Abstract Syntactic obligations are a fragment of LTL formulas that translate to deterministic weak $$\omega $$ ω -automata (DWA). We show that syntactic obligations can be very efficiently converted to minimal DWA represented using multi-terminal binary decision diagrams (MTBDDs), and that synthesis of such specifications can be solved directly on the MTBDD representation on the fly. Our implementation in Spot shows substantial runtime improvements in translation and synthesis.

LTLf synthesis under partial observability requires reasoning about unobservable environment variables, which is typically handled by constructing a belief-state DFA via subset construction that universally quantifies these variables. Existing approaches perform this construction as a separate step prior to game solving, often generating belief states that are unnecessary in practice. We propose an on-the-fly approach to LTLf synthesis under partial observability based on observable progression. Our method incrementally builds the belief-state DFA by progressing the specification with respect to observable variables only, universally quantifying unobservable variables on the fly. We prove the correctness of the construction and show that it naturally enables on-the-fly game solving, leading to a fully on-the-fly synthesis framework. Our implementation leverages DFAs represented using Multi-Terminal Binary Decision Diagrams: a compact representation that has proven highly effective for LTLf synthesis under full observability. Experimental results demonstrate that our approach significantly outperforms existing methods and further highlight the practical benefits of integrating on-the-fly game solving with belief-state construction.

We present an on-the-fly synthesis framework for Linear Temporal Logic over Finite Traces ( LTL \({}_{f}\) ) based on top-down deterministic automata construction. Existing approaches rely on constructing a complete Deterministic Finite Automaton ( DFA ) corresponding to the LTL \({}_{f}\) specification, a process with doubly exponential complexity relative to formula size in the worst case. In this case, the synthesis cannot be conducted until the entire DFA is constructed. This inefficiency is the main bottleneck of existing approaches. To address this challenge, we first present a method for converting LTL \({}_{f}\) into Transition-Based DFA ( TDFA ) by directly leveraging LTL \({}_{f}\) semantics, incorporating intermediate results as direct components of the final automaton to enable parallelized synthesis and automata construction. We then explore the relationship between LTL \({}_{f}\) synthesis and TDFA games and subsequently develop an algorithm for performing LTL \({}_{f}\) synthesis via on-the-fly TDFA game solving. This algorithm traverses the state space in a global forward manner combined with a local backward method, along with detecting strongly connected components. Moreover, we introduce two optimization techniques—model-guided synthesis and state entailment—to enhance the practical efficiency of our approach. Experimental results demonstrate that our on-the-fly approach achieves the best performance on the tested benchmarks and effectively complements existing approaches.

Reactive synthesis from Linear Temporal Logic over finite traces (LTLf) can be reduced to a two-player game over a Deterministic Finite Automaton (DFA) of the LTLf specification. The primary challenge here is DFA construction, which is 2EXPTIME-complete in the worst case. Existing techniques either construct the DFA compositionally before solving the game, leveraging automata minimization to mitigate state-space explosion, or build the DFA incrementally during game solving to avoid full DFA construction. However, neither is dominant. In this paper, we introduce a compositional on-the-fly synthesis framework that integrates the strengths of both approaches, focusing on large conjunctions of smaller LTLf formulas common in practice. This framework applies composition during game solving instead of automata (game arena) construction. While composing all intermediate results may be necessary in the worst case, pruning these results simplifies subsequent compositions and enables early detection of unrealizability. Specifically, the framework allows two composition variants: pruning before composition to take full advantage of minimization or pruning during composition to guide on-the-fly synthesis. Compared to state-of-the-art synthesis solvers, our framework is able to solve a notable number of instances that other solvers cannot handle. A detailed analysis shows that both composition variants have unique merits.

We study a variant of LTLf synthesis that synthesizes adaptive strategies for achieving a multi-tier goal, consisting of multiple increasingly challenging LTLf objectives in nondeterministic planning domains. Adaptive strategies are strategies that at any point of their execution (i) enforce the satisfaction of as many objectives as possible in the multi-tier goal, and (ii) exploit possible cooperation from the environment to satisfy as many as possible of the remaining ones. This happens dynamically: if the environment cooperates (ii) and an objective becomes enforceable (i), then our strategies will enforce it. We provide a game-theoretic technique to compute adaptive strategies that is sound and complete. Notably, our technique is polynomial, in fact quadratic, in the number of objectives. In other words, it handles multi-tier goals with only a minor overhead compared to standard LTLf synthesis.

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.

Abstract There has been a massive interest in utilizing Linear Temporal Logic on finite traces ( $$\textsc {ltl}_f$$ L T L f ) as a specification language in the last decade, particularly in reactive synthesis. This highlights the need for a unified and efficient framework to fulfil the increasing demand for easy-to-use implementations of synthesis (and reasoning) algorithms for $$\textsc {ltl}_f$$ L T L f . To that end, we introduce , an open-source compositional symbolic synthesis framework that integrates efficient data structures and techniques focused on $$\textsc {ltl}_f$$ L T L f specifications. supports both explicit-DFA and symbolic-DFA construction from $$\textsc {ltl}_f$$ L T L f formulas, essential DFA manipulations, and offers an extensible framework for reactive synthesis of $$\textsc {ltl}_f$$ L T L f specifications, accommodating more complex synthesis scenarios. We demonstrate this feasibility by supporting $$\textsc {ltl}_f$$ L T L f synthesis as well as $$\textsc {ltl}_f$$ L T L f synthesis with ltl environment specifications expressed in various forms. is highly efficient and versatile, providing user-friendly C++ interfaces and extensive benchmarks that cater to a diverse audience, including computer scientists, practitioners, students, and the reactive synthesis research community.

In this paper, we study ltl f synthesis under environment specifications for arbitrary reachability and safety properties. We consider both kinds of properties for both agent tasks and environment specifications, providing a complete landscape of synthesis algorithms. For each case, we devise a specific algorithm (optimal wrt complexity of the problem) and prove its correctness. The algorithms combine common building blocks in different ways. While some cases are already studied in literature others are studied here for the first time.

Devising a strategy to make a system mimic behaviors from another system is a problem that naturally arises in many areas of Computer Science. In this work, we interpret this problem in the context of intelligent agents, from the perspective of LTLf, a formalism commonly used in AI for expressing finite-trace properties. Our model consists of two separated dynamic domains, D_A and D_B, and an LTLf specification that formalizes the notion of mimicking by mapping properties on behaviors (traces) of D_A into properties on behaviors of D_B. The goal is to synthesize a strategy that step-by-step maps every behavior of D_A into a behavior of D_B so that the specification is met. We consider several forms of mapping specifications, ranging from simple ones to full LTLf, and for each, we study synthesis algorithms and computational properties.

Abstract With the growing use of temporal logics in areas ranging from robot planning to runtime verification, it is critical that users have a clear understanding of what a specification means. Toward this end, we have been developing a catalog of semantic errors and a suite of test instruments targeting various user-groups. The catalog is of interest to educators, to logic designers, to formula authors, and to tool builders, e.g., to identify mistakes. The test instruments are suitable for classroom teaching or self-study. This paper reports on five sets of survey data collected over a three-year span. We study misconceptions about finite-trace $$\textsc {ltl}_{f}$$ L T L f in three ltl-aware audiences, and misconceptions about standard ltl in novices. We find several mistakes, even among experts. In addition, the data supports several categories of errors in both $$\textsc {ltl}_{f}$$ L T L f and ltl that have not been identified in prior work. These findings, based on data from actual users, offer insights into what specific ways temporal logics are tricky and provide a groundwork for future interventions.

We study best-effort strategies (aka plans) in fully observable nondeterministic domains (FOND) for goals expressed in Linear Temporal Logic on Finite Traces (LTLf). The notion of best-effort strategy has been introduced to also deal with the scenario when no agent strategy exists that fulfills the goal against every possible nondeterministic environment reaction. Such strategies fulfill the goal if possible, and do their best to do so otherwise. We present a game-theoretic technique for synthesizing best-effort strategies that exploit the specificity of nondeterministic planning domains. We formally show its correctness and demonstrate its effectiveness experimentally, exhibiting a much greater scalability with respect to a direct best-effort synthesis approach based on re-expressing the planning domain as generic environment specifications.

Devising a strategy to make a system mimic behaviors from another system is a problem that naturally arises in many areas of Computer Science. In this work, we interpret this problem in the context of intelligent agents, from the perspective of ltlf , a formalism commonly used in AI for expressing finite-trace properties. Our model consists of two separated dynamic domains, DA and DB, and an LTLf specification that formalizes the notion of mimicking by mapping properties on behaviors (traces) of DA into properties on behaviors of DB. The goal is to synthesize a strategy that step-by-step maps every behavior of DA into a behavior of DB so that the specification is met. We consider several forms of mapping specifications, ranging from simple ones to full LTLf , and for each, we study synthesis algorithms and computational properties.

In this paper, we study synthesis of maximally permissive strategies for Linear Temporal Logic on finite traces (LTLf) specifications. That is, instead of computing a single strategy (aka plan, or policy), we aim at computing the entire set of strategies at once and then choosing among them while in execution, without committing to a single one beforehand. Maximally permissive strategies have been introduced and investigated for safety properties, especially in the context of Discrete Event Control Theory. However, the available results for safety properties do not apply to reachability properties (eventually reach a given state of affair) nor to LTLf properties in general. In this paper, we show that maximally permissive strategies do exist also for reachability and general LTLf properties, and can in fact be computed with minimal overhead wrt the computation of a single strategy using state-of-the-art tools.

Synthesis techniques for temporal logic specifications are typically based on exploiting symbolic techniques, as done in model checking. These symbolic techniques typically use backward fixpoint computation. Planning, which can be seen as a specific form of synthesis, is a witness of the success of forward search approaches. In this paper, we develop a forward-search approach to full-fledged Linear Temporal Logic on finite traces (LTLf) synthesis. We show how to compute the Deterministic Finite Automaton (DFA) of an LTLf formula on-the-fly, while performing an adversarial forward search towards the final states, by considering the DFA as a sort of AND-OR graph. Our approach is characterized by branching on suitable propositional formulas, instead of individual evaluations, hence radically reducing the branching factor of the search space. Specifically, we take advantage of techniques developed for knowledge compilation, such as Sentential Decision Diagrams (SDDs), to implement the approach efficiently.

Abstract Linear Temporal Logic (LTL) synthesis aims at automatically synthesizing a program that complies with desired properties expressed in LTL. Unfortunately it has been proved to be too difficult computationally to perform full LTL synthesis. There have been two success stories with LTL synthesis, both having to do with the form of the specification. The first is the GR(1) approach: use safety conditions to determine the possible transitions in a game between the environment and the agent, plus one powerful notion of fairness, Generalized Reactivity(1), or GR(1). The second, inspired by AI planning, is focusing on finite-trace temporal synthesis, with LTL $$_f$$ f (LTL on finite traces) as the specification language. In this paper we take these two lines of work and bring them together. We first study the case in which we have an LTL $$_f$$ f agent goal and a GR(1) environment specification. We then add to the framework safety conditions for both the environment and the agent, obtaining a highly expressive yet still scalable form of LTL synthesis.

We present a new synthesis framework based on the on-the-fly DFA construction for LTL over finite traces (LTLf ). Extant approaches rely heavily on the construction of the complete DFA w.r.t. the input LTLf formula, whose size can be doubly exponential to the size of the formula in the worst case. Under those approaches, the synthesis cannot be conducted unless the whole DFA is completely constructed, which is not only inefficient but also not scalable in practice. Indeed, the DFA construction is the main bottleneck of LTLf synthesis in prior work. To mitigate this challenge, we follow two steps in this paper: Firstly, we present several light-weight pre-processing techniques such that the synthesis result can be obtained even without DFA construction; Secondly, we propose to achieve the synthesis together with the on-the-fly DFA construction such that the synthesis result can be obtained before constructing the whole DFA. The on-the-fly DFA construction is implemented using the SAT-based techniques for automata generation. We compared our new approach with the traditional ones on extensive LTLf synthesis benchmarks. Experimental results showed that the pre-processing techniques have a significant advantage on the synthesis performance in terms of scalability, and the on-the-fly synthesis is able to complement extant approaches on both realizable and unrealizable cases.

Linear Temporal Logic (LTL) synthesis aims at automatically synthesizing a program that complies with desired properties expressed in LTL. Unfortunately it has been proved to be too difficult computationally to perform full LTL synthesis. There have been two success stories with LTL synthesis, both having to do with the form of the specification. The first is the GR(1) approach: use safety conditions to determine the possible transitions in a game between the environment and the agent, plus one powerful notion of fairness, Generalized Reactivity(1), or GR(1). The second, inspired by AI planning, is focusing on finite-trace temporal synthesis, with LTLf (LTL on finite traces) as the specification language. In this paper we take these two lines of work and bring them together. We first study the case in which we have an LTLf agent goal and a GR(1) assumption. We then add to the framework safety conditions for both the environment and the agent, obtaining a highly expressive yet still scalable form of LTL synthesis.

We study the impact of the need for the agent to obligatorily instruct the action stop in her strategies. More specifically we consider synthesis (i.e., planning) for LTLf goals under LTL environment specifications in the case the agent must mandatorily stop at a certain point. We show that this obligation makes it impossible to exploit the liveness part of the LTL environment specifications to achieve her goal, effectively reducing the environment specifications to their safety part only. This has a deep impact on the efficiency of solving the synthesis, which can sidestep handling Buchi determinization associated to LTL synthesis, in favor of finite-state automata manipulation as in LTLf synthesis. Next, we add to the agent goal, expressed in LTLf, a safety goal, expressed in LTL. Safety goals must hold forever, even when the agent stops, since the environment can still continue its evolution. Hence the agent, before stopping, must ensure that her safety goal will be maintained even after she stops. To do synthesis in this case, we devise an effective approach that mixes a synthesis technique based on finite-state automata (as in the case of LTLf goals) and model-checking of nondeterministic Buchi automata. In this way, again, we sidestep Buchi automata determinization, hence getting a synthesis technique that is intrinsically simpler than standard LTL synthesis.

In synthesis, assumptions are constraints on the environment that rule out certain environment behaviors. A key observation here is that even if we consider systems with LTLƒ goals on finite traces, environment assumptions need to be expressed over infinite traces, since accomplishing the agent goals may require an unbounded number of environment action. To solve synthesis with respect to finite-trace LTLƒ goals under infinite-trace assumptions, we could reduce the problem to LTL synthesis. Unfortunately, while synthesis in LTLƒ and in LTL have the same worst-case complexity (both 2EXPTIME-complete), the algorithms available for LTL synthesis are much more difficult in practice than those for LTLƒ synthesis. In this work we show that in interesting cases we can avoid such a detour to LTL synthesis and keep the simplicity of LTLƒ synthesis. Specifically, we develop a BDD-based fixpoint-based technique for handling basic forms of fairness and of stability assumptions. We show, empirically, that this technique performs much better than standard LTL synthesis.

In synthesis, assumption are constraints on the environments that rule out certain environment behaviors. A key observation is that even if we consider system with LTLf goals on finite traces, assumptions need to be expressed considering infinite traces, using LTL on infinite traces, since the decision to stop the trace is controlled by the agent. To solve synthesis of LTLf goals under LTL assumptions, we could reduce the problem to LTL synthesis. Unfortunately, while synthesis in LTLf and in LTL have the same worst-case complexity (both are 2EXPTIME-complete), the algorithms available for LTL synthesis are much harder in practice than those for LTLf synthesis. Recently, it has been shown that in basic forms of fairness and stability assumptions we can avoid such a detour to LTL and keep the simplicity of LTLf synthesis. In this paper, we generalize these results and show how to effectively handle any kind of LTL assumptions. Specifically, we devise a two-stage technique for solving LTLf under general LTL assumptions and show empirically that this technique performs much better than standard LTL synthesis.

Abstract We propose a novel algorithm for the satisfiability problem for linear temporal logic (LTL). Existing automata-based approaches first transform the LTL formula into a Büchi automaton and then perform an emptiness checking of the resulting automaton. Instead, our approach works on-the-fly by inspecting the formula directly, thus enabling to find a satisfying model quickly without constructing the full automaton. This makes our algorithm particularly fast for satisfiable formulas. We construct experiments on different pattern formulas, the experimental results show that our approach is superior to other solvers under automata-based framework.

Formal-verification techniques, such as model checking, are becoming popular in hardware design. SAT-based model checking techniques, such as IC3/PDR, have gained a significant success in the hardware industry. In this paper, we present a new framework for SAT-based safety model checking, named Complementary Approximate Reachability (CAR). CAR is based on standard reachability analysis, but instead of maintaining a single sequence of reachable-state sets, CAR maintains two sequences of over- and under-approximate reachable-state sets, checking safety and unsafety at the same time. To construct the two sequences, CAR uses standard Boolean-reasoning algorithms, based on satisfiability solving, one to find a satisfying cube of a satisfiable Boolean formula, and one to provide a minimal unsatisfiable core of an unsatisfiable Boolean formula. We applied CAR to 548 hardware model-checking instances, and compared its performance with IC3/PDR. Our results show that CAR is able to solve 42 instances that cannot be solved by IC3/PDR. When evaluated against a portfolio that includes IC3/PDR and other approaches, CAR is able to solve 21 instances that the other approaches cannot solve. We conclude that CAR should be considered as a valuable member of any algorithmic portfolio for safety model checking.

LTLf synthesis is the process of finding a strategy that satisfies a linear temporal specification over finite traces. An existing solution to this problem relies on a reduction to a DFA game. In this paper, we propose a symbolic framework for LTLf synthesis based on this technique, by performing the computation over a representation of the DFA as a boolean formula rather than as an explicit graph. This approach enables strategy generation by utilizing the mechanism of boolean synthesis. We implement this symbolic synthesis method in a tool called Syft, and demonstrate by experiments on scalable benchmarks that the symbolic approach scales better than the explicit one.