2 research outputs found
Revisiting Membership Problems in Subclasses of Rational Relations
We revisit the membership problem for subclasses of rational relations over
finite and infinite words: Given a relation R in a class C_2, does R belong to
a smaller class C_1? The subclasses of rational relations that we consider are
formed by the deterministic rational relations, synchronous (also called
automatic or regular) relations, and recognizable relations. For almost all
versions of the membership problem, determining the precise complexity or even
decidability has remained an open problem for almost two decades. In this
paper, we provide improved complexity and new decidability results. (i) Testing
whether a synchronous relation over infinite words is recognizable is
NL-complete (PSPACE-complete) if the relation is given by a deterministic
(nondeterministic) omega-automaton. This fully settles the complexity of this
recognizability problem, matching the complexity of the same problem over
finite words. (ii) Testing whether a deterministic rational binary relation is
recognizable is decidable in polynomial time, which improves a previously known
double exponential time upper bound. For relations of higher arity, we present
a randomized exponential time algorithm. (iii) We provide the first algorithm
to decide whether a deterministic rational relation is synchronous. For binary
relations the algorithm even runs in polynomial time
Ramsey Quantifiers over Automatic Structures: Complexity and Applications to Verification
Automatic structures are infinite structures that are finitely represented by
synchronized finite-state automata. This paper concerns specifically automatic
structures over finite words and trees (ranked/unranked). We investigate the
"directed version" of Ramsey quantifiers, which express the existence of an
infinite directed clique. This subsumes the standard "undirected version" of
Ramsey quantifiers. Interesting connections between Ramsey quantifiers and two
problems in verification are firstly observed: (1) reachability with B\"{u}chi
and generalized B\"{u}chi conditions in regular model checking can be seen as
Ramsey quantification over transitive automatic graphs (i.e., whose edge
relations are transitive), (2) checking monadic decomposability (a.k.a.
recognizability) of automatic relations can be viewed as Ramsey quantification
over co-transitive automatic graphs (i.e., the complements of whose edge
relations are transitive). We provide a comprehensive complexity landscape of
Ramsey quantifiers in these three cases (general, transitive, co-transitive),
all between NL and EXP. In turn, this yields a wealth of new results with
precise complexity, e.g., verification of subtree/flat prefix rewriting, as
well as monadic decomposability over tree-automatic relations. We also obtain
substantially simpler proofs, e.g., for NL complexity for monadic
decomposability over word-automatic relations (given by DFAs)