Better Quality in Synthesis through Quantitative Objectives (0904.2638v3)

Abstract: Most specification languages express only qualitative constraints. However, among two implementations that satisfy a given specification, one may be preferred to another. For example, if a specification asks that every request is followed by a response, one may prefer an implementation that generates responses quickly but does not generate unnecessary responses. We use quantitative properties to measure the "goodness" of an implementation. Using games with corresponding quantitative objectives, we can synthesize "optimal" implementations, which are preferred among the set of possible implementations that satisfy a given specification. In particular, we show how automata with lexicographic mean-payoff conditions can be used to express many interesting quantitative properties for reactive systems. In this framework, the synthesis of optimal implementations requires the solution of lexicographic mean-payoff games (for safety requirements), and the solution of games with both lexicographic mean-payoff and parity objectives (for liveness requirements). We present algorithms for solving both kinds of novel graph games.

• The paper introduces a novel framework using lexicographic mean-payoff games to incorporate quantitative objectives into system synthesis, enabling the optimization of implementation quality beyond simple correctness.
• It presents novel algorithms for solving these quantitative games and demonstrates the feasibility of solving them within NP and coNP complexity classes.
• The practical implications include generating optimal and efficient system implementations, while theoretical impacts extend to quantitative reasoning in AI systems.

Quantitative Objectives in Synthesis with Lexicographic Mean-Payoff Games

The paper presents an advanced framework for system synthesis via quantitative objectives, particularly aiming at optimizing implementations based on their performance measures. Traditionally, specifications for system syntheses have focused on qualitative aspects: an implementation either satisfies a specification or it does not, typically in a Boolean fashion. This overlooks the nuance required when distinguishing between varied implementations that meet a specification but differ in qualitative utility.

This research leverages quantitative properties to enhance specification languages, which can signify preference among different implementations that satisfy a given specification. A prime example is the policy where every request is followed by a response, thus preferring implementations that respond quickly without unnecessary responses.

The authors introduce graph games with novel objectives focusing on lexicographic mean-payoff and parity conditions. This approach brings forward the development of lexicographic mean-payoff games, addressing both safety properties through lexicographic mean-payoff objectives, and liveness properties via a combination of parity and lexicographic mean-payoff objectives.

Key Contributions

1. Lexicographic Mean-Payoff Conditions: The paper introduces automata with lexicographic mean-payoff conditions as a method to express various quantitative properties for reactive systems. This offers an avenue to measure the "goodness" or quality of an implementation beyond mere correctness, evaluating it on a multi-dimensional scale of quantitative rules.
2. Novel Algorithms for Game Solutions: By solving lexicographic mean-payoff games and games combining lexicographic mean-payoff with parity objectives, the authors propose algorithms that determine the optimality of specific implementations in graph games. These solutions are critical in synthesizing implementations that optimize quantitative metrics.
3. Strong Numerical Assertions: The paper provides evidence, suggesting that lexicographic mean-payoff games are determinable within the complexity classes NP and coNP. This corroborates the feasibility of solving complexity issues in potential practical scenarios.
4. Verification and Synthesis: Quantitative verification and synthesis are detailed, with the implications for deciding the satisfiability of quantitative specifications using lexicographic mean-payoff parity automata. These solutions enable not only qualitative checking but also evaluating whether an implementation can satisfy certain quantitative thresholds.

Practical and Theoretical Implications

Practically, these contributions aid in producing optimal implementations that are not only functionally correct but also efficient based on specified criteria, which have real-world applications in designing responsive and efficient software systems. The analytically robust methodologies open pathways for developing systems that optimize user interactions or resource management based on specified priorities.

Theoretically, the exploration of lexicographic objectives and the solutions proposed could influence future developments in quantitative reasoning and decision-making within AI systems, enhancing their ability to integrate complex performance metrics in autonomy and learning.

Future Directions

Future research could delve into expanding these quantitative frameworks with more diverse objective measures, including hybrid metrics such as discounted sum objectives. Application in extended AI scenarios, like reinforcement learning environments where long-term strategic payoffs might define system design, offers a promising direction. Moreover, the exploration of probabilistic measures alongside deterministic approaches could lead to richer models for both system design and verification.

In summary, this paper expands the landscape of system synthesis by incorporating quantitative preferences into specifications, providing a robust framework for generating not only correct but optimal implementations. The implications are significant for fields that require sophisticated and efficient system design, offering a quantitative edge over traditional synthesis practices.