Algebraic Theories, Data Types, and Control Constructs

1986 ◽  
Vol 9 (3) ◽  
pp. 343-370
Author(s):  
Eric G. Wagner
Keyword(s):  
Algebraic Theory ◽  
Data Type ◽  
Theoretical Studies ◽  
Data Types ◽  
Algebraic Theories ◽  
Initial Algebra ◽  
Algebraic Framework ◽  
And Control ◽  
Type Specification ◽  
Mathematical Semantics

The aim of this paper is to model recursive types, equational types, and elementary programming control constructs (such as conditionals and while-do) in one, comparatively simple, algebraic framework, that can be used for theoretical studies and as a basis for data type and program specification. To this end we introduce a new kind of algebraic theory, the RV-theory. We give simple examples of the use of such theories for data type specification. We provide a mathematical semantics for these specifications that extends the initial algebra semantics for equational specification to include recursive types.

Conservativity of Equational Theories in Typed Lambda Calculi1

1993 ◽  
Vol 19 (1-2) ◽  
pp. 1-49
Author(s):  
Val Breazu-Tannen ◽  
Albert R. Meyer
Keyword(s):  
Programming Languages ◽  
Equational Theory ◽  
Data Type ◽  
Data Types ◽  
Functional Programming Languages ◽  
Type Definition ◽  
Algebraic Theories ◽  
A New Technique ◽  
Evaluation Of Data ◽  
Type Specification

In programming languages that feature unrestricted recursion, the equational theory corresponding to evaluation of data type expressions must be distinguished from the classical theory of the data as given by, say, algebraic specifications. Aiming to preserve classical reasoning about the underlying data types, that is, for the equational theory of the programming language to be a conservative extension of the theory given by the data type specification, we investigate, alternative computational settings given by typed lambda calculi, specifically here by the Girard-Reynolds polymorphic lambda calculus ( λ ∀ ). We prove that the addition of just the λ ∀ -constructions to arbitrary specifications, as given by algebraic theories, and even simply typed lambda theories, is conservative. This suggests that polymorphic constructs and reasoning can be superimposed on familiar data-type definition features of programming languages without changing the behavior of these features. Using purely syntactic methods, we give transformational proofs of these results for certain systems of equational reasoning. A new technique for analyzing polymorphic equational proofs is developed to this purpose. Finally, we prove, with a semantics argument, that it is possible to combine arbitrary algebraic data type specifications and the λ ∀ -constructions into functional programming languages that both conserve algebraic reasoning about data. and ensure, over arbitrary algebraically specified observables, a computing power equivalent to that of the pure λ ∀ . The corresponding problem for simply typed specifications remains open.

Hydraulic tomography using joint inversion of head and flux data 

2021 ◽  
Author(s):  
Behzad Pouladiborj ◽  
Olivier Bour ◽  
Niklas Linde ◽  
Laurent Longuevergne
Keyword(s):  
Hydraulic Conductivity ◽  
Joint Inversion ◽  
Data Type ◽  
Data Types ◽  
Hydraulic Tomography ◽  
Flux Data ◽  
Planning And Design ◽  
Constant Rate ◽  
Resolution Analysis ◽  
Constant Head

<p>Hydraulic tomography is a state of the art method for inferring hydraulic conductivity fields using head data. Here, a numerical model is used to simulate a steady-state hydraulic tomography experiment by assuming a Gaussian hydraulic conductivity field (also constant storativity) and generating the head and flux data in different observation points. We employed geostatistical inversion using head and flux data individually and jointly to better understand the relative merits of each data type. For the typical case of a small number of observation points, we find that flux data provide a better resolved hydraulic conductivity field compared to head data when considering data with similar signal-to-noise ratios. In the case of a high number of observation points, we find the estimated fields to be of similar quality regardless of the data type. A resolution analysis for a small number of observations reveals that head data averages over a broader region than flux data, and flux data can better resolve the hydraulic conductivity field than head data. The inversions' performance depends on borehole boundary conditions, with the best performing setting for flux data and head data are constant head and constant rate, respectively. However, the joint inversion results of both data types are insensitive to the borehole boundary type. Considering the same number of observations, the joint inversion of head and flux data does not offer advantages over individual inversions. By increasing the hydraulic conductivity field variance, we find that the resulting increased non-linearity makes it more challenging to recover high-quality estimates of the reference hydraulic conductivity field. Our findings would be useful for future planning and design of hydraulic tomography tests comprising the flux and head data.</p>

Can we build a grammar on the basis of judgments?

2020 ◽  
pp. 165-188
Author(s):  
Sam Featherston
Keyword(s):  
Small Group ◽  
Data Type ◽  
Theory Building ◽  
Final Part ◽  
Data Types ◽  
Ongoing Debate ◽  
Single Person

This chapter is a contribution to the ongoing debate about the necessary quality of the database for theory building in research on syntax. In particular, the focus is upon introspective judgments as a data type or group of data types. In the first part, the chapter lays out some of the evidence for the view that the judgments of a single person or of a small group of people are much less valid than the judgments of a group. In the second part, the chapter criticizes what the author takes to be overstatements and overgeneralizations of findings by Sprouse, Almeida, and Schütze that are sometimes viewed as vindicating an “armchair method” in linguistics. The final part of the chapter attempts to sketch out a productive route forward that empirically grounded syntax could take.

scholarly journals Large algebraic theories with small algebras

1978 ◽  
Vol 19 (3) ◽  
pp. 371-380 ◽  
Author(s):  
Jan Reiterman
Keyword(s):  
Algebraic Theory ◽  
Proper Class ◽  
Algebraic Theories

The aim of the paper is to study the interrelation between several natural smallness conditions on an algebraic theory with a proper class of operations. The conditions concern the existence of sets of data determining algebras, homomorphisms, subalgebras, and congruences.

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