Denotational semantics for intuitionistic type theory using a hierarchy of domains with totality

1999 ◽  
Vol 38 (1) ◽  
pp. 19-60 ◽  
Author(s):  
Geir Waagbø
Keyword(s):  
Type Theory ◽  
Denotational Semantics

The calculus of dependent lambda eliminations

2017 ◽  
Vol 27 ◽  
Author(s):  
AARON STUMP
Keyword(s):  
Type Theory ◽  
Lambda Calculus ◽  
Denotational Semantics ◽  
Prototype Implementation ◽  
Constructive Type Theory ◽  
Typed Lambda Calculus ◽  
Constructive Type

AbstractModern constructive type theory is based on pure dependently typed lambda calculus, augmented with user-defined datatypes. This paper presents an alternative called the Calculus of Dependent Lambda Eliminations, based on pure lambda encodings with no auxiliary datatype system. New typing constructs are defined that enable induction, as well as large eliminations with lambda encodings. These constructs are constructor-constrained recursive types, and a lifting operation to lift simply typed terms to the type level. Using a lattice-theoretic denotational semantics for types, the language is proved logically consistent. The power of CDLE is demonstrated through several examples, which have been checked with a prototype implementation called Cedille.

scholarly journals Denotational semantics of recursive types in synthetic guarded domain theory

2018 ◽  
Vol 29 (3) ◽  
pp. 465-510 ◽  
Author(s):  
RASMUS E. MØGELBERG ◽  
MARCO PAVIOTTI
Keyword(s):  
Programming Languages ◽  
Type Theory ◽  
Operational Semantics ◽  
Lambda Calculus ◽  
Denotational Semantics ◽  
Domain Theory ◽  
Logical Relation ◽  
Dependent Type Theory ◽  
Semantics Of Programming Languages ◽  
Dependent Type

Just like any other branch of mathematics, denotational semantics of programming languages should be formalised in type theory, but adapting traditional domain theoretic semantics, as originally formulated in classical set theory to type theory has proven challenging. This paper is part of a project on formulating denotational semantics in type theories with guarded recursion. This should have the benefit of not only giving simpler semantics and proofs of properties such as adequacy, but also hopefully in the future to scale to languages with advanced features, such as general references, outside the reach of traditional domain theoretic techniques.Working inGuarded Dependent Type Theory(GDTT), we develop denotational semantics for Fixed Point Calculus (FPC), the simply typed lambda calculus extended with recursive types, modelling the recursive types of FPC using the guarded recursive types ofGDTT. We prove soundness and computational adequacy of the model inGDTTusing a logical relation between syntax and semantics constructed also using guarded recursive types. The denotational semantics is intensional in the sense that it counts the number of unfold-fold reductions needed to compute the value of a term, but we construct a relation relating the denotations of extensionally equal terms, i.e., pairs of terms that compute the same value in a different number of steps. Finally, we show how the denotational semantics of terms can be executed inside type theory and prove that executing the denotation of a boolean term computes the same value as the operational semantics of FPC.

scholarly journals Denotational semantics for guarded dependent type theory

2020 ◽  
Vol 30 (4) ◽  
pp. 342-378
Author(s):  
Aleš Bizjak ◽  
Rasmus Ejlers Møgelberg
Keyword(s):  
Type Theory ◽  
Denotational Semantics ◽  
Dependent Types ◽  
Proof Assistants ◽  
New Model ◽  
Dependent Type Theory ◽  
Additional Requirement ◽  
The One ◽  
Dependent Type

AbstractWe present a new model of guarded dependent type theory (GDTT), a type theory with guarded recursion and multiple clocks in which one can program with and reason about coinductive types. Productivity of recursively defined coinductive programs and proofs is encoded in types using guarded recursion and can therefore be checked modularly, unlike the syntactic checks implemented in modern proof assistants. The model is based on a category of covariant presheaves over a category of time objects, and quantification over clocks is modelled using a presheaf of clocks. To model the clock irrelevance axiom, crucial for programming with coinductive types, types must be interpreted as presheaves internally right orthogonal to the object of clocks. In the case of dependent types, this translates to a lifting condition similar to the one found in homotopy theoretic models of type theory, but here with an additional requirement of uniqueness of lifts. Since the universes defined by the standard Hofmann–Streicher construction in this model do not satisfy this property, the universes in GDTT must be indexed by contexts of clock variables. We show how to model these universes in such a way that inclusions of clock contexts give rise to inclusions of universes commuting with type operations on the nose.

Basic Simple Type Theory

1997 ◽  
Author(s):  
J. Roger Hindley
Keyword(s):  
Type Theory ◽  
Simple Type ◽  
Simple Type Theory

Type Theory and Formal Proof

2009 ◽  
Author(s):  
Rob Nederpelt ◽  
Herman Geuvers
Keyword(s):  
Type Theory ◽  
Formal Proof

Computational Models for Multilayered Composite Shells with Application to Tires

1996 ◽  
Vol 24 (1) ◽  
pp. 11-38 ◽  
Author(s):  
G. M. Kulikov
Keyword(s):  
Type Theory ◽  
Computational Models ◽  
Geometrical Nonlinearity ◽  
Higher Order ◽  
Stress Strain ◽  
Strain Fields ◽  
First Order ◽  
Timoshenko Type ◽  
Joint Influence ◽  
Discrete Layer

Abstract This paper focuses on four tire computational models based on two-dimensional shear deformation theories, namely, the first-order Timoshenko-type theory, the higher-order Timoshenko-type theory, the first-order discrete-layer theory, and the higher-order discrete-layer theory. The joint influence of anisotropy, geometrical nonlinearity, and laminated material response on the tire stress-strain fields is examined. The comparative analysis of stresses and strains of the cord-rubber tire on the basis of these four shell computational models is given. Results show that neglecting the effect of anisotropy leads to an incorrect description of the stress-strain fields even in bias-ply tires.

Campus Social Climate Correlates of Environmental Type Dimensions

NASPA Journal ◽  
2004 ◽  
Vol 41 (4) ◽  
Author(s):  
Daniel W. Salter ◽  
Reynol Junco ◽  
Summer D. Irvin
Keyword(s):  
Work Environment ◽  
Personal Growth ◽  
Type Theory ◽  
General Purpose ◽  
Myers Briggs Type Indicator ◽  
Environment Scale ◽  
Type Indicator ◽  
Group Environment ◽  
System Maintenance ◽  
Environment Type

To address the ability of the Salter Environment Type Assessment (SETA) to measure different kinds of campus environments, data from three studies of the SETA with the Work Environment Scale, Group Environment Scale, and University Residence Environment Scale were reexamined (n = 534). Relationship dimension scales were very consistent with extraversion and feeling from environmental type theory. System maintenance and systems change scales were associated with judging and perception on the SETA, respectively. Results from the SETA and personal growth dimension scales were mixed. Based on this analysis, the SETA may serve as a general purpose environmental assessment for use with the Myers-Briggs Type Indicator.

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