scholarly journals Well-definedness and observational equivalence for inductive–coinductive programs

2019 ◽  
Vol 29 (4) ◽  
pp. 419-468
Author(s):  
Henning Basold ◽  
Helle Hvid Hansen
Keyword(s):  
Fixed Point ◽  
Modal Logic ◽  
The Other ◽  
Observational Equivalence ◽  
Other Hand ◽  
Inductive Types ◽  
Point Types ◽  
Usual Topology

Abstract We define notions of well-definedness and observational equivalence for programs of mixed inductive and coinductive types. These notions are defined by means of tests formulas which combine structural congruence for inductive types and modal logic for coinductive types. Tests also correspond to certain evaluation contexts. We define a program to be well-defined if it is strongly normalizing under all tests, and two programs are observationally equivalent if they satisfy the same tests. We show that observational equivalence is sufficiently coarse to ensure that least and greatest fixed point types are initial algebras and final coalgebras, respectively. This yields inductive and coinductive proof principles for reasoning about program behaviour. On the other hand, we argue that observational equivalence does not identify too many terms, by showing that tests induce a topology that, on streams, coincides with usual topology induced by the prefix metric. As one would expect, observational equivalence is, in general, undecidable, but in order to develop some practically useful heuristics we provide coinductive techniques for establishing observational normalization and observational equivalence, along with up-to techniques for enhancing these methods.

scholarly journals Consolidation via Tacit Culpability Measures: Between Explicit and Implicit Degrees of Culpability

2021 ◽  
Author(s):  
Jandson S. Ribeiro ◽  
Matthias Thimm
Keyword(s):  
Fixed Point ◽  
Knowledge Base ◽  
Special Class ◽  
Belief Change ◽  
Minimal Change ◽  
The Other ◽  
Preference Relations ◽  
Other Hand ◽  
The One ◽  
Rationality Postulates

Restoring consistency of a knowledge base, known as consolidation, should preserve as much information as possible of the original knowledge base. On the one hand, the field of belief change captures this principle of minimal change via rationality postulates. On the other hand, within the field of inconsistency measurement, culpability measures have been developed to assess how much a formula participates in making a knowledge base inconsistent. We look at culpability measures as a tool to disclose epistemic preference relations and build rational consolidation functions. We introduce tacit culpability measures that consider semantic counterparts between conflicting formulae, and we define a special class of these culpability measures based on a fixed-point characterisation: the stable tacit culpability measures. We show that the stable tacit culpability measures yield rational consolidation functions and that these are also the only culpability measures that yield rational consolidation functions.

Amalgamation in relation algebras

1998 ◽  
Vol 63 (2) ◽  
pp. 479-484 ◽  
Author(s):  
Maarten Marx
Keyword(s):  
Modal Logic ◽  
Equational Theory ◽  
Weak Form ◽  
Boolean Algebras ◽  
The Other ◽  
Relation Algebras ◽  
Boolean Algebras With Operators ◽  
Other Hand ◽  
Arrow Logic ◽  
Representable Relation

We investigate amalgamation properties of relational type algebras. Besides purely algebraic interest, amalgamation in a class of algebras is important because it leads to interpolation results for the logic corresponding to that class (cf. [15]). The multi-modal logic corresponding to relational type algebras became known under the name of “arrow logic” (cf. [18, 17]), and has been studied rather extensively lately (cf. [10]). Our research was inspired by the following result of Andréka et al. [1].Let K be a class of relational type algebras such that(i) composition is associative,(ii) K is a class of boolean algebras with operators, and(iii) K contains the representable relation algebras RRA.Then the equational theory of K is undecidable.On the other hand, there are several classes of relational type algebras (e.g., NA, WA denned below) whose equational (even universal) theories are decidable (cf. [13]). Composition is not associative in these classes. Theorem 5 indicates that also with respect to amalgamation (a very weak form of) associativity forms a borderline. We first recall the relevant definitions.

scholarly journals Sturmian Sequences and Invertible Substitutions

2011 ◽  
Vol Vol. 13 no. 2 (Combinatorics) ◽  
Author(s):  
Li Peng ◽  
Bo Tan
Keyword(s):  
Fixed Point ◽  
The Other ◽  
Sturmian Sequence ◽  
Other Hand ◽  
Angle Alpha ◽  
International Audience ◽  
Sturmian Sequences ◽  
Irrational Angle

Combinatorics International audience It is known that a Sturmian sequence S can be defined as a coding of the orbit of rho (called the intercept of S) under a rotation of irrational angle alpha (called the slope). On the other hand, a fixed point of an invertible substitution is Sturmian. Naturally, there are two interrelated questions: (1) Given an invertible substitution, we know that its fixed point is Sturmian. What is the slope and intercept? (2) Which kind of Sturmian sequences can be fixed by certain non-trivial invertible substitutions? In this paper we give a unified treatment to the two questions. We remark that though the results are known, our proof is very elementary and concise.

Some Random Fixed Point Theorems and Comparing Random Operator Equations

2011 ◽  
Vol 52-54 ◽  
pp. 127-132
Author(s):  
Ning Chen ◽  
Bao Dan Tian ◽  
Ji Qian Chen
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorems ◽  
Random Operator ◽  
The Other ◽  
Operator Equations ◽  
Random Solution ◽  
Random Fixed Point ◽  
Other Hand ◽  
The Common

In this paper, some new results are given for the common random solution for a class of random operator equations which generalize several results in [4], [5] and [6] in Banach space. On the other hand, Altman’s inequality is also extending into the type of the determinant form. And comparing some solution for several examples, main results are theorem 2.3, theorem 3.3-3.4, theorem 4.1 and theorem 4.3.

The problem of interpreting modal logic

1947 ◽  
Vol 12 (2) ◽  
pp. 43-48 ◽  
Author(s):  
W. V. Quine
Keyword(s):  
Modal Logic ◽  
The Other ◽  
Material Objects ◽  
Quantification Theory ◽  
Other Hand

There are logicians, myself among them, to whom the ideas of modal logic (e. g. Lewis's) are not intuitively clear until explained in non-modal terms. But so long as modal logic stops short of quantification theory, it is possible (as I shall indicate in §2) to provide somewhat the type of explanation desired. When modal logic is extended (as by Miss Barcan1) to include quantification theory, on the other hand, serious obstarles to interpretation are encountered—particularly if one cares to avoid a curiously idealistic ontology which repudiates material objects. Such are the matters which it is the purpose of the present paper to set forth.

scholarly journals The non-SUSY AdS6 and AdS7 fixed points are brane-jet unstable

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Minwoo Suh
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Decay Channel ◽  
Scalar Potential ◽  
The Other ◽  
Other Hand ◽  
Weak Gravity

Abstract In six- and seven-dimensional gauged supergravity, each scalar potential has one supersymmetric and one non-supersymmetric fixed points. The non-supersymmetric AdS7 fixed point is perturbatively unstable. On the other hand, the non-supersymmetric AdS6 fixed point is known to be perturbatively stable. In this note we examine the newly proposed non-perturbative decay channel, called brane-jet instabilities of the AdS6 and AdS7 vacua. We find that when they are uplifted to massive type IIA and eleven- dimensional supergravity, respectively, the non-supersymmetric AdS6 and AdS7 vacua are both brane-jet unstable, in fond of the weak gravity conjecture.

On axiomatising products of Kripke frames

2000 ◽  
Vol 65 (2) ◽  
pp. 923-945 ◽  
Author(s):  
Ágnes Kurucz
Keyword(s):  
Modal Logic ◽  
The Other ◽  
Modal Language ◽  
First Order ◽  
Kripke Frames ◽  
Other Hand ◽  
The Many ◽  
By Products

AbstractIt is shown that the many-dimensional modal logic Kn, determined by products of n-many Kripke frames, is not finitely axiomatisable in the n-modal language, for any n > 2. On the other hand, Kn is determined by a class of frames satisfying a single first-order sentence.

Infinitary combinatorics and modal logic

1990 ◽  
Vol 55 (2) ◽  
pp. 761-778 ◽  
Author(s):  
Andreas Blass
Keyword(s):  
Modal Logic ◽  
Propositional Logic ◽  
The Other ◽  
Ordinal Numbers ◽  
Mahlo Cardinal ◽  
Other Hand ◽  
Modal Propositional Logic

AbstractWe show that the modal propositional logic G, originally introduced to describe the modality “it is provable that”, is also sound for various interpretations using filters on ordinal numbers, for example the end-segment filters, the club filters, or the ineffable filters. We also prove that G is complete for the interpretation using end-segment filters. In the case of club filters, we show that G is complete if Jensen's principle □κ holds for all κ < ℵω; on the other hand, it is consistent relative to a Mahlo cardinal that G be incomplete for the club filter interpretation.

scholarly journals On the Costs of Classical Logic

Erkenntnis ◽  
2021 ◽  
Author(s):  
Luca Castaldo
Keyword(s):  
Fixed Point ◽  
Classical Logic ◽  
The Other ◽  
Theories Of Truth ◽  
Other Hand ◽  
Fixed Point Semantics ◽  
Theoretical Equivalence ◽  
The One

AbstractThis article compares classical (or -like) and nonclassical (or -like) axiomatisations of the fixed-point semantics developed by Kripke (J Philos 72(19): 690–716, 1975). Following the line of investigation of Halbach and Nicolai (J Philos Logic 47(2): 227–257, 2018), we do not compare and qua theories of truth simpliciter, but rather qua axiomatisations of the Kripkean conception of truth. We strengthen the central results of Halbach and Nicolai (2018) and Nicolai (Stud Log 106(1): 101–130, 2018), showing that, on the one hand, there is a stronger sense in which some variants of and some variants of can be taken to be, truth-theoretically, equivalent. On the other hand, we show that this truth-theoretical equivalence is not preserved by some other variants of and , arguing that the variants are more adequate axiomatisations of the fixed-point semantics than the corresponding variants.

On Saint Venant’s Principle: Elastic Shells and Plates

1960 ◽  
Vol 27 (3) ◽  
pp. 417-422 ◽  
Author(s):  
P. M. Naghdi
Keyword(s):  
Fixed Point ◽  
Integral Formula ◽  
Middle Surface ◽  
The Other ◽  
Elastic Shells ◽  
Partial Derivatives ◽  
Other Hand ◽  
Derivatives Of ◽  
Surface Loads

This investigation is concerned with an examination of the validity of Saint Venant’s principle in the theory of thin elastic shells and plates. With the aid of an integral formula derived for the displacements and their relevant partial derivatives of all orders at a fixed point of the shell middle surface, the conclusions reached may be roughly stated as follows: If the loads acting on the shell maintained in equilibrium are purely edge loads, then the orders of magnitude of the displacements and stresses are in accord with the traditional statement of Saint Venant’s principle. On the other hand, if the loads on the shell are purely surface loads, then the conclusions concerning the orders of magnitude of the displacements and stresses are the same as those of the modified Saint Venant principle.

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