Related papers: An Equational Logical Framework for Type Theories
We describe a Martin-L\"of-style dependent type theory, called Cocon, that allows us to mix the intensional function space that is used to represent higher-order abstract syntax (HOAS) trees with the extensional function space that…
We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we…
Brouwer's constructivist foundations of mathematics is based on an intuitively meaningful notion of computation shared by all mathematicians. Martin-L\"of's meaning explanations for constructive type theory define the concept of a type in…
We prove an extensionality theorem for the "type-in-type" dependent type theory with Sigma-types. We suggest that the extensional equality type be identified with the logical equivalence relation on the free term model of type theory.
This paper introduces a novel type theory and logic for probabilistic reasoning. Its logic is quantitative, with fuzzy predicates. It includes normalisation and conditioning of states. This conditioning uses a key aspect that distinguishes…
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…
Beginning in the 1970s, statistician-cum-logician Per Martin-L\"of wrote a series of papers developing what became Martin-L\"of type theory, realizing a system where the distinction between mathematics and programming disappears. Inspired…
The definitional equality of an intensional type theory is its test of type compatibility. Today's systems rely on ordinary evaluation semantics to compare expressions in types, frustrating users with type errors arising when evaluation…
We prove completeness results for a wide variety of intuitionistic conditional logics. We do so by first using a canonical model construction obtain completeness with respect to descriptive conditional frames, and then introducing the…
A logic-enriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named LTTO and LTTO*, which we claim correspond closely to the classical predicative…
This paper continues the series of papers that develop a new approach to syntax and semantics of dependent type theories. Here we study the interpretation of the rules of the identity types in the intensional Martin-Lof type theories on the…
We introduce a modification of standard Martin-Lof type theory in which we eliminate definitional equality and replace all computation rules by propositional equalities. We show that type checking for such a system can be done in quadratic…
This paper presents a new system of logic, LF, that is intended to be used as the foundation of the formalization of science. That is, deductive validity according to LF is to be used as the criterion for assessing what follows from the…
We argue that locally Cartesian closed categories form a suitable doctrine for defining dependent type theories, including non-extensional ones. Using the theory of sketches, one may define syntactic categories for type theories in a style…
This book introduces a temporal type theory, the first of its kind as far as we know. It is based on a standard core, and as such it can be formalized in a proof assistant such as Coq or Lean by adding a number of axioms. Well-known…
Cubical type theory is an extension of Martin-L\"of type theory recently proposed by Cohen, Coquand, M\"ortberg and the author which allows for direct manipulation of $n$-dimensional cubes and where Voevodsky's Univalence Axiom is provable.…
Large language models (LLMs) have proven to be highly effective for solving complex reasoning tasks. Surprisingly, their capabilities can often be improved by iterating on previously generated solutions. In this context, a reasoning plan…
The use of logical systems for problem-solving may be as diverse as in proving theorems in mathematics or in figuring out how to meet up with a friend. In either case, the problem solving activity is captured by the search for an…
We present a logic named L_{LF} whose intended use is to formalize properties of specifications developed in the dependently typed lambda calculus LF. The logic is parameterized by the LF signature that constitutes the specification. Atomic…
Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of heigher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hibert-style Axiomatic Method. The new notion of Axiomatic…