in monoidal categories. Inspired by well-known structural width measures for graphs, like tree width and rank width, monoidal width is based on a notion of syntactic decomposition: a monoidal decomposition of a morphism is an expression in the language of monoidal categories, where operations are monoidal products and compositions, that specifies this morphism. Monoidal width penalises the composition operation along ``big'' objects, while it encourages the use of monoidal products. We show that, by choosing the correct categorical algebra for decomposing graphs, we can capture tree width and rank width. For matrices, monoidal width is related to the rank. These examples suggest monoidal width as a good measure for structural complexity of processes modelled as morphisms in monoidal categories.

Game comonads provide a categorical syntax-free approach to finite model theory, and their Eilenberg-Moore coalgebras typically encode important combinatorial parameters of structures. In this paper, we develop a framework whereby the essential properties of these categories of coalgebras are captured in a purely axiomatic fashion. To this end, we introduce arboreal categories, which have an intrinsic process structure, allowing dynamic notions such as bisimulation and back-and-forth games, and resource notions such as number of rounds of a game, to be defined. These are related to extensional or "static" structures via arboreal covers, which are resource-indexed comonadic adjunctions. These ideas are developed in a general, axiomatic setting, and applied to relational structures, where the comonadic constructions for pebbling, Ehrenfeucht-Fra\"iss\'e and modal bisimulation games recently introduced by Abramsky et al. are recovered, showing that many of the fundamental notions of finite model theory and descriptive complexity arise from instances of arboreal covers.

Linear Temporal Logic (LTL) is one of the most popular temporal logics, that comes into play in a variety of branches of computer science. Among the various reasons of its widespread use there are its strong foundational properties: LTL is equivalent to counter-free omega-automata, to star-free omega-regular expressions, and (by Kamp's theorem) to the First-Order Theory of Linear Orders (FO-TLO). Safety and co-safety languages, where a finite prefix suffices to establish whether a word does not belong or belongs to the language, respectively, play a crucial role in lowering the complexity of problems like model checking and reactive synthesis for LTL. SafetyLTL (resp., coSafetyLTL) is a fragment of LTL where only universal (resp., existential) temporal modalities are allowed, that recognises safety (resp., co-safety) languages only. The main contribution of this paper is the introduction of a fragment of FO-TLO, called SafetyFO, and of its dual coSafetyFO, which are expressively complete with respect to the LTL-definable safety and co-safety languages. We prove that they exactly characterize SafetyLTL and coSafetyLTL, respectively, a result that joins Kamp's theorem, and provides a clearer view of the characterization of (fragments of) LTL in terms of first-order languages. In addition, it gives a direct, compact, and self-contained proof that any safety language definable in LTL is definable in SafetyLTL as well. As a by-product, we obtain some interesting results on the expressive power of the weak tomorrow operator of SafetyLTL, interpreted over finite and infinite words. Moreover, we prove that, when interpreted over finite words, SafetyLTL (resp. coSafetyLTL) devoid of the tomorrow (resp., weak tomorrow) operator captures the safety (resp., co-safety) fragment of LTL over finite words.

Regular expressions with capture variables, also known as regex-formulas, extract relations of spans (intervals identified by their start and end indices) from text. In turn, the class of regular document spanners is the closure of the regex formulas under the Relational Algebra. We investigate the computational complexity of querying text by aggregate functions, such as sum, average, and quantile, on top of regular document spanners. To this end, we formally define aggregate functions over regular document spanners and analyze the computational complexity of exact and approximate computation. More precisely, we show that in a restricted case, all studied aggregate functions can be computed in polynomial time. In general, however, even though exact computation is intractable, some aggregates can still be approximated with fully polynomial-time randomized approximation schemes (FPRAS).

We investigate trade-offs in static and dynamic evaluation of hierarchical queries with arbitrary free variables. In the static setting, the trade-off is between the time to partially compute the query result and the delay needed to enumerate its tuples. In the dynamic setting, we additionally consider the time needed to update the query result under single-tuple inserts or deletes to the database. Our approach observes the degree of values in the database and uses different computation and maintenance strategies for high-degree (heavy) and low-degree (light) values. For the latter it partially computes the result, while for the former it computes enough information to allow for on-the-fly enumeration. We define the preprocessing time, the update time, and the enumeration delay as functions of the light/heavy threshold. By appropriately choosing this threshold, our approach recovers a number of prior results when restricted to hierarchical queries. We show that for a restricted class of hierarchical queries, our approach achieves worst-case optimal update time and enumeration delay conditioned on the Online Matrix-Vector Multiplication Conjecture.

What payoffs are positionally determined for deterministic two-player antagonistic games on finite directed graphs? In this paper we study this question for payoffs that are continuous. The main reason why continuous positionally determined payoffs are interesting is that they include the multi-discounted payoffs. We show that for continuous payoffs, positional determinacy is equivalent to a simple property called prefix-monotonicity. We provide three proofs of it, using three major techniques of establishing positional determinacy -- inductive technique, fixed point technique and strategy improvement technique. A combination of these approaches provides us with better understanding of the structure of continuous positionally determined payoffs as well as with some algorithmic results.

We introduce a meta-model based on formal languages, dubbed formal choreographic languages, to study message-passing systems. Our framework allows us to generalise standard constructions from the literature and to compare them. In particular, we consider notions such as global view, local view, and projections from the former to the latter. The correctness of local views projected from global views is characterised in terms of a closure property. We consider a number of communication properties -- such as (dead)lock-freedom -- and give conditions on formal choreographic languages to guarantee them. Finally, we show how formal choreographic languages can capture existing formalisms; specifically we consider communicating finite-state machines, choreography automata, and multiparty session types. Notably, formal choreographic languages, differently from most approaches in the literature, can naturally model systems exhibiting non-regular behaviour.

Collective Adaptive Systems often consist of many heterogeneous components typically organised in groups. These entities interact with each other by adapting their behaviour to pursue individual or collective goals. In these systems, the distribution of these entities determines a space that can be either physical or logical. The former is defined in terms of a physical relation among components. The latter depends on logical relations, such as being part of the same group. In this context, specification and verification of spatial properties play a fundamental role in supporting the design of systems and predicting their behaviour. For this reason, different tools and techniques have been proposed to specify and verify the properties of space, mainly described as graphs. Therefore, the approaches generally use model spatial relations to describe a form of proximity among pairs of entities. Unfortunately, these graph-based models do not permit considering relations among more than two entities that may arise when one is interested in describing aspects of space by involving interactions among groups of entities. In this work, we propose a spatial logic interpreted on simplicial complexes. These are topological objects, able to represent surfaces and volumes efficiently that generalise graphs with higher-order edges. We discuss how the satisfaction of logical formulas can be verified by a correct and complete model checking algorithm, which is linear to the dimension of the simplicial complex and logical formula. The expressiveness of the proposed logic is studied in terms of the spatial variants of classical bisimulation and branching bisimulation relations defined over simplicial complexes.