Measuring and Synthesizing Systems in Probabilistic Environments

Abstract

The traditional synthesis question given a specification asks for the automatic construction of a system that satisfies the specification, whereas often there exists a preference order among the different systems that satisfy the given specification. Under a probabilistic assumption about the possible inputs, such a preference order is naturally expressed by a weighted automaton, which assigns to each word a value, such that a system is preferred if it generates a higher expected value. We solve the following optimal synthesis problem: given an omega-regular specification, a Markov chain that describes the distribution of inputs, and a weighted automaton that measures how well a system satisfies the given specification under the input assumption, synthesize a system that optimizes the measured value.For safety specifications and quantitative measures that are defined by mean-payoff automata, the optimal synthesis problem reduces to finding a strategy in a Markov decision process (MDP) that is optimal for a long-run average reward objective, which can be achieved in polynomial time. For general omega-regular specifications along with mean-payoff automata, the solution rests on a new, polynomial-time algorithm for computing optimal strategies in MDPs with mean-payoff parity objectives. Our algorithm constructs optimal strategies that consist of two memoryless strategies and a counter. The counter is in general not bounded. To obtain a finite-state system, we show how to construct an ε-optimal strategy with a bounded counter, for all ε > 0. Furthermore, we show how to decide in polynomial time if it is possible to construct an optimal finite-state system (i.e., a system without a counter) for a given specification.We have implemented our approach and the underlying algorithms in a tool that takes qualitative and quantitative specifications and automatically constructs a system that satisfies the qualitative specification and optimizes the quantitative specification, if such a system exists. We present some experimental results showing optimal systems that were automatically generated in this way.

Language and Automata Theory and Applications

“…Weighted (word) automata define the class of recognizable word functions, first introduced in the study of stochastic automata by A. Heller [39]. Weighted automata are used in verification, [6,52], in program synthesis, [13,14], in digital image compression, [19], and speech processing, [53,28,1]. For a comprehensive survey, see the Handbook of Weighted Automata [27].…”

Bridging boolean and quantitative synthesis using smoothed proof search

“…Quantitative synthesis has been studied in the past by the automata-theoretic synthesis community [3][4][5][6]. Specifically, the statement for our problem is derived from prior work by Chatterjee et al [6].…”

“…Specifically, the statement for our problem is derived from prior work by Chatterjee et al [6]. However, the solutions provided by these papers are restricted to finite-state systems, whereas our programs and specifications are infinite-state.…”

“…However, there can be many implementations that meet these properties, and some of them are more desirable than others. Given this, it is appropriate to consider synthesis tasks where the synthesized implementation must not only meet a boolean specification, but also be optimal with respect to a quantitative objective [3,6].…”

“…One interpretation is that the programmer wishes to minimize the worstcase behavior of the system; alternatively, Chatterjee et al [6] have argued that a more natural goal is to optimize the expected value of the output. This is potentially a harder problem, because it requires knowledge of the distribution of its inputs, and it requires an analysis capable of deriving the distribution of the outputs from this input distribution.…”