Abstract
A challenging problem for autonomous systems is to synthesize a reactive controller that conforms to a set of given correctness properties. Linear temporal logic (LTL) provides a formal language to specify the desired behavioral properties of systems. In applications in which the specifications originate from various aspects of the system design, or consist of a large set of formulas, the overall system specification may be unrealizable. Driven by this fact, we develop an optimization variant of synthesis from LTL formulas, where the goal is to design a controller that satisfies a set of hard specifications and minimally violates a set of soft specifications. To that end, we introduce a value function that, by exploiting the LTL semantics, quantifies the level of violation of properties. Inspired by the idea of bounded synthesis, we fix a bound on the implementation size and search for an implementation that is optimal with respect to the said value function. We propose a novel maximum satisfiability encoding of the search for an optimal implementation (within the given bound on the implementation size). We iteratively increase the bound on the implementation size until a termination criterion, such as a threshold over the value function, is met.
Highlights
We start by an overview of syntax and semantics of linear temporal logic (LTL) and languageequivalent automata representation
Maximum realizability problem: Given an LTL formula φ and formulas φ1, . . . , φn, where each φi is a safety LTL formula, the maximum realizability problem asks to determine if there exists a transition system T such that T |= φ, and if the answer is positive, to synthesize a transition system T such that T |= φ, and for every transition system T ′ with T ′ |= φ it holds that val (T, φ1 ∧ . . . ∧ φn) ≥ val (T ′, φ1 ∧ . . . ∧ φn)
We considered settings in which a system’s requirements are categorized as hard and soft linear temporal logic (LTL) specifications and the goal is to design a controller that satisfies the hard specifications while maximally realizing the soft specifications
Summary
Maximum realizability and several closely related problems have attracted significant attention in recent years. Lahijanian et al [13] studied the problem of partial satisfaction of guarantees in an unknown environment, where, unlike in our work, no relaxations of the soft specifications are considered, but the number of those that are satisfied is maximized. When applied to multiple soft specifications, the method by Tomita et al combines the corresponding mean-payoff terms in a weighted sum, and synthesizes an implementation optimizing the value of this sum. Our approach to maximum realizability can prove useful for specification analysis, since instead of providing an optimal value, it computes an optimal relaxation of the given specification in the form of another LTL formula
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