scholarly journals Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences

10.37236/5752 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Luke Schaeffer ◽  
Jeffrey Shallit
Keyword(s):  
Period Doubling ◽  
Fibonacci Sequence ◽  
Order Logic ◽  
First Order Logic ◽  
Characteristic Functions ◽  
First Order ◽  
Automatic Sequences ◽  
Automatic Sequence ◽  
Doubling Sequence

We prove that the property of being closed (resp., palindromic, rich, privileged trapezoidal, balanced) is expressible in first-order logic for automatic (and some related) sequences. It therefore follows that the characteristic function of those $n$ for which an automatic sequence $\bf x$ has a closed (resp., palindromic, privileged, rich, trapezoidal, balanced) factor of length  $n$ is itself automatic. For privileged words this requires a new characterization of the privileged property. We compute the corresponding characteristic functions for various famous sequences, such as the Thue-Morse sequence, the Rudin-Shapiro sequence, the ordinary  paperfolding sequence, the period-doubling sequence, and the Fibonacci sequence. Finally, we also show that the function counting the total number of palindromic factors in the prefix of length $n$ of a $k$-automatic sequence is not $k$-synchronized.

SKEPTICAL ABDUCTION

1993 ◽  
Vol 02 (04) ◽  
pp. 511-540 ◽  
Author(s):  
P. MARQUIS
Keyword(s):  
Propositional Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Real Problem ◽  
Logical Framework ◽  
First Order ◽  
Selective Generation ◽  
The Given ◽  
Reasonable Prospect

Abduction is the process of generating the best explanation as to why a fact is observed given what is already known. A real problem in this area is the selective generation of hypotheses that have some reasonable prospect of being valid. In this paper, we propose the notion of skeptical abduction as a model to face this problem. Intuitively, the hypotheses pointed out by skeptical abduction are all the explanations that are consistent with the given knowledge and that are minimal, i.e. not unnecessarily general. Our contribution is twofold. First, we present a formal characterization of skeptical abduction in a logical framework. On this ground, we address the problem of mechanizing skeptical abduction. A new method to compute minimal and consistent hypotheses in propositional logic is put forward. The extent to which skeptical abduction can be mechanized in first—order logic is also investigated. In particular, two classes of first-order formulas in which skeptical abduction is effective are provided. As an illustration, we finally sketch how our notion of skeptical abduction applies as a theoretical tool to some artificial intelligence problems (e.g. diagnosis, machine learning).

Towards a characterization of order-invariant queries over tame graphs

2009 ◽  
Vol 74 (1) ◽  
pp. 168-186 ◽  
Author(s):  
Michael Benedikt ◽  
Luc Segoufin
Keyword(s):  
Linear Order ◽  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
Finite Graphs ◽  
Bounded Treewidth ◽  
First Order ◽  
A New Technique ◽  
Order Invariant

AbstractThis work deals with the expressive power of logics on finite graphs with access to an additional “arbitrary” linear order. The queries that can be expressed this way are the order-invariant queries for the logic. For the standard logics used in computer science, such as first-order logic, it is known that access to an arbitrary linear order increases the expressiveness of the logic. However, when we look at the separating examples, we find that they have satisfying models whose Gaifman Graph is complex – unbounded in valence and in treewidth. We thus explore the expressiveness of order-invariant queries over well-behaved graphs. We prove that first-order order-invariant queries over strings and trees have no additional expressiveness over first-order logic in the original signature. We also prove new upper bounds on order-invariant queries over bounded treewidth and bounded valence graphs. Our results make use of a new technique of independent interest: the application of algebraic characterizations of definability to show collapse results.

Fraïssé–Hintikka theorem in institutions

2020 ◽  
Vol 30 (7) ◽  
pp. 1377-1399
Author(s):  
Daniel Găină ◽  
Tomasz Kowalski
Keyword(s):  
Classical Case ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Specification Languages ◽  
Higher Order Logic ◽  
Elementary Equivalence ◽  
First Order

Abstract We generalize the characterization of elementary equivalence by Ehrenfeucht–Fraïssé games to arbitrary institutions whose sentences are finitary. These include many-sorted first-order logic, higher-order logic with types, as well as a number of other logics arising in connection to specification languages. The gain for the classical case is that the characterization is proved directly for all signatures, including infinite ones.

scholarly journals A Modal Characterization Theorem for a Probabilistic Fuzzy Description Logic

2019 ◽  
Author(s):  
Paul Wild ◽  
Lutz Schröder ◽  
Dirk Pattinson ◽  
Barbara König
Keyword(s):  
Description Logic ◽  
Characterization Theorem ◽  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Type Spaces ◽  
Suitable Notion ◽  
Fuzzy Description

The fuzzy modality probably is interpreted over probabilistic type spaces by taking expected truth values. The arising probabilistic fuzzy description logic is invariant under probabilistic bisimilarity; more informatively, it is non-expansive wrt. a suitable notion of behavioural distance. In the present paper, we provide a characterization of the expressive power of this logic based on this observation: We prove a probabilistic analogue of the classical van Benthem theorem, which states that modal logic is precisely the bisimulation-invariant fragment of first-order logic. Specifically, we show that every formula in probabilistic fuzzy first-order logic that is non-expansive wrt. behavioural distance can be approximated by concepts of bounded rank in probabilistic fuzzy description logic.

A Logical Characterization of Small 2NFAs

2017 ◽  
Vol 28 (05) ◽  
pp. 445-464
Author(s):  
Christos A. Kapoutsis ◽  
Lamana Mulaffer
Keyword(s):  
Transitive Closure ◽  
Closure Operator ◽  
Finite Automata ◽  
Disjunctive Normal Form ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Polynomial Size ◽  
Fixed Input

Let 2N be the class of families of problems solvable by families of two-way nondeterministic finite automata of polynomial size. We characterize 2N in terms of families of formulas of transitive-closure logic. These formulas apply the transitive-closure operator on a quantifier-free disjunctive normal form of first-order logic with successor and constants, where (i) apart from two special variables, all others are equated to constants in every clause, and (ii) no clause simultaneously relates these two special variables and refers to fixed input cells. We prove that automata with polynomially many states are as powerful as formulas with polynomially many clauses and polynomially large constants. This can be seen as a refinement of Immerman’s theorem that nondeterministic logarithmic space matches positive transitive-closure logic ([Formula: see text]).

scholarly journals CONTINUOUS FIRST ORDER LOGIC FOR UNBOUNDED METRIC STRUCTURES

2008 ◽  
Vol 08 (02) ◽  
pp. 197-223 ◽  
Author(s):  
ITAÏ BEN YAACOV
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Single Point ◽  
Order Logic ◽  
First Order Logic ◽  
Space Structures ◽  
First Order ◽  
Metric Structures ◽  
The Common

We present an adaptation of continuous first order logic to unbounded metric structures. This has the advantage of being closer in spirit to C. Ward Henson's logic for Banach space structures than the unit ball approach (which has been the common approach so far to Banach space structures in continuous logic), as well as of applying in situations where the unit ball approach does not apply (i.e. when the unit ball is not a definable set). We also introduce the process of single point emboundment (closely related to the topological single point compactification), allowing to bring unbounded structures back into the setting of bounded continuous first order logic. Together with results from [4] regarding perturbations of bounded metric structures, we prove a Ryll–Nardzewski style characterization of theories of Banach spaces which are separably categorical up to small perturbation of the norm. This last result is motivated by an unpublished result of Henson.

First-Order Logic Reasoning Support for the Semantic Web

2009 ◽  
Vol 19 (12) ◽  
pp. 3091-3099 ◽  
Author(s):  
Gui-Hong XU ◽  
Jian ZHANG
Keyword(s):  
Semantic Web ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Logic Reasoning

Boolean-valued structures

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Rules ◽  
Mathematical Concepts ◽  
Semantic Framework ◽  
First Order ◽  
Standard Models ◽  
Boolean Valued Model ◽  
Semantic Values

Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin down the semantic frameworks of language. It transpires that there are not just non-standard models, but non-standard ways of doing model theory itself. In more detail: whilst we normally outline a two-valued semantics which makes sentences True or False in a model, the inference rules for first-order logic are compatible with a four-valued semantics; or a semantics with countably many values; or what-have-you. The appropriate level of generality here is that of a Boolean-valued model, which we introduce. And the plurality of possible semantic values gives rise to perhaps the ‘deepest’ level of indeterminacy questions: How can humans pin down the semantic framework for their languages? We consider three different ways for inferentialists to respond to this question.

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