A Focused Sequent Calculus Framework for Proof Search in Pure Type Systems

Basic proof-search tactics in logic and type theory can be seen as the root-first applications of rules in an appropriate sequent calculus, preferably without the redundancies generated by permutation of rules. This paper addresses the issues of defining such sequent calculi for Pure Type Systems (PTS, which were originally presented in natural deduction style) and then organizing their rules for effective proof-search. We introduce the idea of Pure Type Sequent Calculus with meta-variables (PTSCalpha), by enriching the syntax of a permutation-free sequent calculus for propositional logic due to Herbelin, which is strongly related to natural deduction and already well adapted to proof-search. The operational semantics is adapted from Herbelin's and is defined by a system of local rewrite rules as in cut-elimination, using explicit substitutions. We prove confluence for this system. Restricting our attention to PTSC, a type system for the ground terms of this system, we obtain the Subject Reduction property and show that each PTSC is logically equivalent to its corresponding PTS, and the former is strongly normalising iff the latter is. We show how to make the logical rules of PTSC into a syntax-directed system PS for proof-search, by incorporating the conversion rules as in syntax-directed presentations of the PTS rules for type-checking. Finally, we consider how to use the explicitly scoped meta-variables of PTSCalpha to represent partial proof-terms, and use them to analyse interactive proof construction. This sets up a framework PE in which we are able to study proof-search strategies, type inhabitant enumeration and (higher-order) unification

Cut-Simulation and Impredicativity

We investigate cut-elimination and cut-simulation in impredicative (higher-order) logics. We illustrate that adding simple axioms such as Leibniz equations to a calculus for an impredicative logic -- in our case a sequent calculus for classical type theory -- is like adding cut. The phenomenon equally applies to prominent axioms like Boolean- and functional extensionality, induction, choice, and description. This calls for the development of calculi where these principles are built-in instead of being treated axiomatically.Comment: 21 page

Denotational Semantics of the Simplified Lambda-Mu Calculus and a New Deduction System of Classical Type Theory

Classical (or Boolean) type theory is the type theory that allows the type inference $\sigma \to \bot) \to \bot => \sigma$ (the type counterpart of double-negation elimination), where $\sigma$ is any type and $\bot$ is absurdity type. This paper first presents a denotational semantics for a simplified version of Parigot's lambda-mu calculus, a premier example of classical type theory. In this semantics the domain of each type is divided into infinitely many ranks and contains not only the usual members of the type at rank 0 but also their negative, conjunctive, and disjunctive shadows in the higher ranks, which form an infinitely nested Boolean structure. Absurdity type $\bot$ is identified as the type of truth values. The paper then presents a new deduction system of classical type theory, a sequent calculus called the classical type system (CTS), which involves the standard logical operators such as negation, conjunction, and disjunction and thus reflects the discussed semantic structure in a more straightforward fashion.Comment: In Proceedings CL&C 2016, arXiv:1606.0582

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

Combinatorial structure of type dependency

We give an account of the basic combinatorial structure underlying the notion of type dependency. We do so by considering the category of all dependent sequent calculi, and exhibiting it as the category of algebras for a monad on a presheaf category. The objects of the presheaf category encode the basic judgements of a dependent sequent calculus, while the action of the monad encodes the deduction rules; so by giving an explicit description of the monad, we obtain an explicit account of the combinatorics of type dependency. We find that this combinatorics is controlled by a particular kind of decorated ordered tree, familiar from computer science and from innocent game semantics. Furthermore, we find that the monad at issue is of a particularly well-behaved kind: it is local right adjoint in the sense of Street--Weber. In future work, we will use this fact to describe nerves for dependent type theories, and to study the coherence problem for dependent type theory using the tools of two-dimensional monad theory.Comment: 35 page

On polymorphic sessions and functions: A tale of two (fully abstract) encodings

This work exploits the logical foundation of session types to determine what kind of type discipline for the -calculus can exactly capture, and is captured by, -calculus behaviours. Leveraging the proof theoretic content of the soundness and completeness of sequent calculus and natural deduction presentations of linear logic, we develop the first mutually inverse and fully abstract processes-as-functions and functions-as-processes encodings between a polymorphic session -calculus and a linear formulation of System F. We are then able to derive results of the session calculus from the theory of the -calculus: (1) we obtain a characterisation of inductive and coinductive session types via their algebraic representations in System F; and (2) we extend our results to account for value and process passing, entailing strong normalisation

Multi-dimensional Type Theory: Rules, Categories, and Combinators for Syntax and Semantics

We investigate the possibility of modelling the syntax and semantics of natural language by constraints, or rules, imposed by the multi-dimensional type theory Nabla. The only multiplicity we explicitly consider is two, namely one dimension for the syntax and one dimension for the semantics, but the general perspective is important. For example, issues of pragmatics could be handled as additional dimensions. One of the main problems addressed is the rather complicated repertoire of operations that exists besides the notion of categories in traditional Montague grammar. For the syntax we use a categorial grammar along the lines of Lambek. For the semantics we use so-called lexical and logical combinators inspired by work in natural logic. Nabla provides a concise interpretation and a sequent calculus as the basis for implementations.Comment: 20 page

A Fibrational Framework for Substructural and Modal Logics

We define a general framework that abstracts the common features of many intuitionistic substructural and modal logics / type theories. The framework is a sequent calculus / normal-form type theory parametrized by a mode theory, which is used to describe the structure of contexts and the structural properties they obey. In this sequent calculus, the context itself obeys standard structural properties, while a term, drawn from the mode theory, constrains how the context can be used. Product types, implications, and modalities are defined as instances of two general connectives, one positive and one negative, that manipulate these terms. Specific mode theories can express a range of substructural and modal connectives, including non-associative, ordered, linear, affine, relevant, and cartesian products and implications; monoidal and non-monoidal functors, (co)monads and adjunctions; n-linear variables; and bunched implications. We prove cut (and identity) admissibility independently of the mode theory, obtaining it for many different logics at once. Further, we give a general equational theory on derivations / terms that, in addition to the usual beta/eta-rules, characterizes when two derivations differ only by the placement of structural rules. Additionally, we give an equivalent semantic presentation of these ideas, in which a mode theory corresponds to a 2-dimensional cartesian multicategory, the framework corresponds to another such multicategory with a functor to the mode theory, and the logical connectives make this into a bifibration. Finally, we show how the framework can be used both to encode existing existing logics / type theories and to design new ones