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Partial recursive functions in MartinLöf Type Theory
 Logical Approaches to Computational Barriers: Second Conference on Computability in Europe, CiE 2006
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A Categorical Semantics for InductiveInductive Definitions
"... Abstract. Inductioninduction is a principle for defining data types in MartinLöf Type Theory. An inductiveinductive definition consists of a set A, together with an Aindexed 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 ..."
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Abstract. Inductioninduction is a principle for defining data types in MartinLöf Type Theory. An inductiveinductive definition consists of a set A, together with an Aindexed 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 inductiveinductive setting by considering dialgebras instead of ordinary algebras. This gives a new and compact formalisation of inductiveinductive definitions, which we prove is equivalent to the usual formulation with elimination rules. 1
A data type of partial recursive functions in MartinLöf Type Theory
, 2006
"... In this article we investigate how to represent partialrecursive functions in MartinLöf’s type theory. Our representation will be based on the approach by Bove and Capretta, which makes use of indexed inductiverecursive definitions (IIRD). We will show how to restrict the IIRD used so that we obt ..."
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In this article we investigate how to represent partialrecursive functions in MartinLöf’s type theory. Our representation will be based on the approach by Bove and Capretta, which makes use of indexed inductiverecursive 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 MartinLöf’s type theory, and that therefore the functions defined by this data type are in fact partialrecursive. 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 partialrecursive functions. Both versions will be large types. Next we prove a Kleenestyle 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 selfevaluation 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.
Chapter 1 Coalgebras as Types determined by their Elimination Rules
"... 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 op ..."
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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 MartinLöf on the occasion of his retirement. 1.1
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"... 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... ..."
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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...
Algebraic MetaTheories and . . .
"... 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 [ ..."
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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 multisorted 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 namebinding 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 secondorder extension of the Maude system [37], a firstorder semantic and logical framework used in formal software engineering for specification and programming. Our research strategy can be visualised as follows: (I)
Indexed InductionRecursion
"... An indexed inductive definition (IID) is a simultaneous inductive definition of an indexed family of sets. An inductiverecursive definition (IRD) is a simultaneous inductive definition of a set and a recursive definition of a function from that set into another type. An indexed inductiverecursive ..."
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An indexed inductive definition (IID) is a simultaneous inductive definition of an indexed family of sets. An inductiverecursive definition (IRD) is a simultaneous inductive definition of a set and a recursive definition of a function from that set into another type. An indexed inductiverecursive definition (IIRD) is a combination of both. We present a closed theory which allows us to introduce all IIRDs in a natural way without much encoding. By specialising it we also get a closed theory of IID. Our theory of IIRDs includes essentially all definitions of sets which occur in MartinLöf type theory. We show in particular that MartinLöf’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 and more restricted one formalises a principle for introducing meaningful IIRD by using the dataconstruct in the original version of the proof assistant Agda for MartinLöf 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 restricted one. 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.
Universes in Type Theory Part II – Autonomous Mahlo
, 2009
"... We introduce the autonomous Mahlo universe which is an extension of MartinLöf type theory which we consider as predicatively justified and which has a strength which goes substantially beyond that of the Mahlo universe, which is before writing this paper the strongest predicatively justified publis ..."
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We introduce the autonomous Mahlo universe which is an extension of MartinLöf type theory which we consider as predicatively justified and which has a strength which goes substantially beyond that of the Mahlo universe, which is before writing this paper the strongest predicatively justified published extension of MartinLöf type theory. We conjecture it to have the same proof theoretic strength as KripkePlatek set theory extended by one recursively autonomous Mahlo ordinal and finitely many admissibles above it. Here a recursively autonomous Mahlo universe ordinal is an ordinal κ which is recursively hyper αMahlo for all α < κ. We introduce as well as intermediate steps the hyperMahlo and hyper αMahlo universes, and give meaning explanations for these theories as well as for the super and the Mahlo universe. We introduce a model for the autonomous Mahlo universe, and determine an upper bound for its proof theoretic strength, therefore establishing one half of the conjecture mentioned before. The autonomous Mahlo universe is the crucial intermediate step for understanding the Π3reflecting universe, which will be published in a successor of this article and which is even stronger and will slightly exceed the strength of KripkePlatek set theory plus the principle of Π3reflection. 1