Alessandro Facchini

Rabin-Mostowski Index Problem: A Step beyond Deterministic Automata

For a given regular language of infinite trees, one can ask about the minimal number of priorities needed to recognise this language with a non-deterministic or alternating parity automaton. These questions are known as, respectively, the non-deterministic and the alternating Rabin-Mostowski index problems. Whether they can be answered effectively is a long-standing open problem, solved so far only for languages recognisable by deterministic automata (the alternating variant trivialises). We investigate a wider class of regular languages, recognisable by so-called game automata, which can be seen as the closure of deterministic ones under complementation and composition. Game automata are known to recognise languages arbitrarily high in the alternating Rabin-Mostowski index hierarchy, i.e., the alternating index problem does not trivialise any more. Our main contribution is that both index problems are decidable for languages recognisable by game automata. Additionally, we show that it is decidable whether a given regular language can be recognised by a game automaton.

Definable Operations On Weakly Recognizable Sets of Trees

Alternating automata on infinite trees induce operations on languages which do not preserve natural equivalence relations, like having the same Mostowski--Rabin index, the same Borel rank, or being continuously reducible to each other (Wadge equivalence). In order to prevent this, alternation needs to be restricted to the choice of direction in the tree. For weak alternating automata with restricted alternation a small set of computable operations generates all definable operations, which implies that the Wadge degree of a given automaton is computable. The weak index and the Borel rank coincide, and are computable. An equivalent automaton of minimal index can be computed in polynomial time (if the productive states of the automaton are given).

Weak MSO: automata and expressiveness modulo bisimilarity

We prove that the bisimulation-invariant fragment of weak monadic second-order logic (WMSO) is equivalent to the fragment of the modal μ-calculus where the application of the least fixpoint operator μp.φ is restricted to formulas φ that are continuous in p. Our proof is automata-theoretic in nature; in particular, we introduce a class of automata characterizing the expressive power of WMSO over tree models of arbitrary branching degree. The transition map of these automata is defined in terms of a logic FOE1∞ that is the extension of first-order logic with a generalized quantifier ∃∞, where ∃∞x.φ means that there are infinitely many objects satisfying φ. An important part of our work consists of a model-theoretic analysis of FOE1∞.

A Characterization Theorem for the Alternation-Free Fragment of the Modal µ-Calculus

We provide a characterization theorem, in the style of van Benthem and Janin-Walukiewicz, for the alternation-free fragment of the modal μ-calculus. For this purpose we introduce a variant of standard monadic second-order logic (MSO), which we call well-founded monadic second-order logic (WFMSO). When interpreted in a tree model, the second-order quantifiers of WFMSO range over subsets of conversely well-founded subtrees. The first main result of the paper states that the expressive power of WFMSO over trees exactly corresponds to that of weak MSO-automata. Using this automata-theoretic characterization, we then show that, over the class of all transition structures, the bisimulation-invariant fragment of WFMSO is the alternation-free fragment of the modal μ-calculus. As a corollary, we find that the logics WFMSO and WMSO (weak monadic second-order logic, where second-order quantification concerns finite subsets), are incomparable in expressive power.

Characterizing EF over infinite trees and modal logic on transitive graphs

We provide several effective equivalent characterizations of EF (the modal logic of the descendant relation) on arbitrary trees. More specifically, we prove that, for EF-bisimulation invariant properties of trees, being definable by an EF formula, being a Borel set, and being definable in weak monadic second order logic, all coincide. The proof builds upon a known algebraic characterization of EF for the case of finitely branching trees due to Bojanczyk and Idziaszek. We furthermore obtain characterizations of modal logic on transitive Kripke structures as a fragment of weak monadic second order logic and of the µ-calculus.

A Gleason-Type Theorem for Any Dimension Based on a Gambling Formulation of Quantum Mechanics

Based on a gambling formulation of quantum mechanics, we derive a Gleason-type theorem that holds for any dimension n of a quantum system, and in particular for

Deciding the Borel Complexity of Regular Tree Languages

n=2

Index Problems for Game Automata

n=2. The theorem states that the only logically consistent probability assignments are exactly the ones that are definable as the trace of the product of a projector and a density matrix operator. In addition, we detail the reason why dispersion-free probabilities are actually not valid, or rational, probabilities for quantum mechanics, and hence should be excluded from consideration.