Intensional Models for the Theory of Types

In this paper we define intensional models for the classical theory of types, thus arriving at an intensional type logic ITL. Intensional models generalize Henkin's general models and have a natural definition. As a class they do not validate the axiom of Extensionality. We give a cut-free sequent calculus for type theory and show completeness of this calculus with respect to the class of intensional models via a model existence theorem. After this we turn our attention to applications. Firstly, it is argued that, since ITL is truly intensional, it can be used to model ascriptions of propositional attitude without predicting logical omniscience. In order to illustrate this a small fragment of English is defined and provided with an ITL semantics. Secondly, it is shown that ITL models contain certain objects that can be identified with possible worlds. Essential elements of modal logic become available within classical type theory once the axiom of Extensionality is given up.Comment: 25 page

Models of Type Theory Based on Moore Paths

This paper introduces a new family of models of intensional Martin-L\"of type theory. We use constructive ordered algebra in toposes. Identity types in the models are given by a notion of Moore path. By considering a particular gros topos, we show that there is such a model that is non-truncated, i.e. contains non-trivial structure at all dimensions. In other words, in this model a type in a nested sequence of identity types can contain more than one element, no matter how great the degree of nesting. Although inspired by existing non-truncated models of type theory based on simplicial and cubical sets, the notion of model presented here is notable for avoiding any form of Kan filling condition in the semantics of types.Comment: This is a revised and expanded version of a paper with the same name that appeared in the proceedings of the 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017

Quotient types in type theory

Martin-Lof's intuitionistic type theory (Type Theory) is a formal system that serves not only as a foundation of constructive mathematics but also as a dependently typed programming language. Dependent types are types that depend on values of other types. Type Theory is based on the Curry-Howard isomorphism which relates computer programs with mathematical proofs so that we can do computer-aided formal reasoning and write certified programs in programming languages like Agda, Epigram etc. Martin Lof proposed two variants of Type Theory which are differentiated by the treatment of equality. In Intensional Type Theory, propositional equality defined by identity types does not imply definitional equality, and type checking is decidable. In Extensional Type Theory, propositional equality is identified with definitional equality which makes type checking undecidable. Because of the good computational properties, Intensional Type Theory is more popular, however it lacks some important extensional concepts such as functional extensionality and quotient types. This thesis is about quotient types. A quotient type is a new type whose equality is redefined by a given equivalence relation. However, in the usual formulation of Intensional Type Theory, there is no type former to create a quotient. We also lose canonicity if we add quotient types into Intensional Type Theory as axioms. In this thesis, we first investigate the expected syntax of quotient types and explain it with categorical notions. For quotients which can be represented as a setoid as well as defined as a set without a quotient type former, we propose to define an algebraic structure of quotients called definable quotients. It relates the setoid interpretation and the set definition via a normalisation function which returns a normal form (canonical choice) for each equivalence class. It can be seen as a simulation of quotient types and it helps theorem proving because we can benefit from both representations. However this approach cannot be used for all quotients. It seems that we cannot define a normalisation function for some quotients in Type Theory, e.g. Cauchy reals and finite multisets. Quotient types are indeed essential for formalisation of mathematics and reasoning of programs. Then we consider some models of Type Theory where types are interpreted as structured objects such as setoids, groupoids or weak omega-groupoids. In these models equalities are internalised into types which means that it is possible to redefine equalities. We present an implementation of Altenkirch's setoid model and show that quotient types can be defined within this model. We also describe a new extension of Martin-Lof type theory called Homotopy Type Theory where types are interpreted as weak omega-groupoids. It can be seen as a generalisation of the groupoid model which makes extensional concepts including quotient types available. We also introduce a syntactic encoding of weak omega-groupoids which can be seen as a first step towards building a weak omega-groupoids model in Intensional Type Theory. All of these implementations were performed in the dependently typed programming language Agda which is based on intensional Martin-Lof type theory

Models of Type Theory Based on Moore Paths

This paper introduces a new family of models of intensional Martin-Löf type theory. We use constructive ordered algebra in toposes. Identity types in the models are given by a notion of Moore path. By considering a particular gros topos, we show that there is such a model that is non-truncated, i.e. contains non-trivial structure at all dimensions. In other words, in this model a type in a nested sequence of identity types can contain more than one element, no matter how great the degree of nesting. Although inspired by existing non-truncated models of type theory based on simplicial and on cubical sets, the notion of model presented here is notable for avoiding any form of Kan filling condition in the semantics of types.EPSRC Studentshi

Predicativity, the Russell-Myhill Paradox, and Church's Intensional Logic

This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church's intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also model other axioms of Church's intensional logic that have been criticized by Parsons and Klement: this, it turns out, is due to resources which also permit an interpretation of a fragment of Gallin's intensional logic. Finally, the relation between the predicative response to the Russell-Myhill paradox of propositions and the Russell paradox of sets is discussed, and it is shown that the predicative conception of set induced by this predicative intensional logic allows one to respond to the Wehmeier problem of many non-extensions.Comment: Forthcoming in The Journal of Philosophical Logi

Quotient types in type theory

Martin-Lof's intuitionistic type theory (Type Theory) is a formal system that serves not only as a foundation of constructive mathematics but also as a dependently typed programming language. Dependent types are types that depend on values of other types. Type Theory is based on the Curry-Howard isomorphism which relates computer programs with mathematical proofs so that we can do computer-aided formal reasoning and write certified programs in programming languages like Agda, Epigram etc. Martin Lof proposed two variants of Type Theory which are differentiated by the treatment of equality. In Intensional Type Theory, propositional equality defined by identity types does not imply definitional equality, and type checking is decidable. In Extensional Type Theory, propositional equality is identified with definitional equality which makes type checking undecidable. Because of the good computational properties, Intensional Type Theory is more popular, however it lacks some important extensional concepts such as functional extensionality and quotient types. This thesis is about quotient types. A quotient type is a new type whose equality is redefined by a given equivalence relation. However, in the usual formulation of Intensional Type Theory, there is no type former to create a quotient. We also lose canonicity if we add quotient types into Intensional Type Theory as axioms. In this thesis, we first investigate the expected syntax of quotient types and explain it with categorical notions. For quotients which can be represented as a setoid as well as defined as a set without a quotient type former, we propose to define an algebraic structure of quotients called definable quotients. It relates the setoid interpretation and the set definition via a normalisation function which returns a normal form (canonical choice) for each equivalence class. It can be seen as a simulation of quotient types and it helps theorem proving because we can benefit from both representations. However this approach cannot be used for all quotients. It seems that we cannot define a normalisation function for some quotients in Type Theory, e.g. Cauchy reals and finite multisets. Quotient types are indeed essential for formalisation of mathematics and reasoning of programs. Then we consider some models of Type Theory where types are interpreted as structured objects such as setoids, groupoids or weak omega-groupoids. In these models equalities are internalised into types which means that it is possible to redefine equalities. We present an implementation of Altenkirch's setoid model and show that quotient types can be defined within this model. We also describe a new extension of Martin-Lof type theory called Homotopy Type Theory where types are interpreted as weak omega-groupoids. It can be seen as a generalisation of the groupoid model which makes extensional concepts including quotient types available. We also introduce a syntactic encoding of weak omega-groupoids which can be seen as a first step towards building a weak omega-groupoids model in Intensional Type Theory. All of these implementations were performed in the dependently typed programming language Agda which is based on intensional Martin-Lof type theory

Classical Opacity

Philosophy and Phenomenological Research, EarlyView

Models of Type Theory Based on Moore Paths

This paper introduces a new family of models of intensional Martin-Löf type theory. We use constructive ordered algebra in toposes. Identity types in the models are given by a notion of Moore path. By considering a particular gros topos, we show that there is such a model that is non-truncated, i.e. contains non-trivial structure at all dimensions. In other words, in this model a type in a nested sequence of identity types can contain more than one element, no matter how great the degree of nesting. Although inspired by existing non-truncated models of type theory based on simplicial and cubical sets, the notion of model presented here is notable for avoiding any form of Kan filling condition in the semantics of types.Orton was supported by a PhD studentship from the UK EPSRC funded by grants EP/L504920/1 and EP/M506485/1

Natural models of homotopy type theory

The notion of a natural model of type theory is defined in terms of that of a representable natural transfomation of presheaves. It is shown that such models agree exactly with the concept of a category with families in the sense of Dybjer, which can be regarded as an algebraic formulation of type theory. We determine conditions for such models to satisfy the inference rules for dependent sums, dependent products, and intensional identity types, as used in homotopy type theory. It is then shown that a category admits such a model if it has a class of maps that behave like the abstract fibrations in axiomatic homotopy theory: they should be stable under pullback, closed under composition and relative products, and there should be weakly orthogonal factorizations into the class. It follows that many familiar settings for homotopy theory also admit natural models of the basic system of homotopy type theory.Comment: 51 page

Monoidal computer III: A coalgebraic view of computability and complexity

Monoidal computer is a categorical model of intensional computation, where many different programs correspond to the same input-output behavior. The upshot of yet another model of computation is that a categorical formalism should provide a much needed high level language for theory of computation, flexible enough to allow abstracting away the low level implementation details when they are irrelevant, or taking them into account when they are genuinely needed. A salient feature of the approach through monoidal categories is the formal graphical language of string diagrams, which supports visual reasoning about programs and computations. In the present paper, we provide a coalgebraic characterization of monoidal computer. It turns out that the availability of interpreters and specializers, that make a monoidal category into a monoidal computer, is equivalent with the existence of a *universal state space*, that carries a weakly final state machine for any pair of input and output types. Being able to program state machines in monoidal computers allows us to represent Turing machines, to capture their execution, count their steps, as well as, e.g., the memory cells that they use. The coalgebraic view of monoidal computer thus provides a convenient diagrammatic language for studying computability and complexity.Comment: 34 pages, 24 figures; in this version: added the Appendi