The notion of comparison between system runs is fundamental in formal verification. This concept is implicitly present in the verification of qualitative systems, and is more pronounced in the verification of quantitative systems. In this work, we identify a novel mode of comparison in quantitative systems: the online comparison of the aggregate values of two sequences of quantitative weights. This notion is embodied by comparator automata (comparators, in short), a new class of automata that read two infinite sequences of weights synchronously and relate their aggregate values.We show that aggregate functions that can be represented with Büchi automaton result in comparators that are finite-state and accept by the Büchi condition as well. Such ωregular comparators further lead to generic algorithms for a number of well-studied problems, including the quantitative inclusion and winning strategies in quantitative graph games with incomplete information, as well as related non-decision problems, such as obtaining a finite representation of all counterexamples in the quantitative inclusion problem.We study comparators for two aggregate functions: discounted-sum and limit-average. We prove that the discounted-sum comparator is ω-regular iff the discount-factor is an integer. Not every aggregate function, however, has an ω-regular comparator. Specifically, we show that the language of sequence-pairs for which limit-average aggregates exist is neither ω-regular nor ω-context-free. Given this result, we introduce the notion of prefixaverage as a relaxation of limit-average aggregation, and show that it admits ω-context-free comparators.Does an ω-regular comparator for an aggregate function and a relation imply that the aggregate function is also ω-regular? Furthermore, we show that ω-regular comparators lead to generic algorithms for a number of well-studied problems including the quantitative inclusion problem, and in showing existence of winning strategies in incomplete-information quantitative games. Our algorithm yields PSPACE-completeness of quantitative inclusion when the ω-regular comparator is provided. The same algorithm extends to obtaining finite-state representations of counterexample words in inclusion.Next, we show that the discounted-sum aggregation function admits an ω-regular comparator for all relations R iff the discount-factor d > 1 is an integer. We use this result to prove that discounted-sum aggregate function for discount-factor d > 1 is ω-regular iff d is an integer. Furthermore, we use properties of ω-regular comparators to conclude that the discounted-sum inclusion is PSPACE-complete, hence resolving the complexity gap. Finally, we investigate the limit-average comparator. Since limit-average is only defined for sequences in which the average of prefixes converge, limit-average comparison is not welldefined. We show that even a Büchi pushdown automaton cannot separate sequences for which limit-average exists from those for which it does not. Hence, we introduce the novel notion of prefix-avera...

LTLf synthesis is the automated construction of a reactive system from a high-level description, expressed in LTLf, of its finite-horizon behavior. So far, the conversion of LTLf formulas to deterministic finite-state automata (DFAs) has been identified as the primary bottleneck to the scalabity of synthesis. Recent investigations have also shown that the size of the DFA state space plays a critical role in synthesis as well.Therefore, effective resolution of the bottleneck for synthesis requires the conversion to be time and memory performant, and prevent state-space explosion. Current conversion approaches, however, which are based either on explicit-state representation or symbolic-state representation, fail to address these necessities adequately at scale: Explicit-state approaches generate minimal DFA but are slow due to expensive DFA minimization. Symbolic-state representations can be succinct, but due to the lack of DFA minimization they generate such large state spaces that even their symbolic representations cannot compensate for the blow-up.This work proposes a hybrid representation approach for the conversion. Our approach utilizes both explicit and symbolic representations of the state-space, and effectively leverages their complementary strengths. In doing so, we offer an LTLf to DFA conversion technique that addresses all three necessities, hence resolving the bottleneck. A comprehensive empirical evaluation on conversion and synthesis benchmarks supports the merits of our hybrid approach.

Reinforcement learning has been shown to be an effective strategy for automatically training policies for challenging control problems. Focusing on non-cooperative multi-agent systems, we propose a novel reinforcement learning framework for training joint policies that form a Nash equilibrium. In our approach, rather than providing low-level reward functions, the user provides high-level specifications that encode the objective of each agent. Then, guided by the structure of the specifications, our algorithm searches over policies to identify one that provably forms an$$\epsilon $$ϵ-Nash equilibrium (with high probability). Importantly, it prioritizes policies in a way that maximizes social welfare across all agents. Our empirical evaluation demonstrates that our algorithm computes equilibrium policies with high social welfare, whereas state-of-the-art baselines either fail to compute Nash equilibria or compute ones with comparatively lower social welfare.

The problem of quantitative inclusion formalizes the goal of comparing quantitative dimensions between systems such as worst-case execution time, resource consumption, and the like. Such systems are typically represented by formalisms such as weighted logics or weighted automata. Despite its significance in analyzing the quality of computing systems, the study of quantitative inclusion has mostly been conducted from a theoretical standpoint. In this work, we conduct the first empirical study of quantitative inclusion for discounted-sum weighted automata (DS-inclusion, in short). Currently, two contrasting approaches for DS-inclusion exist: the linear-programming based DetLP and the purely automata-theoretic BCV. Theoretical complexity of DetLP is exponential in time and space while of BCV is PSPACE-complete. All practical implementations of BCV, however, are also exponential in time and space. Hence, it is not clear which of the two algorithms renders a superior implementation. In this work we present the first implementations of these algorithms, and perform extensive experimentation to compare between the two approaches. Our empirical analysis shows how the two approaches complement each other. This is a nuanced picture that is much richer than the one obtained from the theoretical study alone.

Discounted-sum inclusion (DS-inclusion, in short) formalizes the goal of comparing quantitative dimensions of systems such as cost, resource consumption, and the like, when the mode of aggregation for the quantitative dimension is discounted-sum aggregation. Discounted-sum comparator automata, or DS-comparators in short, are Büchi automata that read two infinite sequences of weights synchronously and relate their discounted-sum. Recent empirical investigations have shown that while DS-comparators enable competitive algorithms for DS-inclusion, they still suffer from the scalability bottleneck of Büchi operations. Motivated by the connections between discounted-sum and Büchi automata, this paper undertakes an investigation of language-theoretic properties of DS-comparators in order to mitigate the challenges of Büchi DS-comparators to achieve improved scalability of DS-inclusion. Our investigation uncovers that DS-comparators possess safety and co-safety language-theoretic properties. As a result, they enable reductions based on subset construction-based methods as opposed to higher complexity Büchi complementation, yielding tighter worst-case complexity and improved empirical scalability for DS-inclusion.

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Asynchronous interactions are ubiquitous in computing systems and complicate design and programming. Automatic construction of asynchronous programs from specifications ("synthesis") could ease the difficulty, but known methods are complex, and intractable in practice. This work develops substantially simpler synthesis methods. A direct, exponentially more compact automaton construction is formulated for the reduction of asynchronous to synchronous synthesis. Experiments with a prototype implementation of the new method demonstrate feasibility. Furthermore, it is shown that for several useful classes of temporal properties, automaton-based methods can be avoided altogether and replaced with simpler Boolean constraint solving.

In the Adapter Design Pattern, a programmer implements a Target interface by constructing an Adapter that accesses an existing Adaptee code. In this work, we present a reactive synthesis interpretation to the adapter design pattern, wherein an algorithm takes an Adaptee and a Target transducers, and the aim is to synthesize an Adapter transducer that, when composed with the Adaptee, generates a behavior that is equivalent to the behavior of the Target. One use of such an algorithm is to synthesize controllers that achieve similar goals on different hardware platforms. While this problem can be solved with existing synthesis algorithms, current state-of-the-art tools fail to scale. To cope with the computational complexity of the problem, we introduce a special form of specification format, called Separated GR(k), which can be solved with a scalable synthesis algorithm but still allows for a large set of realistic specifications. We solve the realizability and the synthesis problems for Separated GR(k), and show how to exploit the separated nature of our specification to construct better algorithms, in terms of time complexity, than known algorithms for GR(k) synthesis. We then describe a tool, called SGR(k), that we have implemented based on the above approach and show, by experimental evaluation, how our tool outperforms current state-of-the-art tools on various benchmarks and test-cases.