818 research outputs found
From Linear to Branching-Time Temporal Logics: Transfer of Semantics and Definability
This paper investigates logical aspects of combining linear orders as semantics for modal and temporal logics, with modalities for possible paths, resulting in a variety of branching time logics over classes of trees. Here we adopt a unified approach to the Priorean, Peircean and Ockhamist semantics for branching time logics, by considering them all as fragments of the latter, obtained as combinations, in various degrees, of languages and semantics for linear time with a modality for possible paths. We then consider a hierarchy of natural classes of trees and bundled trees arising from a given class of linear orders and show that in general they provide different semantics. We also discuss transfer of definability from linear orders to trees and introduce a uniform translation from Priorean to Peircean formulae which transfers definability of properties of linear orders to definability of properties of all paths in tree
Finite-Degree Predicates and Two-Variable First-Order Logic
We consider two-variable first-order logic on finite words with a fixed
number of quantifier alternations. We show that all languages with a neutral
letter definable using the order and finite-degree predicates are also
definable with the order predicate only. From this result we derive the
separation of the alternation hierarchy of two-variable logic on this
signature
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
The Church Synthesis Problem with Parameters
For a two-variable formula ψ(X,Y) of Monadic Logic of Order (MLO) the
Church Synthesis Problem concerns the existence and construction of an operator
Y=F(X) such that ψ(X,F(X)) is universally valid over Nat.
B\"{u}chi and Landweber proved that the Church synthesis problem is
decidable; moreover, they showed that if there is an operator F that solves the
Church Synthesis Problem, then it can also be solved by an operator defined by
a finite state automaton or equivalently by an MLO formula. We investigate a
parameterized version of the Church synthesis problem. In this version ψ
might contain as a parameter a unary predicate P. We show that the Church
synthesis problem for P is computable if and only if the monadic theory of
is decidable. We prove that the B\"{u}chi-Landweber theorem can be
extended only to ultimately periodic parameters. However, the MLO-definability
part of the B\"{u}chi-Landweber theorem holds for the parameterized version of
the Church synthesis problem
The Church Problem for Countable Ordinals
A fundamental theorem of Buchi and Landweber shows that the Church synthesis
problem is computable. Buchi and Landweber reduced the Church Problem to
problems about ω-games and used the determinacy of such games as one of
the main tools to show its computability. We consider a natural generalization
of the Church problem to countable ordinals and investigate games of arbitrary
countable length. We prove that determinacy and decidability parts of the
Bu}chi and Landweber theorem hold for all countable ordinals and that its full
extension holds for all ordinals < \omega\^\omega
Logic Meets Algebra: the Case of Regular Languages
The study of finite automata and regular languages is a privileged meeting
point of algebra and logic. Since the work of Buchi, regular languages have
been classified according to their descriptive complexity, i.e. the type of
logical formalism required to define them. The algebraic point of view on
automata is an essential complement of this classification: by providing
alternative, algebraic characterizations for the classes, it often yields the
only opportunity for the design of algorithms that decide expressibility in
some logical fragment.
We survey the existing results relating the expressibility of regular
languages in logical fragments of MSO[S] with algebraic properties of their
minimal automata. In particular, we show that many of the best known results in
this area share the same underlying mechanics and rely on a very strong
relation between logical substitutions and block-products of pseudovarieties of
monoid. We also explain the impact of these connections on circuit complexity
theory.Comment: 37 page
Non-Definability Results for Randomised First-Order Logic
We investigate the expressive power of randomised first-order logic
(BPFO) on restricted classes of structures. While BPFO is
stronger than FO in general, even on structures with a built-in
addition relation, we show that BPFO is not stronger than FO
on structures with a unary vocabulary, nor on the class of
equivalence relations. The same techniques can be applied to show
that evenness of a linear order, and therefore graph connectivity,
can not be defined in BPFO. Finally, we show that there is an
FO[<]-definable query on word structures which can not be
defined in BPFO[+1]
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