Generalizing generalized tries

2000 ◽  
Vol 10 (4) ◽  
pp. 327-351 ◽  
Author(s):  
RALF HINZE
Keyword(s):  
Type System ◽  
Search Tree ◽  
Second Order ◽  
First Order ◽  
Programming Paradigm ◽  
Arbitrary Signature ◽  
One Step ◽  
Rank 2 ◽  
Definition Of

A trie is a search tree scheme that employs the structure of search keys to organize information. Tries were originally devised as a means to represent a collection of records indexed by strings over a fixed alphabet. Based on work by C. P. Wadsworth and others, R. H. Connelly and F. L. Morris generalized the concept to permit indexing by elements built according to an arbitrary signature. Here we go one step further, and define tries and operations on tries generically for arbitrary datatypes of first-order kind, including parameterized and nested datatypes. The derivation employs techniques recently developed in the context of polytypic programming and can be regarded as a comprehensive case study in this new programming paradigm. It is well known that for the implementation of generalized tries, nested datatypes and polymorphic recursion are needed. Implementing tries for first-order kinded datatypes places even greater demands on the type system: it requires rank-2 type signatures and second-order nested datatypes. Despite these requirements, the definition of tries is surprisingly simple, which is mostly due to the framework of polytypic programming.

scholarly journals Macroscopic and Multi-Scale Models for Multi-Class Vehicular Dynamics with Uneven Space Occupancy: A Case Study

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 102
Author(s):  
Maya Briani ◽  
Emiliano Cristiani ◽  
Paolo Ranut
Keyword(s):  
Road Network ◽  
Real Data ◽  
Second Order ◽  
Heavy Vehicles ◽  
First Order ◽  
Multi Scale ◽  
Stop And Go ◽  
Scale Models ◽  
Road Space

In this paper, we propose two models describing the dynamics of heavy and light vehicles on a road network, taking into account the interactions between the two classes. The models are tailored for two-lane highways where heavy vehicles cannot overtake. This means that heavy vehicles cannot saturate the whole road space, while light vehicles can. In these conditions, the creeping phenomenon can appear, i.e., one class of vehicles can proceed even if the other class has reached the maximal density. The first model we propose couples two first-order macroscopic LWR models, while the second model couples a second-order microscopic follow-the-leader model with a first-order macroscopic LWR model. Numerical results show that both models are able to catch some second-order (inertial) phenomena such as stop and go waves. Models are calibrated by means of real data measured by fixed sensors placed along the A4 Italian highway Trieste–Venice and its branches, provided by Autovie Venete S.p.A.

One step up and two steps back? The Italian debate on secularization, heteronormativity and LGBTQ citizenship

2018 ◽  
Vol 65 (5) ◽  
pp. 591-607 ◽  
Author(s):  
Elisa Bellè ◽  
Caterina Peroni ◽  
Elisa Rapetti
Keyword(s):  
Critical Discourse Analysis ◽  
Political Conflict ◽  
Interesting Case ◽  
Critical Discourse ◽  
Political Events ◽  
The Family ◽  
One Step ◽  
Definition Of ◽  
Shed Light

The aim of this article is to furnish insights of the Italian public debate on the recognition of LGBTQ rights, which can be understood as an interesting case study of the complex relationship between (multi)secularisation processes and re/definition of citizenship models. More specifically, the article analyses two political events related to this debate that took place in Rome in June 2015. The first is the Family Day demonstration, promoted by conservative Catholic groups; the second is the LGBTQ Pride parade, promoted by various gay, lesbian and transsexual/gender associations. We analyse the official statements issued by the two organising committees of the demonstrations, adopting the framework and methods of the Critical Discourse Analysis. Above and beyond an evident political conflict between the two discourses, we try to shed light on their mutual construction on the basis of what we call ‘naturalization’ and ‘universalization’ processes.

Barack Obama, being sharp

Gesture ◽  
2011 ◽  
Vol 11 (3) ◽  
pp. 241-270 ◽  
Author(s):  
Michael Lempert
Keyword(s):  
Information Structure ◽  
Precision Grip ◽  
Barack Obama ◽  
Second Order ◽  
First Order ◽  
Political Oratory ◽  
The Way

Gesture in political oratory and debate is renowned for its nonreferential indexical functions, for the way it purportedly can indicate qualities of speaker and materialize acts of persuasion — functions famously addressed in Quintilian’s classic writings but understudied today. I revisit this problematic through a case study of precision-grip (especially thumb to tip of forefinger) in Barack Obama’s debate performances (2004–2008). Cospeech gesture can index valorized attributes of speaker — not directly but through orders of semiotic motivation. In terms of first-order indexicality, precision-grip highlights discourse in respect of information structure, indicating focus. In debate, precision grip has undergone a degree of conventionalization and has reemerged as a second-order pragmatic resource for performatively “making a ‘sharp’, effective point.” Repetitions and parallelisms of precision grip in debate can, in turn, exhibit speaker-attributes, such as being argumentatively ‘sharp’, and from there may even partake in candidate branding.

Development of the female flower and gynecandrous partial inflorescence of Myrica californica

1979 ◽  
Vol 57 (2) ◽  
pp. 141-151 ◽  
Author(s):  
Alastair D. Macdonald
Keyword(s):  
Fourth Order ◽  
Female Flower ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
Unique Structure ◽  
Inflorescence Axis ◽  
Definition Of ◽  
Fifth Order ◽  
Partial Inflorescence

Organogenesis of the female flower and gynecandrous partial inflorescence is described. Approximately 25 first-order inflorescence bracts are formed in an acropetal sequence. A second-order inflorescence axis, the partial inflorescence, develops in the axil of each bract. Third-, fourth-, and fifth-order axes arise in the axils of second-, third-, and fourth-order bracts. A gynoecium terminates a second-order axis and sometimes a distal third-order axis. A gynoecium consists of two stigmas and one basal, unitegmic, orthotropous ovule. The wall enclosing the ovule, the circumlocular wall, is comprised distally of gynoecial tissue and proximally of tissue of the inflorescence axis and its appendages. The latter portion of the wall is formed by zonal growth. Androecial members, formed proximal to the gynoecium on the partial inflorescence, are carried onto the circumlocular wall by zonal growth. A stamen may develop from the last-formed primordium before gynoecial inception or from a potentially stigmatic primordium. The papillae of the flower and fruit arise as emergences and from potentially bracteate, axial, and staminate primorida during the development of the circumlocular wall. The term circumlocular wall is used in a neutral sense to describe this unique structure. Since the gynoecium is composed of gynoecial appendages and inflorescence axis and appendages, a functional definition of gynoecium must be expanded to include any tissue, including an inflorescence, that surrounds the ovule(s) and forms the fruit(s).

A theorem on deducibility for second-order functions

1939 ◽  
Vol 4 (2) ◽  
pp. 77-79 ◽  
Author(s):  
C. H. Langford
Keyword(s):  
Second Order ◽  
Order Function ◽  
Usual Definition ◽  
First Order ◽  
Free Variables ◽  
Usual Convention ◽  
Definition Of ◽  
Image Position

It is known that the usual definition of a dense series without extreme elements is complete with respect to first-order functions, in the sense that any first-order function on the base of a set of postulates defining such a series either is implied by the postulates or is inconsistent with them. It is here understood, in accordance with the usual convention, that when we speak of a function on the base , the function shall be such as to place restrictions only upon elements belonging to the class determined by f; or, more exactly, every variable with a universal prefix shall occur under the hypothesis that its values satisfy f, while every variable with an existential prefix shall have this condition categorically imposed upon it.Consider a set of postulates defining a dense series without extreme elements, and add to this set the condition of Dedekind section, to be formulated as follows. Let the conjunction of the three functions,be written H(ϕ), where the free variables f and g, being parameters throughout, are suppressed. This is the hypothesis of Dedekind's condition, and the conclusion iswhich may be written C(ϕ).

Forcing and reducibilities. III. Forcing in fragments of set theory

1983 ◽  
Vol 48 (4) ◽  
pp. 1013-1034
Author(s):  
Piergiorgio Odifreddi
Keyword(s):  
Set Theory ◽  
Recursion Theory ◽  
Second Order ◽  
Analytic Hierarchy ◽  
Second Order Arithmetic ◽  
First Order ◽  
Order Language ◽  
Order Arithmetic ◽  
Definition Of ◽  
Generic Set

We conclude here the treatment of forcing in recursion theory begun in Part I and continued in Part II of [31]. The numbering of sections is the continuation of the numbering of the first two parts. The bibliography is independent.In Part I our language was a first-order language: the only set we considered was the (set constant for the) generic set. In Part II a second-order language was introduced, and we had to interpret the second-order variables in some way. What we did was to consider the ramified analytic hierarchy, defined by induction as:A0 = {X ⊆ ω: X is arithmetic},Aα+1 = {X ⊆ ω: X is definable (in 2nd order arithmetic) over Aα},Aλ = ⋃α<λAα (λ limit),RA = ⋃αAα.We then used (a relativized version of) the fact that (Kleene [27]). The definition of RA is obviously modeled on the definition of the constructible hierarchy introduced by Gödel [14]. For this we no longer work in a language for second-order arithmetic, but in a language for (first-order) set theory with membership as the only nonlogical relation:L0 = ⊘,Lα+1 = {X: X is (first-order) definable over Lα},Lλ = ⋃α<λLα (λ limit),L = ⋃αLα.

A model theoretic approach to Malcev conditions

1977 ◽  
Vol 42 (2) ◽  
pp. 277-288 ◽  
Author(s):  
John T. Baldwin ◽  
Joel Berman
Keyword(s):  
Extensive Study ◽  
Second Order ◽  
Theoretic Approach ◽  
Easy Consequence ◽  
First Order ◽  
Congruence Relations ◽  
Individual Variables ◽  
Definition Of ◽  
The Individual ◽  
Necessary And Sufficient

A varietyV(equational class of algebras) satisfies a strong Malcev condition ∃f1,…, ∃fnθ(f1, …,fn,x1, …,xm) where θ is a conjunction of equations in the function variablesf1, …,fnand the individual variablesx1, …,xm, if there are polynomial symbolsp1, …,pnin the language ofVsuch that ∀x1, …,xmθ(p1…,pn,x1, …,xm) is a law ofV. Thus a strong Malcev condition involves restricted second order quantification of a strange sort. The quantification is restricted to functions which are “polynomially definable”. This notion was introduced by Malcev [6] who used it to describe those varieties all of whose members have permutable congruence relations. The general formal definition of Malcev conditions is due to Grätzer [1]. Since then and especially since Jónsson's [3] characterization of varieties with distributive congruences there has been extensive study of strong Malcev conditions and the related concepts: Malcev conditions and weak Malcev conditions.In [9], Taylor gives necessary and sufficient semantic conditions for a class of varieties to be defined by a (strong) Malcev condition. A key to the proof is the translation of the restricted second order concepts into first order concepts in a certain many sorted language. In this paper we show that, given this translation, Taylor's theorem is an easy consequence of a result of Tarski [8] and the standard preservation theorems of first order logic.

ON THE CONSERVATIVITY OF LEIBNIZ EQUALITY

1998 ◽  
Vol 09 (04) ◽  
pp. 431-454
Author(s):  
M. P. A. SELLINK
Keyword(s):  
Type System ◽  
Type Systems ◽  
Order Theory ◽  
Second Order ◽  
First Order ◽  
Pure Type ◽  
First Order Theory ◽  
Pure Type Systems ◽  
Leibniz Equality

We embed a first order theory with equality in the Pure Type System λMON2 that is a subsystem of the well-known type system λPRED2. The embedding is based on the Curry-Howard isomorphism, i.e. → and ∀ coincide with → and Π. Formulas of the form [Formula: see text] are treated as Leibniz equalities. That is, [Formula: see text] is identified with the second order formula ∀ P. P(t1)→ P(t2), which contains only →'s and ∀'s and can hence be embedded straightforwardly. We give a syntactic proof — based on enriching typed λ-calculus with extra reduction steps — for the equivalence between derivability in the logic and inhabitance in λMNO2. Familiarity with Pure Type Systems is assumed.

Type assignment and termination of interaction nets

1998 ◽  
Vol 8 (6) ◽  
pp. 593-636 ◽  
Author(s):  
MARIBEL FERNÁNDEZ
Keyword(s):  
Type System ◽  
Term Rewriting ◽  
Term Rewriting Systems ◽  
Rewriting Systems ◽  
Programming Paradigm ◽  
Type Assignment ◽  
Computational Aspects ◽  
Rank 2 ◽  
Intersection Types ◽  
Simple Interaction

Interaction nets have proved to be a useful tool for the study of computational aspects of various formalisms (e.g. λ-calculus, term rewriting systems), but they are also a programming paradigm in themselves, and this is actually how they were introduced by Lafont. In this paper we consider semi-simple interaction nets as a programming language, and present a type assignment system using intersection types. First we show that interactions preserve types (i.e., the system enjoys subject reduction), and we compare this type assignment system with the intersection systems for λ-calculus and term rewriting systems. Then we define a recursion scheme that ensures termination of all interaction sequences. By relaxing the scheme and using the type assignment system, we derive another sufficient condition for termination of interaction nets. Finally, we show that although the type system based on general intersection types is not decidable, its restriction to rank 2 types is, and we give an algorithm that computes principal types for nets.

scholarly journals Defining a New Normal for Extremes in a Warming World

2017 ◽  
Vol 98 (6) ◽  
pp. 1139-1151 ◽  
Author(s):  
Sophie C. Lewis ◽  
Andrew D. King ◽  
Sarah E. Perkins-Kirkpatrick
Keyword(s):  
Climate Models ◽  
New Normal ◽  
The Future ◽  
Reference Event ◽  
Emissions Scenarios ◽  
One Step ◽  
Definition Of ◽  
Warming World ◽  
Climate Events

Abstract The term “new normal” has been used in scientific literature and public commentary to contextualize contemporary climate events as an indicator of a changing climate due to enhanced greenhouse warming. A new normal has been used broadly but tends to be descriptive and ambiguously defined. Here we review previous studies conceptualizing this idea of a new climatological normal and argue that this term should be used cautiously and with explicit definition in order to avoid confusion. We provide a formal definition of a new climate normal relative to present based around record-breaking contemporary events and explore the timing of when such extremes become statistically normal in the future model simulations. Applying this method to the record-breaking global-average 2015 temperatures as a reference event and a suite of model climate models, we determine that 2015 global annual-average temperatures will be the new normal by 2040 in all emissions scenarios. At the regional level, a new normal can be delayed through aggressive greenhouse gas emissions reductions. Using this specific case study to investigate a climatological new normal, our approach demonstrates the greater value of the concept of a climatological new normal for understanding and communicating climate change when the term is explicitly defined. This approach moves us one step closer to understanding how current extremes will change in the future in a warming world.

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