Related papers: Type-Theoretic Signatures for Algebraic Theories a…
In this paper, we propose an abstract definition of dependent type theories as essentially algebraic theories. One of the main advantages of this definition is its composability: simple theories can be combined into more complex ones, and…
Quotient inductive-inductive types (QIITs) are generalized inductive types which allow sorts to be indexed over previously declared sorts, and allow usage of equality constructors. QIITs are especially useful for algebraic descriptions of…
Higher inductive-inductive types (HIITs) generalize inductive types of dependent type theories in two ways. On the one hand they allow the simultaneous definition of multiple sorts that can be indexed over each other. On the other hand they…
Type theory plays an important role in foundations of mathematics as a framework for formalizing mathematics and a base for proof assistants providing semi-automatic proof checking and construction. Derivation of each theorem in type theory…
Homotopy type theory is an interpretation of Martin-L\"of's constructive type theory into abstract homotopy theory. There results a link between constructive mathematics and algebraic topology, providing topological semantics for…
Ten years ago, it was shown that nominal techniques can be used to design coalgebraic data types with variable binding, so that alpha-equivalence classes of infinitary terms are directly endowed with a corecursion principle. We introduce…
We give a polymorphic account of the relational algebra. We introduce a formalism of ``type formulas'' specifically tuned for relational algebra expressions, and present an algorithm that computes the ``principal'' type for a given…
This paper introduces an expressive class of quotient-inductive types, called QW-types. We show that in dependent type theory with uniqueness of identity proofs, even the infinitary case of QW-types can be encoded using the combination of…
In this paper, I establish the categorical structure necessary to interpret dependent inductive and coinductive types. It is well-known that dependent type theories \`a la Martin-L\"of can be interpreted using fibrations. Modern theorem…
At the heart of intuitionistic type theory lies an intuitive semantics called the "meaning explanations"; crucially, when meaning explanations are taken as definitive for type theory, the core notion is no longer "proof" but "verification".…
We define a general class of dependent type theories, encompassing Martin-L\"of's intuitionistic type theories and variants and extensions. The primary aim is pragmatic: to unify and organise their study, allowing results and constructions…
We develop a dependent type theory that is based purely on inductive and coinductive types, and the corresponding recursion and corecursion principles. This results in a type theory with a small set of rules, while still being fairly…
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…
Domain theory has been developed as a mathematical theory of computation and to give a denotational semantics to programming languages. It helps us to fix the meaning of language concepts, to understand how programs behave and to reason…
Algebraic specifications of data types provide a natural basis for testing data types implementations. In this framework, the conformance relation is based on the satisfaction of axioms. This makes it possible to formally state the…
We define a monoidal semantics for algebraic theories. The basis for the definition is provided by the analysis of the structural rules in the term calculus of algebraic languages. Models are described both explicitly, in a form that…
We develop a taxonomy of different behavioral specification theories and expose their algebraic properties. We start by clarifying what precisely constitutes a behavioral specification theory and then introduce logical and structural…
Algebraic theories with dependency between sorts form the structural core of Martin-L\"of type theory and similar systems. Their denotational semantics are typically studied using categorical techniques; many different categorical…
The introduction of first-class type classes in the Coq system calls for re-examination of the basic interfaces used for mathematical formalization in type theory. We present a new set of type classes for mathematics and take full advantage…
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…