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Abstract. Higher inductive types (HITs) in Homotopy TypeTheory allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types, and allow to define types with non-trivial higher equality types, such as spheres, suspensions and the torus. However, there are also interesting uses of HITs to define types satisfying uniqueness of equality proofs, such as the Cauchy reals, the partiality monad, and the well-typed syntax of type theory. In each of these examples we define several types that depend on each other mutually, i.e. they are inductive-inductive definitions. We call those HITs quotient inductive-inductive types (QIITs). Although there has been recent progress on a general theory of HITs, there is not yet a theoretical foundation for the combination of equality constructors and induction-induction, despite many interesting applications. In the present paper we present a first step towards a semantic definition of QIITs. In particular, we give an initial-algebra semantics. We further derive a section induction principle, stating that every algebra morphism into the algebra in question has a section, which is close to the intuitively expected elimination rules.

Enterprise Application Integration is the centerpiece of current on-premise, cloud and device integration scenarios. We describe optimization strategies that help reduce the model complexity, and improve the process execution using design time techniques. In order to achieve this, we formalize compositions of Enterprise Integration Patterns based on their characteristics, and propose a realization of optimization strategies using graph rewriting. The framework is successfully evaluated on a real-world catalog of pattern compositions, containing over 900 integration scenarios.

Abstract. We present a principle for introducing new types in type theory which generalises strictly positive indexed inductive data types. In this new principle a set A is defined inductively simultaneously with an A-indexed set B, which is also defined inductively. Compared to indexed inductive definitions, the novelty is that the index set A is generated inductively simultaneously with B. In other words, we mutually define two inductive sets, of which one depends on the other. Instances of this principle have previously been used in order to formalise type theory inside type theory. However the consistency of the framework used (the theorem prover Agda) is not so clear, as it allows the definition of a universe containing a code for itself. We give an axiomatisation of the new principle in such a way that the resulting type theory is consistent, which we prove by constructing a set-theoretic model.

Compositional Game Theory is a new, recently introduced model of economic games based upon the computer science idea of compositionality. In it, complex and irregular games can be built up from smaller and simpler games, and the equilibria of these complex games can be defined recursively from the equilibria of their simpler subgames. This paper extends the model by providing a final coalgebra semantics for infinite games. In the course of this, we introduce a new operator on games to model the economic concept of subgame perfection.

Abstract. This article is concerned with the application of the program extraction technique to a new class of problems: the synthesis of decision procedures for the classical satisfiability problem that are correct by construction. To this end, we formalize a completeness proof for the DPLL proof system and extract a SAT solver from it. When applied to a propositional formula in conjunctive normal form the program produces either a satisfying assignment or a DPLL derivation showing its unsatisfiability. We use non-computational quantifiers to remove redundant computational content from the extracted program and translate it into Haskell to improve performance. We also prove the equivalence between the resolution proof system and the DPLL proof system with a bound on the size of the resulting resolution proof. This demonstrates that it is possible to capture quantitative information about the extracted program on the proof level. The formalization is carried out in the interactive proof assistant Minlog.

Abstract. Induction-induction is a principle for defining data types in Martin-Löf Type Theory. An inductive-inductive definition consists of a set A, together with an A-indexed family B : A Ñ Set, where both A and B are inductively defined in such a way that the constructors for A can refer to B and vice versa. In addition, the constructors for B can refer to the constructors for A. We extend the usual initial algebra semantics for ordinary inductive data types to the inductive-inductive setting by considering dialgebras instead of ordinary algebras. This gives a new and compact formalisation of inductive-inductive definitions, which we prove is equivalent to the usual formulation with elimination rules.

This paper combines reflexive-graph-category structure for relational parametricity with fibrational models of impredicative polymorphism. To achieve this, we modify the definition of fibrational model of impredicative polymorphism by adding one further ingredient to the structure: comprehension in the sense of Lawvere. Our main result is that such comprehensive models, once further endowed with reflexive-graphcategory structure, enjoy the expected consequences of parametricity. This is proved using a type-theoretic presentation of the category-theoretic structure, within which the desired consequences of parametricity are derived. The formalisation requires new techniques because equality relations are not available, and standard arguments that exploit equality need to be reworked.