In this article we investigate how to represent partial-recursive functions in Martin-Löf’s type theory. Our representation will be based on the approach by Bove and Capretta, which makes use of indexed inductive-recursive definitions (IIRD). We will show how to restrict the IIRD used so that we obtain directly executable partial recursive functions, Then we introduce a data type of partial recursive functions. We show how to evaluate elements of this data type inside Martin-Löf’s type theory, and that therefore the functions defined by this data type are in fact partial-recursive. The data type formulates a very general schema for defining functions recursively in dependent type theory. The initial version of this data type, for which we introduce an induction principle, needs to be expanded, in order to obtain closure under composition. We will obtain two versions of this expanded data type, and prove that they define the same set of partial-recursive functions. Both versions will be large types. Next we prove a Kleene-style normal form theorem. Using it we will show how to obtain a data type of partial recursive functions which is a small set. Finally, we show how to define self-evaluation as a partial recursive function. We obtain a correct version of this evaluation function, which not only computes recursively a result, but as well a proof that the result is correct. Keywords: Martin-Löf type theory, computability theory, recursion theory, Kleene index, Kleene brackets, Kleene’s normal form theorem, partial recursive functions, inductive-recursive definitions, indexed induction-recursion, self-evaluation. 1

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. 1

Fiore and Hur [18] recently introduced a novel methodology—henceforth referred to as Sol—for the Synthesis of equational and rewriting logics from mathematical models. In [18], Sol was successfully applied to rationally reconstruct the traditional equational logic for universal algebra of Birkhoff [3] and its multi-sorted version [26], and also to synthesise a new version of the Nominal Algebra of Gabbay and Mathijssen [41] and the Nominal Equational Logic of Clouston and Pitts [8] for reasoning about languages with name-binding operators. Based on these case studies and further preliminary investigations, we contend that Sol can make an impact in the problem of engineering logics for modern computational languages. For example, our proposed research on secondorder equational logic will provide foundations for designing a second-order extension of the Maude system [37], a first-order semantic and logical framework used in formal software engineering for specification and programming. Our research strategy can be visualised as follows: (I)

Abstract We develop rules for coalgebras in type theory, and give meaning explanations for them. We show that elements of coalgebras are determined by their elimination rules, whereas the introduction rules can be considered as derived. This is in contrast with algebraic data types, for which the opposite is true: elements are determined by their introduction rules, and the elimination rules can be considered as derived. In this sense, the function type from the logical framework is more like a coalgebraic data type, the elements of which are determined by the elimination rule. We illustrate why the simplest form of guarded recursion is nothing but the introduction rule originating from the formulation of coalgebras in category theory. We discuss restrictions needed in order to preserve decidability of equality. Dedicated to Per Martin-Löf on the occasion of his retirement. 1.1

categorical semantics for inductive-inductive definitions

the introduction rules for A may also refer to B. So we have formation rules A ∶ Set, B ∶ A → Set and typical introduction rules might take the form a ∶ A b ∶ B(a)... introA(a, b,...) ∶ A a0 ∶ A b ∶ B(a0) a1 ∶ A...