We study the problem of realizing strategies for an LTLf goal specification while ensuring that at least an LTLf backup specification is satisfied in case of unreliability of certain input variables. We formally define the problem and characterize its worst-case complexity as 2EXPTIME-complete, like standard LTLf synthesis. Then we devise three different solution techniques: one based on direct automata manipulation, which is 2EXPTIME, one disregarding unreliable input variables by adopting a belief construction, which is 3EXPTIME, and one leveraging second-order quantified LTLf (QLTLf), which is 2EXPTIME and allows for a direct encoding into monadic second-order logic, which in turn is worst-case nonelementary. We prove their correctness and evaluate them against each other empirically. Interestingly, theoretical worst-case bounds do not translate into observed performance; the MSO technique performs best, followed by belief construction and direct automata manipulation. As a byproduct of our study, we provide a general synthesis procedure for arbitrary QLTLf specifications.

Abstract Artemov and Protopopescu proposed intuitionistic epistemic logic (IEL) to capture an intuitionistic conception of knowledge. By establishing completeness, they provided the base for a meta-theoretic investigation of IEL, which was continued by Krupski with a proof of cut-elimination, and Su and Sano establishing semantic cut-elimination and the finite model property. However, no analysis of these results in a constructive meta-logic has been conducted, arguably impeding the intuitionistic justification of IEL. We aim to close this gap and investigate IEL in the constructive-type theory of the Coq proof assistant. Concretely, we present a constructive and mechanised completeness proof for IEL, employing a syntactic decidability proof based on cut-elimination to constructivise the ideas from the literature. Following Su and Sano, we then also give constructive versions of semantic cut-elimination and the finite model property. Given our constructive and mechanised setting, all these results now bear executable algorithms. Our particular strategy to establish constructive completeness exploiting syntactic decidability can be used for similar modal logics, which we illustrate with the examples of the classical modal logics K, D and T. For modal logics including the four axioms, however, the method seems not to apply immediately.