Recently, the Manna-Pnueli Hierarchy has been used to define the temporal logics LTLf+ and PPLTL+, which allow to use finite-trace LTLf/PPLTL techniques in infinite-trace settings while achieving the expressiveness of full LTL. In this paper, we present the first actual solvers for reactive synthesis in these logics. These are based on games on graphs that leverage DFA-based techniques from LTLf/PPLTL to construct the game arena. We start with a symbolic solver based on Emerson-Lei games, which reduces lower-class properties (guarantee, safety) to higher ones (recurrence, persistence) before solving the game. We then introduce Manna-Pnueli games, which natively embed Manna-Pnueli objectives into the arena. These games are solved by composing solutions to a DAG of simpler Emerson-Lei games, resulting in a provably more efficient approach. We implemented the solvers and practically evaluated their performance on a range of representative formulas. The results show that Manna-Pnueli games often offer significant advantages, though not universally, indicating that combining both approaches could further enhance practical performance.

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.

Obliging games have been introduced in the context of the game perspective on reactive synthesis in order to enforce a degree of cooperation between the to-be-synthesized system and the environment. Previous approaches to the analysis of obliging games have been small-step in the sense that they have been based on a reduction to standard (non-obliging) games in which single moves correspond to single moves in the original (obliging) game. Here, we propose a novel, large-step view on obliging games, reducing them to standard games in which single moves encode long-term behaviors in the original game. This not only allows us to give a meaningful definition of the environment winning in obliging games, but also leads to significantly improved bounds on both strategy sizes and the solution runtime for obliging games.

Abstract Emerson-Lei conditions have recently attracted attention due to both their succinctness and their favorable closure properties. In the current work, we show how infinite-duration games with Emerson-Lei objectives can be analyzed in two different ways. First, we show that the Zielonka tree of the Emerson-Lei condition naturally gives rise to a new reduction to parity games. This reduction, however, does not result in optimal analysis. Second, we show based on the first reduction (and the Zielonka tree) how to provide a direct fixpoint-based characterization of the winning region. The fixpoint-based characterization allows for symbolic analysis. It generalizes the solutions of games with known winning conditions such as Büchi, GR[1], parity, Streett, Rabin and Muller objectives, and in the case of these conditions reproduces previously known symbolic algorithms and complexity results. We also show how the capabilities of the proposed algorithm can be exploited in reactive synthesis, suggesting a new expressive fragment of LTL that can be handled symbolically. Our fragment combines a safety specification and a liveness part. The safety part is unrestricted and the liveness part allows to define Emerson-Lei conditions on occurrences of letters. The symbolic treatment is enabled due to the simplicity of determinization in the case of safety languages and by using our new algorithm for game solving. This approach maximizes the number of steps solved symbolically in order to maximize the potential for efficient symbolic implementations.

Abstract We consider two-player games over finite graphs in which both players are restricted by fairness constraints on their moves. Given a two player game graph $$G=(V,E)$$ G = ( V , E ) and a set of fair moves $$E_f\subseteq E$$ E f ⊆ E a player is said to play fair in G if they choose an edge $$e\in E_f$$ e ∈ E f infinitely often whenever the source node of e is visited infinitely often. Otherwise, they play unfair . We equip such games with two $$\omega $$ ω -regular winning conditions $$\alpha $$ α and $$\beta $$ β deciding the winner of mutually fair and mutually unfair plays, respectively. Whenever one player plays fair and the other plays unfair, the fairly playing player wins the game. The resulting games are called fair $$\alpha /\beta $$ α / β games . We formalize fair $$\alpha /\beta $$ α / β games and show that they are determined. For fair parity/parity games, i.e., fair $$\alpha /\beta $$ α / β games where $$\alpha $$ α and $$\beta $$ β are given each by a parity condition over G , we provide a polynomial reduction to (normal) parity games via a gadget construction inspired by the reduction of stochastic parity games to parity games. We further give a direct symbolic fixpoint algorithm to solve fair parity/parity games. On a conceptual level, we illustrate the translation between the gadget-based reduction and the direct symbolic algorithm which uncovers the underlying similarities of solution algorithms for fair and stochastic parity games, as well as for the recently considered class of fair games in which only one player is restricted by fair moves.

The coalgebraic $\mu$-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 $\mu$-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 $\mu$-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 $\mu$-calculus with polynomial inequalities (including positive Presburger arithmetic), as well as an extension of the (two-valued) probabilistic $\mu$-calculus with polynomial inequalities.

Abstract 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 $$\mu $$ μ -calculus and the alternating-time $$\mu $$ 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 $$\mu $$ μ -calculus.

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.

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 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 $$\mathcal {O}(n^{\frac{k}{2}})$$ O ( n 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 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 $$\mu $$ μ -calculus, finite latticed $$\mu $$ μ -calculi, and the graded and the (two-valued) probabilistic $$\mu $$ 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 $$\mu $$ μ -calculi is in $$\textsc {QP}$$ QP . Moreover, we improve the exponent in known exponential bounds on satisfiability checking.

The coalgebraic mu-calculus is a generic framework for fixpoint logics with varying branching types that subsumes, besides the standard relational mu-calculus, such diverse logics as the graded mu-calculus, the monotone mu-calculus, the probabilistic mu-calculus, and the alternating-time mu-calculus. In the present work, we give a local model checking algorithm for the coalgebraic mu-calculus using a coalgebraic variant of parity games that runs, under mild assumptions on the complexity of the so-called one-step satisfaction problem, in time p^k where p is a polynomial in the formula and model size and where k is the alternation depth of the formula. We show moreover that under the same assumptions, the model checking problem is in both NP and coNP, improving the complexity in all mentioned non-relational cases. If one-step satisfaction can be solved by means of small finite games, we moreover obtain standard parity games, ensuring quasi-polynomial run time. This applies in particular to the monotone mu-calculus, the alternating-time mu-calculus, and the graded mu-calculus with grades coded in unary.

Abstract The coalgebraic $$\mu $$ -calculus provides a generic semantic framework for fixpoint logics with branching types beyond the standard relational setup, e.g. probabilistic, weighted, or game-based. Previous work on the coalgebraic $$\mu $$ -calculus includes an exponential time upper bound on satisfiability checking, which however requires a well-behaved set of tableau rules for the next-step modalities. Such rules are not available in all cases of interest, in particular ones involving either integer weights as in the graded $$\mu $$ -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. Example applications include new exponential-time upper bounds for satisfiability checking in an extension of the graded $$\mu $$ -calculus with polynomial inequalities (including positive Presburger arithmetic), as well as an extension of the (two-valued) probabilistic $$\mu $$ -calculus with polynomial inequalities.

We introduce a natural notion of limit-deterministic parity automata and present a method that uses such automata to construct satisfiability games for the weakly aconjunctive fragment of the $$\mu $$ -calculus. To this end we devise a method that determinizes limit-deterministic parity automata of size n with k priorities through limit-deterministic Büchi automata to deterministic parity automata of size $$\mathcal {O}((nk)!)$$ and with $$\mathcal {O}(nk)$$ priorities. The construction relies on limit-determinism to avoid the full complexity of the Safra/Piterman-construction by using partial permutations of states in place of Safra-Trees. By showing that limit-deterministic parity automata can be used to recognize unsuccessful branches in pre-tableaux for the weakly aconjunctive $$\mu $$ -calculus, we obtain satisfiability games of size $$\mathcal {O}((nk)!)$$ with $$\mathcal {O}(nk)$$ priorities for weakly aconjunctive input formulas of size n and alternation-depth k. A prototypical implementation that employs a tableau-based global caching algorithm to solve these games on-the-fly shows promising initial results.

We present a sound, complete, and optimal single-pass tableau algorithm for the alternation-free mu-calculus. The algorithm supports global caching with intermediate propagation and runs in time 2^O(n). In game-theoretic terms, our algorithm integrates the steps for constructing and solving the Büchi game arising from the input tableau into a single procedure; this is done on-the-fly, i.e. may terminate before the game has been fully constructed. This suggests a slogan to the effect that global caching = game solving on-the-fly. A prototypical implementation shows promising initial results.

Branching-time temporal logics generalizing relational temporal logics such as CTL have been proposed for various system types beyond the purely relational world. This includes, e.g., alternating-time logics, which talk about winning strategies over concurrent game structures, and Parikh's game logic, which is interpreted over monotone neighbourhood frames, as well as probabilistic fixpoint logics. Coalgebraic logic has emerged as a unifying semantic and algorithmic framework for logics featuring generalized modalities of this type. Here, we present a generic global caching algorithm for satisfiability checking in the flat coalgebraic mu-calculus, which realizes known tight exponential-time upper complexity bounds but offers potential for heuristic optimization. It is based on a tableau system that makes do without additional labelling of nodes beyond formulas from the standard Fischer-Ladner closure, such as foci or termination counters for eventualities. Moreover, the tableau system is single-pass, i.e. avoids building an exponential-sized structure in a first pass, to our best knowledge, optimal single-pass systems without numeric time-outs were not previously available even for CTL.

Conditional logics capture default entailment in a modal framework in which non-monotonic implication is a first-class citizen, and in particular can be negated and nested. There is a wide range of axiomatizations of conditionals in the literature, from weak systems such as the basic conditional logic CK, which allows only for equivalent exchange of conditional antecedents, to strong systems such as Burgess' system 𝒮, which imposes the full Kraus-Lehmann-Magidor properties of preferential logic. While tableaux systems implementing the actual complexity of the logic at hand have recently been developed for several weak systems, strong systems including in particular disjunction elimination or cautious monotonicity have so far eluded such efforts; previous results for strong systems are limited to semantics-based decision procedures and completeness proofs for Hilbert-style axiomatizations. Here, we present tableaux systems of optimal complexity PSPACE for several strong axiom systems in conditional logic, including system 𝒮; the arising decision procedure for system 𝒮 is implemented in the generic reasoning tool CoLoSS.