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Inductive-Inductive Definitions [chapter]

Fredrik Nordvall Forsberg, Anton Setzer
2010 Lecture Notes in Computer Science  
Compared to indexed inductive definitions, the novelty is that the index set A is generated inductively simultaneously with B.  ...  In this new principle a set A is defined inductively simultaneously with an A-indexed set B, which is also defined inductively.  ...  - inductive definitions.  ... 
doi:10.1007/978-3-642-15205-4_35 fatcat:hrog3qrvhfhkdkon3o3yqxvg3q

Partial inductive definitions

Lars Hallnäs
1991 Theoretical Computer Science  
We will in a certain sense think of such data types as inductively defined. A class of inductive definitions will be interpreted as partial definitions: partial inductive definitions.  ...  ., Partial inductive definitions, Theoretical Computer Science 87 (1991) 115-142. An attempt to consider partial definitions of semantically oriented data types will be described.  ...  It is easy to see that definitions of level <~1 are all total. So ordinary monotone inductive definitions are the canonical examples of total inductive definitions.  ... 
doi:10.1016/s0304-3975(06)80007-1 fatcat:uusnzfuyifdrzekgl2ohqknz2a

Elaborating Inductive Definitions [article]

Pierre-Evariste Dagand, Conor McBride
2012 arXiv   pre-print
about inductive definitions could be carried in the type theory, not in the meta-theory as it is usually the case.  ...  By elaborating an inductive definition -- a syntactic artifact -- to its code -- its semantics -- we obtain an internalized account of inductives inside the type theory itself: we claim that reasoning  ...  Acknowledgements We would like to thank Pierre Boutillier for pointing us to the relevant literature on Coq's treatment of inductives.  ... 
arXiv:1210.6390v2 fatcat:oe7zdyq4zffunbyqtvrpkwvgmu

A Categorical Semantics for Inductive-Inductive Definitions [chapter]

Thorsten Altenkirch, Peter Morris, Fredrik Nordvall Forsberg, Anton Setzer
2011 Lecture Notes in Computer Science  
This covers many forms of induction and data type definitions such as indexed inductive definitions [5] and induction-recursion [7] (Dybjer and Setzer [8] also give a direct proof for induction-recursion  ...  This gives a new and compact formalisation of inductive-inductive definitions, which we prove is equivalent to the usual formulation with elimination rules.  ...  The Generic Eliminator for an Inductive-Inductive Definition Let us now generalise the preceding discussion from inductive definitions (i.e. endofunctors on Set) to inductive-inductive definitions (i.e  ... 
doi:10.1007/978-3-642-22944-2_6 fatcat:hn6wrqk65rfrxjj3kgs2ohqx2q

Positive Inductive-Recursive Definitions

Neil Ghani, Fredrik Nordvall Forsberg, Lorenzo Malatesta, Stefan Milius
2015 Logical Methods in Computer Science  
This theory, which we call positive inductive-recursive definitions, is a generalisation of Dybjer and Setzer's theory of inductive-recursive definitions within which C had to be discrete -- our work can  ...  Finally we justify the existence of positive inductive-recursive definitions by adapting Dybjer and Setzer's set-theoretic model to our setting.  ...  Inductive-recursive definitions In increasing complexity and sophistication, inductive definitions, indexed inductive definitions and inductive-recursive definitions encode more and more information about  ... 
doi:10.2168/lmcs-11(1:13)2015 fatcat:2ubrqz3hh5fgxecftuxqctvife

Non-deterministic inductive definitions [article]

Benno van den Berg
2012 arXiv   pre-print
We study a new proof principle in the context of constructive Zermelo-Fraenkel set theory based on what we will call "non-deterministic inductive definitions".  ...  Suppose R is a non-deterministic inductive definition on a set X. If * is an element not in X, then R can also be considered as a non-deterministic inductive definition on X ∪ { * }.  ...  Then consider the following deterministic inductive definition on M: {t(b) : b ∈ B a } {sup a t} a ∈ A, t: B a → M By Proposition 3.6, this inductive definition has a least fixed point.  ... 
arXiv:1104.2744v2 fatcat:ivqrqi6trvcnjg7vh7xhwottgi

Non-deterministic inductive definitions

Benno van den Berg
2012 Archive for Mathematical Logic  
We study a new proof principle in the context of constructive Zermelo-Fraenkel set theory based on what we will call "non-deterministic inductive definitions".  ...  Suppose R is a non-deterministic inductive definition on a set X. If * is an element not in X, then R can also be considered as a non-deterministic inductive definition on X ∪ { * }.  ...  Then consider the following deterministic inductive definition on M: {t(b) : b ∈ B a } {sup a t} a ∈ A, t: B a → M By Proposition 3.6, this inductive definition has a least fixed point.  ... 
doi:10.1007/s00153-012-0309-4 fatcat:x23a2m2rerhg3are4ggksstztu

Inductive Definitions and Computability

Thomas J. Grilliot
1971 Transactions of the American Mathematical Society  
Sets inductively defined with respect to Il0, Sx, (nonmonotonic) II t and S2 predicates are characterized in terms of the four chief notions of abstract recursion.  ...  Add to the inductive definition of S the following clauses : Theorem 3.  ...  These problems are handled by adding some auxiliary clauses to the inductive definition ofS. Theorem 2.  ... 
doi:10.2307/1995907 fatcat:6zzauhczcfglhhxi2hmfg65pxi

Positive Inductive-Recursive Definitions [chapter]

Neil Ghani, Lorenzo Malatesta, Fredrik Nordvall Forsberg
2013 Lecture Notes in Computer Science  
This theory, which we call positive inductive-recursive definitions, is a generalisation of Dybjer and Setzer's theory of inductive-recursive definitions within which C had to be discrete -our work can  ...  We then give an application of positive inductive-recursive definitions to the theory of nested data types.  ...  This leads us to consider a new variation IR + of inductive-recursive definitions which we call positive inductive-recursive definitions.  ... 
doi:10.1007/978-3-642-40206-7_3 fatcat:bmbq64zvbzdzjaj45xl5fyfbje

Functional interpretation and inductive definitions [article]

Jeremy Avigad, Henry Towsner
2009 arXiv   pre-print
Extending Gödel's Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals  ...  Iterating the interpretation In this section, we consider theories ID n of finitely iterated inductive definitions.  ...  The interpretation can be further extended to theories of transfinitely iterated inductive definitions, as described in [9] .  ... 
arXiv:0802.1938v4 fatcat:yxwafhc5rze7zcijmmxaidtbx4

Structuring Metatheory on Inductive Definitions

David Basin, Seán Matthews
2000 Information and Computation  
We show how metatheories based on inductive definitions allow theories and general metatheorems to be organized this way and report on a case study using the theory FS 0 . ]  ...  INDUCTIVE DEFINITIONS AND THE THEORY FS 0 We use Feferman's FS 0 for our work since it is a particularly simple minimal theory of inductive definitions.  ...  Other Metatheories Supporting Inductive Definitions We have indicated how parameterized inductive definitions in FS 0 support structured metatheoretic development.  ... 
doi:10.1006/inco.2000.2858 fatcat:am3mm6zv2vegjf4syvnx2c7ifu

Inductive definitions over finite structures

Daniel Leivant
1990 Information and Computation  
INTRODUCTION Inductive definitions have played a central role in the foundations of mathematics for over a century.  ...  The inductive closure, F", of an arbitrary operator F can be defined as the union of the increasing chain F<') = Df uj< i F[F("'], If F is inductive, then FCil = F<'> (by induction on i), so this definition  ... 
doi:10.1016/0890-5401(90)90006-4 fatcat:s4rvi2bbwfd7hin4ku4jldgl7y

Elaborating Inductive Definitions and Course-of-Values Induction in Cedille [article]

Christopher Jenkins, Colin McDonald, Aaron Stump
2019 arXiv   pre-print
This datatype subsystem and elaborator are implemented in Cedille, establishing for the first time a complete translation of inductive definitions to a small pure typed λ-calculus.  ...  In the Calculus of Dependent Lambda Eliminations (CDLE), a pure Curry-style type theory, it is possible to generically λ-encode inductive datatypes which support course-of-values (CoV) induction.  ...  Inductive-recursive and inductive-inductive definitions are not part of this proposal.  ... 
arXiv:1903.08233v3 fatcat:rgzfv32azfcwfmhrfi5kv5mwbe

Induction and inductive definitions in fragments of second order arithmetic

Klaus Aehlig
2005 Journal of Symbolic Logic (JSL)  
The amount of induction on arithmetical formulae only affects the arithmetical consequences of these theories, whereas adding induction for arbitrary formulae increases the strength by one inductive definition  ...  A fragment with the same provably recursive functions as n iterated inductive definitions is obtained by restricting second order arithmetic in the following way.  ...  Induction along the inductive Definition 5.2 of d n,(K) Γ = ∀X n A(X n ).  ... 
doi:10.2178/jsl/1129642116 fatcat:qs6hrkuxlvc2hamlvy6foqdwfa

Proof-theoretic strengths of weak theories for positive inductive definitions [article]

Toshiyasu Arai
2018 arXiv   pre-print
definitions over natural numbers.  ...  Conjunctions of negative and positive formulas in the transfinite induction axiom of ID_1 are shown to be weak, and disjunctions are strong.  ...  Inductive definition of H ⊢ a d Γ. Let Γ be a sequent, a < Ω · ε 0 and d ≤ 3.  ... 
arXiv:1603.01342v7 fatcat:sddiwn2iunenlje3fjcsfeqmzi
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