1,271 research outputs found
First-order definable string transformations
The connection between languages defined by computational models and logic
for languages is well-studied. Monadic second-order logic and finite automata
are shown to closely correspond to each-other for the languages of strings,
trees, and partial-orders. Similar connections are shown for first-order logic
and finite automata with certain aperiodicity restriction. Courcelle in 1994
proposed a way to use logic to define functions over structures where the
output structure is defined using logical formulas interpreted over the input
structure. Engelfriet and Hoogeboom discovered the corresponding "automata
connection" by showing that two-way generalised sequential machines capture the
class of monadic-second order definable transformations. Alur and Cerny further
refined the result by proposing a one-way deterministic transducer model with
string variables---called the streaming string transducers---to capture the
same class of transformations. In this paper we establish a transducer-logic
correspondence for Courcelle's first-order definable string transformations. We
propose a new notion of transition monoid for streaming string transducers that
involves structural properties of both underlying input automata and variable
dependencies. By putting an aperiodicity restriction on the transition monoids,
we define a class of streaming string transducers that captures exactly the
class of first-order definable transformations.Comment: 31 page
On the definability of properties of finite graphs
AbstractThis paper considers the definability of graph-properties by restricted second-order and first-order sentences. For example, it is shown that the class of Hamiltonian graphs cannot be defined by monadic second-order sentences (i.e., if quantification over the subsets of vertices is allowed); any first-order sentence that defines Hamiltonian graphs on n vertices must contain at least 12n quantifiers. The proofs use Fraïssé-Ehrenfeucht games and ultraproducts
Logic and operator algebras
The most recent wave of applications of logic to operator algebras is a young
and rapidly developing field. This is a snapshot of the current state of the
art.Comment: A minor chang
Hidden variables in quantum mechanics: Generic models, set-theoretic forcing, and the emergence of probability
The hidden-variables premise is shown to be equivalent to the existence of
generic filters for algebras of commuting propositions and for certain more
general propositional systems. The significance of this equivalence is
interpreted in light of the theory of generic filters and boolean-valued models
in set theory (the method of forcing). The apparent stochastic nature of
quantum observation is derived for these hidden-variables models.Comment: 67 pages. Corrected formulas for conditional and joint probabilities
per comment of J. Malle
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