scholarly journals Inductive inference of recursive functions: Complexity bounds

2005 ◽  
pp. 111-155 ◽  
Author(s):  
Rūsiņš Freivalds ◽  
Jānis Bārzdiņš ◽  
Kārlis Podnieks
Keyword(s):  
Inductive Inference ◽  
Complexity Bounds ◽  
Recursive Functions

scholarly journals Reflective inductive inference of recursive functions

2008 ◽  
Vol 397 (1-3) ◽  
pp. 57-69 ◽  
Author(s):  
Gunter Grieser
Keyword(s):  
Inductive Inference ◽  
Recursive Functions

INDUCTIVE INFERENCE OF LIMITING PROGRAMS WITH BOUNDED NUMBER OF MIND CHANGES

1996 ◽  
Vol 07 (03) ◽  
pp. 187-208
Author(s):  
JURIS VĪKSNA
Keyword(s):  
Finite Number ◽  
Inductive Inference ◽  
Recursive Functions ◽  
Bounded Number

We consider inductive inference of total recursive functions in the case, when produced hypotheses are allowed some finite number of times to change “their mind” about each value of identifiable function. Such type of identification, which we call inductive inference of limiting programs with bounded number of mind changes, by its power lies somewhere between the traditional criteria of inductive inference and recently introduced inference of limiting programs. We consider such model of inductive inference for EX and BC types of identification, and we study • tradeoffs between the number of allowed mind changes and the number of anomalies, and • relations between classes of functions identifiable with different probabilities. For the case of probabilistic identification we establish probabilistic hierarchies which are quite unusual for EX and BC types of inference.

scholarly journals Plausible Reasoning about EL-Ontologies using Concept Interpolation

2020 ◽  
Author(s):  
Yazmín Ibáñez-García ◽  
Víctor Gutiérrez-Basulto ◽  
Steven Schockaert
Keyword(s):  
Computational Complexity ◽  
Knowledge Representation ◽  
Deductive Reasoning ◽  
Inductive Inference ◽  
Description Logics ◽  
Data Driven ◽  
Commonsense Reasoning ◽  
Inference Mechanism ◽  
Complexity Bounds ◽  
Reasoning Mechanism

Description logics (DLs) are standard knowledge representation languages for modelling ontologies, i.e. knowledge about concepts and the relations between them. Unfortunately, DL ontologies are difficult to learn from data and time-consuming to encode manually. As a result, ontologies for broad domains are almost inevitably incomplete. In recent years, several data-driven approaches have been proposed for automatically extending such ontologies. One family of methods rely on characterizations of concepts that are derived from text descriptions. While such characterizations do not capture ontological knowledge directly, they encode information about the similarity between different concepts that can be exploited for filling in the gaps in existing ontologies. To this end, several inductive inference mechanisms have already been proposed, but these have been defined and used in a heuristic fashion. In this paper, we instead propose an inductive inference mechanism which is based on a clear model-theoretic semantics, and can thus be tightly integrated with standard deductive reasoning. We particularly focus on interpolation, a powerful commonsense reasoning mechanism which is closely related to cognitive models of category-based induction. Apart from the formalization of the underlying semantics, as our main technical contribution we provide computational complexity bounds for reasoning in EL with this interpolation mechanism.

How is it that infinitary methods can be applied to finitary mathematics? Gödel's T: a case study

1998 ◽  
Vol 63 (4) ◽  
pp. 1348-1370 ◽  
Author(s):  
Andreas Weiermann
Keyword(s):  
Recursive Function ◽  
Term Rewriting ◽  
Careful Analysis ◽  
Recursion Theory ◽  
Cut Elimination ◽  
Complexity Bounds ◽  
Recursive Functions ◽  
Collapsing Function ◽  
Primitive Recursive

AbstractInspired by Pohlers' local predicativity approach to Pure Proof Theory and Howard's ordinal analysis of bar recursion of type zero we present a short, technically smooth and constructive strong normalization proof for Gödel's system T of primitive recursive functionals of finite types by constructing an ε0-recursive function []0: T → ω so that a reduces to b implies [a]0 > [b]0. The construction of [ ]0 is based on a careful analysis of the Howard-Schütte treatment of Gödel's T and utilizes the collapsing function ψ: ε0 → ω which has been developed by the author for a local predicativity style proof-theoretic analysis of PA. The construction of [ ]0 is also crucially based on ideas developed in the 1995 paper “A proof of strongly uniform termination for Gödel's T by the method of local predicativity” by the author. The results on complexity bounds for the fragments of T which are obtained in this paper strengthen considerably the results of the 1995 paper.Indeed, for given n let Tn be the subsystem of T in which the recursors have type level less than or equal to n + 2. (By definition, case distinction functionals for every type are also contained in Tn.) As a corollary of the main theorem of this paper we obtain (reobtain?) optimal bounds for the Tn-derivation lengths in terms of ω+2-descent recursive functions. The derivation lengths of type one functionals from Tn (hence also their computational complexities) are classified optimally in terms of <ωn+2 -descent recursive functions.In particular we obtain (reobtain?) that the derivation lengths function of a type one functional a ∈ T0 is primitive recursive, thus any type one functional a in T0 defines a primitive recursive function. Similarly we also obtain (reobtain?) a full classification of T1 in terms of multiple recursion.As proof-theoretic corollaries we reobtain the classification of the IΣn+1-provably recursive functions. Taking advantage from our finitistic and constructive treatment of the terms of Gödel's T we reobtain additionally (without employing continuous cut elimination techniques) that PRA + PRWO(ε0) ⊢ Π20 − Refl(PA) and PRA + PRWO(ωn+2) ⊢ Π20 − Refl(IΣn+1), hence PRA + PRWO(ε0) ⊢ Con(PA) and PRA + PRWO(ωn+2) ⊢ Con(IΣn+1).For programmatic reasons we outline in the introduction a vision of how to apply a certain type of infinitary methods to questions of finitary mathematics and recursion theory. We also indicate some connections between ordinals, term rewriting, recursion theory and computational complexity.

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