Semirings of Evidence

In traditional justification logic, evidence terms have the syntactic form of polynomials, but they are not equipped with the corresponding algebraic structure. We present a novel semantic approach to justification logic that models evidence by a semiring. Hence justification terms can be interpreted as polynomial functions on that semiring. This provides an adequate semantics for evidence terms and clarifies the role of variables in justification logic. Moreover, the algebraic structure makes it possible to compute with evidence. Depending on the chosen semiring this can be used to model trust, probabilities, cost, etc. Last but not least the semiring approach seems promising for obtaining a realization procedure for modal fixed point logics

Decidability of Conversion for Type Theory in Type Theory

Type theory should be able to handle its own meta-theory, both to justify its foundational claims and to obtain a verified implementation. At the core of a type checker for intensional type theory lies an algorithm to check equality of types, or in other words, to check whether two types are convertible. We have formalized in Agda a practical conversion checking algorithm for a dependent type theory with one universe \ue0 la Russell, natural numbers, and η-equality for Π types. We prove the algorithm correct via a Kripke logical relation parameterized by a suitable notion of equivalence of terms. We then instantiate the parameterized fundamental lemma twice: once to obtain canonicity and injectivity of type formers, and once again to prove the completeness of the algorithm. Our proof relies on inductive-recursive definitions, but not on the uniqueness of identity proofs. Thus, it is valid in variants of intensional Martin-L\uf6f Type Theory as long as they support induction-recursion, for instance, Extensional, Observational, or Homotopy Type Theory

A Logical Account of PSPACE

International audienceThe Soft Type Assignment system has been introduced as a language for Polytime Computations. In this work we extend the language by boolean additives constants obtaining a system named BSTA which we will show to be correct and complete for the complexity class of problem decidable in polynomial space

On Induction, Coinduction and Equality in Martin-L\uf6f and Homotopy Type Theory

Martin L\uf6f Type Theory, having put computation at the center of logicalreasoning, has been shown to be an effective foundation for proof assistants,with applications both in computer science and constructive mathematics. Oneambition though is for MLTT to also double as a practical general purposeprogramming language. Datatypes in type theory come with an induction orcoinduction principle which gives a precise and concise specification of theirinterface. However, such principles can interfere with how we would like toexpress our programs. In this thesis, we investigate more flexible alternativesto direct uses of the (co)induction principles.As a first contribution, we consider the n-truncation of a type in Homo-topy Type Theory. We derive in HoTT an eliminator into (n+1)-truncatedtypes instead of n-truncated ones, assuming extra conditions on the underlyingfunction.As a second contribution, we improve on type-based criteria for terminationand productivity. By augmenting the types with well-foundedness information,such criteria allow function definitions in a style closer to general recursion.We consider two criteria: guarded types, and sized types.Guarded types introduce a modality ”later” to guard the availability ofrecursive calls provided by a general fixed-point combinator. In Guarded Cu-bical Type Theory we equip the fixed-point combinator with a propositionalequality to its one-step unfolding, instead of a definitional equality that wouldbreak normalization. The notion of path from Cubical Type Theory allows usto do so without losing canonicity or decidability of conversion.Sized types, on the other hand, explicitly index datatypes with size boundson the height or depth of their elements. The sizes however can get in theway of the reasoning principles we expect. Our approach is to introduce newquantifiers for ”irrelevant” size quantification. We present a type theory withparametric quantifiers where irrelevance arises as a “free theorem”. We alsodevelop a conversion checking algorithm for a more specific theory where thenew quantifiers are restricted to sizes.Finally, our third contribution is about the operational semantics of typetheory. For the extensions above we would like to devise a practical conversionchecking algorithm suitable for integration into a proof assistant. We formal-ized the correctness of such an algorithm for a small but challenging corecalculus, proving that conversion is decidable. We expect this development toform a good basis to verify more complex theories.The ideas discussed in this thesis are already influencing the developmentof Agda, a proof assistant based on type theory