876 research outputs found
Wreath Products of Forest Algebras, with Applications to Tree Logics
We use the recently developed theory of forest algebras to find algebraic
characterizations of the languages of unranked trees and forests definable in
various logics. These include the temporal logics CTL and EF, and first-order
logic over the ancestor relation. While the characterizations are in general
non-effective, we are able to use them to formulate necessary conditions for
definability and provide new proofs that a number of languages are not
definable in these logics
?-Forest Algebras and Temporal Logics
We use the algebraic framework for languages of infinite trees introduced in [A. Blumensath, 2020] to derive effective characterisations of various temporal logics, in particular the logic EF (a fragment of CTL) and its counting variant cEF
Tree Languages Defined in First-Order Logic with One Quantifier Alternation
We study tree languages that can be defined in \Delta_2 . These are tree
languages definable by a first-order formula whose quantifier prefix is forall
exists, and simultaneously by a first-order formula whose quantifier prefix is
. For the quantifier free part we consider two signatures, either the
descendant relation alone or together with the lexicographical order relation
on nodes. We provide an effective characterization of tree and forest languages
definable in \Delta_2 . This characterization is in terms of algebraic
equations. Over words, the class of word languages definable in \Delta_2 forms
a robust class, which was given an effective algebraic characterization by Pin
and Weil
A Characterization for Decidable Separability by Piecewise Testable Languages
The separability problem for word languages of a class by
languages of a class asks, for two given languages and
from , whether there exists a language from that
includes and excludes , that is, and . In this work, we assume some mild closure properties for
and study for which such classes separability by a piecewise
testable language (PTL) is decidable. We characterize these classes in terms of
decidability of (two variants of) an unboundedness problem. From this, we
deduce that separability by PTL is decidable for a number of language classes,
such as the context-free languages and languages of labeled vector addition
systems. Furthermore, it follows that separability by PTL is decidable if and
only if one can compute for any language of the class its downward closure wrt.
the scattered substring ordering (i.e., if the set of scattered substrings of
any language of the class is effectively regular).
The obtained decidability results contrast some undecidability results. In
fact, for all (non-regular) language classes that we present as examples with
decidable separability, it is undecidable whether a given language is a PTL
itself.
Our characterization involves a result of independent interest, which states
that for any kind of languages and , non-separability by PTL is
equivalent to the existence of common patterns in and
Logic and Automata
Mathematical logic and automata theory are two scientific disciplines with a fundamentally close relationship. The authors of Logic and Automata take the occasion of the sixtieth birthday of Wolfgang Thomas to present a tour d'horizon of automata theory and logic. The twenty papers in this volume cover many different facets of logic and automata theory, emphasizing the connections to other disciplines such as games, algorithms, and semigroup theory, as well as discussing current challenges in the field
Logics for Unranked Trees: An Overview
Labeled unranked trees are used as a model of XML documents, and logical
languages for them have been studied actively over the past several years. Such
logics have different purposes: some are better suited for extracting data,
some for expressing navigational properties, and some make it easy to relate
complex properties of trees to the existence of tree automata for those
properties. Furthermore, logics differ significantly in their model-checking
properties, their automata models, and their behavior on ordered and unordered
trees. In this paper we present a survey of logics for unranked trees
Recognisable languages over monads
The principle behind algebraic language theory for various kinds of
structures, such as words or trees, is to use a compositional function from the
structures into a finite set. To talk about compositionality, one needs some
way of composing structures into bigger structures. It so happens that category
theory has an abstract concept for this, namely a monad. The goal of this paper
is to propose monads as a unifying framework for discussing existing algebras
and designing new algebras
!-Logic: First-order reasoning for families of non-commutative string diagrams
Equational reasoning with string diagrams provides an intuitive method for proving equations between morphisms in various forms of monoidal category. !-Graphs were introduced with the intention of reasoning with infinite families of string diagrams by allowing repetition of sub-diagrams. However, their combinatoric nature only allows commutative nodes. The aim of this thesis is to extend the !-graph formalism to remove the restriction of commutativity and replace the notion of equational reasoning with a natural deduction system based on first order logic. The first major contribution is the syntactic !-tensor formalism, which enriches Penrose’s abstract tensor notation to allow repeated structure via !-boxes. This will allow us to work with many noncommutative theories such as bialgebras, Frobenius algebras, and Hopf algebras, which have applications in quantum information theory. A more subtle consequence of switching to !-tensors is the ability to definitionally extend a theory. We will demonstrate how noncommutativity allows us to define nodes which encapsulate entire diagrams, without inherently assuming the diagram is commutative. This is particularly useful for recursively defining arbitrary arity nodes from fixed arity nodes. For example, we can construct a !-tensor node representing the family of left associated trees of multiplications in a monoid. The ability to recursively define nodes goes hand in hand with proof by induction. This leads to the second major contribution of this thesis, which is !-Logic (!L). We extend previous attempts at equational reasoning to a fully fledged natural deduction system based on positive intuitionistic first order logic, with conjunction, implication, and universal quantification over !-boxes. The key component of !L is the principle of !-box induction. We demonstrate its application by proving how we can transition from fixed to arbitrary arity theories for monoids, antihomomorphisms, bialgebras, and various forms of Frobenius algebras. We also define a semantics for !L, which we use to prove its soundness. Finally, we reintroduce commutativity as an optional property of a morphism, along with another property called symmetry, which describes morphisms which are not affected by cyclic permutations of their edges. Implementing these notions in the !-tensor language allows us to more easily describe theories involving symmetric or commutative morphisms, which we then demonstrate for recursively defined Frobenius algebra nodes
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