Anton Setzer

A Finite Axiomatization of Inductive-Recursive Definitions

Induction-recursion is a schema which formalizes the principles for introducing new sets in Martin-Lofs type theory. It states that we may inductively define a set while simultaneously defining a function from this set into an arbitrary type by structural recursion. This extends the notion of an inductively defined set substantially and allows us to introduce universes and higher order universes (but not a Mahlo universe). In this article we give a finite axiomatization of inductive-recursive definitions. We prove consistency by constructing a set-theoretic model which makes use of one Mahlo cardinal.

Indexed induction–recursion

Abstract An indexed inductive definition (IID) is a simultaneous inductive definition of an indexed family of sets. An inductive–recursive definition (IRD) is a simultaneous inductive definition of a set and a recursive definition of a function on that set. An indexed inductive–recursive definition (IIRD) is a combination of both. We present a closed theory which allows us to introduce all IIRD in a natural way without much encoding. By specialising it we also get a closed theory of IID. Our theory of IIRD includes essentially all definitions of sets which occur in Martin–Lof type theory. We show in particular that Martin–Lof’s computability predicates for dependent types and Palmgren’s higher order universes are special kinds of IIRD and thereby clarify why they are constructively acceptable notions. We give two axiomatisations. The first formalises a principle for introducing meaningful IIRD by using the data-construct in the original version of the proof assistant Agda for Martin–Lof type theory. The second one admits a more general form of introduction rule, including the introduction rule for the intensional identity relation, which is not covered by the first axiomatisation. If we add an extensional identity relation to our logical framework, we show that the theories of restricted and general IIRD are equivalent by interpreting them in each other. Finally, we show the consistency of our theories by constructing a model in classical set theory extended by a Mahlo cardinal.

Copatterns: programming infinite structures by observations

Inductive datatypes provide mechanisms to define finite data such as finite lists and trees via constructors and allow programmers to analyze and manipulate finite data via pattern matching. In this paper, we develop a dual approach for working with infinite data structures such as streams. Infinite data inhabits coinductive datatypes which denote greatest fixpoints. Unlike finite data which is defined by constructors we define infinite data by observations. Dual to pattern matching, a tool for analyzing finite data, we develop the concept of copattern matching, which allows us to synthesize infinite data. This leads to a symmetric language design where pattern matching on finite and infinite data can be mixed. We present a core language for programming with infinite structures by observations together with its operational semantics based on (co)pattern matching and describe coverage of copatterns. Our language naturally supports both call-by-name and call-by-value interpretations and can be seamlessly integrated into existing languages like Haskell and ML. We prove type soundness for our language and sketch how copatterns open new directions for solving problems in the interaction of coinductive and dependent types.

Induction–recursion and initial algebras

Induction-recursion is a powerful definition method in intuitionistic type theory. It extends (generalized) inductive definitions and allows us to define all standard sets of Martin-Lof type theory as well as a large collection of commonly occurring inductive data structures. It also includes a variety of universes which are constructive analogues of inaccessibles and other large cardinals below the first Mahlo cardinal. In this article we give a new compact formalization of inductive-recursive definitions by modeling them as initial algebras in slice categories. We give generic formation, introduction, elimination, and equality rules generalizing the usual rules of type theory. Moreover, we prove that the elimination and equality rules are equivalent to the principle of the existence of initial algebras for certain endofunctors. We also show the equivalence of the current formulation with the formulation of induction-recursion as a reflection principle given in Dybjer and Setzer (Lecture Notes in Comput. Sci. 2183 (2001) 93). Finally, we discuss two type-theoretic analogues of Mahlo cardinals in set theory: an external Mahlo universe which is defined by induction-recursion and captured by our formalization, and an internal Mahlo universe, which goes beyond induction-recursion. We show that the external Mahlo universe, and therefore also the theory of inductive-recursive definitions, have proof-theoretical strength of at least Rathjens theory KPM.

Automated Verification of Signalling Principles in Railway Interlocking Systems

In this paper we present a verification strategy for signalling principles for the control of a railway interlocking system written in ladder logic. All translation steps have been implemented and tested on a real-world example of a railway interlocking system. The steps in this translation are as follows: 1. The development of a mathematical model of a railway interlocking system and the translation from ladder logic into this model. 2. The development of verification conditions guaranteeing the correctness of safety conditions. 3. The verification of safety conditions using a satisfiability solver. 4. The generation of safety conditions from signalling principles using a topological model of a railway yard.

Extending Martin-Löf Type Theory by one Mahlo-universe

Abstract. We define a type theory MLM, which has proof theoretical strength slightly greater then Rathjens theory KPM. This is achieved by replacing the universe in Martin-Löfs Type Theory by a new universe V having the property that for every function f, mapping families of sets in V to families of sets in V, there exists a universe inside V closed under f. We show that the proof theoretical strength of MLM is

The proof-theoretic analysis of transfinitely iterated fixed point theories

\geq \psi_{\Omega_1}\Omega_{{\rm M}+\omega}

Inductive-inductive definitions

. This is slightly greater than

Well-ordering proofs for Martin-Löf type theory☆

|{\rm KPM}|

Verification of Solid State Interlocking Programs

, and shows that V can be considered to be a Mahlo-universe. Together with [Se96a] it follows