Optimal Direct Translation from Past Linear Temporal Logic to Deterministic Rabin Automata

Core Concepts

The paper presents an asymptotically optimal direct translation from past linear temporal logic (pLTL) to deterministic Rabin automata, avoiding the intermediate step of constructing nondeterministic automata.

Abstract

The paper presents a direct translation from past linear temporal logic (pLTL) to deterministic Rabin automata. The key contributions are: Adaptation of the "after-function" from LTL to pLTL, which allows encoding the history contained in the prefix of a word directly into the formula being translated. This is achieved by rewriting strong past operators into their weak counterparts based on the consumed input. Extension of the Master Theorem, which establishes that the language of a pLTL formula can be decomposed into a Boolean combination of simpler languages. This allows constructing deterministic automata with simple acceptance conditions for these simpler languages and combining them into a deterministic Rabin automaton. Construction of the final deterministic Rabin automaton, which checks the three premises of the extended Master Theorem using a combination of Büchi and co-Büchi automata. The resulting automaton has a doubly exponential number of states in the size of the input formula and at most an exponential number of Rabin pairs. The direct translation avoids the intermediate step of constructing nondeterministic automata, which is required in the classic approach of translating pLTL to nondeterministic Büchi automata and then determinizing. This results in an asymptotically optimal translation.

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A Direct Translation from LTL with Past to Deterministic Rabin Automata

by Shaun Azzopa... at arxiv.org 05-03-2024

https://arxiv.org/pdf/2405.01178.pdf

Deeper Inquiries

How can the presented translation be extended to handle other temporal logic formalisms, such as branching-time logics or timed temporal logics

The presented translation from pLTL to deterministic Rabin automata can be extended to handle other temporal logic formalisms by adapting the concepts and techniques used in the translation process. For branching-time logics, such as CTL (Computation Tree Logic) or CTL* (CTL extended with temporal operators), the translation would involve incorporating the branching nature of the logic into the automata construction. This could be achieved by modifying the after-function to account for different paths in the computation tree and adjusting the acceptance conditions of the automata to capture branching behaviors. For timed temporal logics, such as Timed CTL or Timed LTL, the translation would need to consider the temporal aspects related to time constraints and durations. This could involve introducing clock variables to track time progress and incorporating timing constraints into the automata transitions. The after-function would need to handle temporal operators related to timing requirements, and the automata construction would need to account for timing constraints in the acceptance conditions. Overall, extending the translation to handle other temporal logic formalisms would require a deep understanding of the specific characteristics and semantics of each logic, as well as the ability to adapt the translation techniques to capture the unique features of the respective logics.

What are the practical implications of the optimal translation from pLTL to deterministic Rabin automata, and how can it be leveraged in applications like reactive synthesis and probabilistic model checking

The optimal translation from pLTL to deterministic Rabin automata has significant practical implications in various applications, such as reactive synthesis and probabilistic model checking. In reactive synthesis, where a system is automatically generated from a temporal logic specification, the translation allows for efficient synthesis of reactive systems with past-time requirements. By directly translating pLTL formulas to deterministic Rabin automata, the synthesis process can be streamlined, leading to faster and more reliable system generation. In probabilistic model checking, where the behavior of probabilistic systems is analyzed against temporal logic properties, the translation enables the efficient verification of systems with past-time specifications. The deterministic Rabin automata provide a structured way to check the satisfaction of complex temporal properties, enhancing the scalability and accuracy of probabilistic model checking algorithms. Overall, the optimal translation can be leveraged to improve the efficiency and effectiveness of system synthesis and verification processes in various domains, ultimately leading to more robust and reliable systems.

Are there any connections between the techniques used in this work and recent advancements in automata-based decision procedures for other logics, such as the ones based on the antichain approach

There are connections between the techniques used in this work and recent advancements in automata-based decision procedures for other logics, particularly those based on the antichain approach. The antichain approach is a powerful method for handling large state spaces in automata-based verification, especially for branching-time logics like CTL. The techniques used in the presented translation, such as the decomposition of languages into simpler components and the construction of deterministic automata, share similarities with the principles of the antichain approach. Both approaches aim to manage the complexity of automata constructions by breaking down the problem into smaller, more manageable parts. By leveraging the insights and methodologies from the antichain approach, advancements in automata-based decision procedures for other logics can benefit from the efficient and scalable techniques developed in the context of the optimal translation from pLTL to deterministic Rabin automata. This cross-pollination of ideas and methodologies can lead to further improvements in automata-based verification and synthesis techniques across different logics and applications.