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An algebraic view of structural induction
[chapter]

1995
*
Lecture Notes in Computer Science
*

B over B , we consider a logical predicate lifting

doi:10.1007/bfb0022272
fatcat:mffbyj4ohrcafind7abxlrpbda
*of*T to the total category P. Then, a predicate is*inductive*precisely when it carries*an**algebra**structure*for such lifted endofunctor. ... We propose a uniform, category-theoretic account*of**structural**induction*for*inductively*de ned data types. ... The key point is that, given a T-*algebra*TX x / / X and a predicate P on X, i.e. pP = X, P is*inductive*, meaning that it sati es the premise*of*the*structural**induction*principle for the`type*structure*...##
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Reasoning about modular datatypes with Mendler induction

2015
*
Electronic Proceedings in Theoretical Computer Science
*

In functional programming, datatypes a la carte provide a convenient modular representation

doi:10.4204/eptcs.191.13
fatcat:hjx4q5zskrf35or3j7xx5rjt2q
*of*recursive datatypes, based on their initial*algebra*semantics. ... The known work-around*of*impredicative encodings is problematic, insofar as it impedes conventional*inductive*reasoning. ... The writing*of*this paper has been supported by EU funding (Horizon 2020, grant 640954) to KU Leuven for the GRACEFUL project. ...##
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Equivariant Syntax and Semantics
[chapter]

2002
*
Lecture Notes in Computer Science
*

The notion

doi:10.1007/3-540-45465-9_3
fatcat:j5yhzgv4sbhxndrjnfqfltbbim
*of*symmetry in mathematical*structures*is a powerful tool in many branches*of*mathematics. The talk presents*an*application*of*this notion to programming language theory. ... The initial*algebra*property also has useful logical consequences, since it gives rise to principles*of**structural*recursion and*induction*that are fundamental for proving properties*of*the programming ... In doing so, one looses the initial*algebra*property and hence looses automatically generated principles*of**structural*recursion/*induction*that apply directly to parse trees modulo α-equivalence. ...##
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Deductively Definable Logics of Induction

2010
*
Journal of Philosophical Logic
*

A broad class

doi:10.1007/s10992-010-9146-2
fatcat:5zz2mlhcpre4jpx4w7x5c3ou7y
*of**inductive*logics that includes the probability calculus is defined by the conditions that the*inductive*strengths [A|B] are defined fully in terms*of*deductive relations in preferred partitions ... If the presence*of*preferred partitions is not presumed, no*inductive*logic is definable. This no-go result precludes many possible*inductive*logics, including versions*of*hypothetico-deductivism. ... Acknowledgements For helpful discussion, I thank the Visiting Fellows in the Center for Philosophy*of*Science in Notes ...##
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Page 6993 of Mathematical Reviews Vol. , Issue 96k
[page]

1996
*
Mathematical Reviews
*

Summary: “We formalise proofs by

*structural**induction*on sets, specified as*an**algebraic*data type. ... Within*an*appropriate framework*of*dynamic*structures*(called d-oids), which play the same role as*algebras*in the static case, we define a language*of*dynamic terms, also enjoying the property*of*unique ...##
###
Structural Induction and Coinduction in a Fibrational Setting

1998
*
Information and Computation
*

After giving

doi:10.1006/inco.1998.2725
fatcat:qwvx6yx3j5evdn7zrtdb6rox3i
*an*alternative formulation*of**induction*in terms*of*binary relations, we combine both principles and obtain a mixed*induction*/coinduction principle which allows us to reason about minimal ... Our main results provide su cient criteria for the validity*of*such principles: in the presence*of*comprehension, the*induction*principle for initial*algebras*is admissible, and dually, in the presence ... In our analysis, validity*of*this*induction*principle follows from comprehension: the*algebra**structure*on P in a category*of*predicates can be transferred to*an**algebra**structure*on the associated set ...##
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Initial Algebra Semantics for Cyclic Sharing Structures
[chapter]

2009
*
Lecture Notes in Computer Science
*

Since they can be naturally defined by

doi:10.1007/978-3-642-02273-9_11
fatcat:ktvhen52arccxbxt7ytyspjjyq
*an**inductive*type, they offer data*structures*in functional programming and mechanised reasoning with useful principles such as*structural**induction*and*structural*... In the case*of*graphs or "tree-like" structurestrees involving cycles and sharing -however, it is not clear what kind*of**inductive**structures*exists and how we can faithfully assign a term representation ... The basis for this work, which was motivated by a question*of*Zhenjiang Hu, was done while the author visited IPL, University*of*Tokyo during April 2007 -March 2008. ...##
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What inductive explanations could not be

2017
*
Synthese
*

Thus, neither is

doi:10.1007/s11229-017-1457-1
fatcat:7yzcb5hefzgkbfjrys3mkfuqty
*an*explanation. I argue that there is no important asymmetry between the two cases because they are two presentations*of*the same explanation. ... This criterion can be expressed in two equivalent ways: one uses the language*of*homotopy type theory, and the second assigns*algebraic*representatives to proofs. ... and its contents are solely the responsibility*of*the authors and do not necessarily represent the official*views**of*the John Templeton Foundation. ...##
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Homotopy theory of the master equation package applied to algebra and geometry: a sketch of two interlocking programs
[article]

2008
*
arXiv
*
pre-print

to describe the

arXiv:0803.0588v1
fatcat:pklswzt54fetld5wohkqsw3y2a
*algebraic**structure*and one says that B has the*algebraic**structure*in question. ... We sketch two general applications: I to the theory*of*the definition and homotopy theory*of*infinity versions*of*general*algebraic**structures*including noncompact frobenius*algebras*and Lie bialgebras ... Forming composition with the resolution map associates with one particular instance*of**an**algebraic**structure*a particular instance*of**an*infinity*algebraic**structure*. ...##
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On Sahlqvist theory for hybrid logics

2015
*
Journal of Logic and Computation
*

SQEMA based on order-theoretic and

doi:10.1093/logcom/exv045
fatcat:3vknyc43znd4zp7ohuo5uu2fuu
*algebraic*insights, which effectively computes first-order correspondents*of*input formulas/inequalities, and is guaranteed to succeed on the Sahlqvist and*inductive*... Dating back to [20, 21] , the Sahlqvist theorem gives a syntactic definition*of*a class*of*modal formulas, the Sahlqvist class, each member*of*which defines*an*elementary (i.e. first-order definable) class ...*algebra**of*regular open subsets B RO (when*viewing*the possibility frame as a possibility frame itself) and the Boolean*algebra**of*arbitrary subsets When it comes to canonicity, we use the fact that filter-canonicity ...##
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Page 8926 of Mathematical Reviews Vol. , Issue 2004k
[page]

2004
*
Mathematical Reviews
*

For a family

*of*Hilbert A-bimodules { X; } endowed with*an**inductive*system, they first show the existence*of**an**inductive*limit*of*such a family. ... (E) = Phy E,.dx can be*viewed*as continuous analogues*of*Fock spaces, and their associated C*-*algebras*C*(E) generated by {W,W; | f,g € L'(E)} can be*viewed*as analogues*of*the Toeplitz extensions*of*the ...##
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Page 624 of Mathematical Reviews Vol. , Issue 91B
[page]

1991
*
Mathematical Reviews
*

*An*implicative

*algebra*

*of*finite predicates

*of*arbitrary order. (Russian) Problems

*of*bionics, No. 43 (Russian), 35-38, ““Vyshcha Shkola”’, Kharkov, 1989. ... for the signifi- cance

*of*specific mathematical

*structures*. ...

##
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Algebraically closed structures in Positive Logic
[article]

2019
*
arXiv
*
pre-print

In this paper we extend

arXiv:1911.02997v1
fatcat:hwolrlt75jaerpiofeuvdkbtnq
*of*the notion*of**algebraically*closed given in the case*of*groups and skew fields to*an*arbitrary h-*inductive*theory. ... The main subject*of*this paper is the study*of*the notion*of*positive*algebraic*closedness and its relationship with the notion*of*positive closedness and the amalgamation property. ... Given that A ≺ e C, then i•p is*an*immersion, so p is*an*immersion. By the proposition (4) we have T i (A) = T i (C) = T i (B) over the language L(A). ...##
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An Axiomatic Theory of Inductive Inference

2018
*
Philosophy of Science
*

This paper develops

doi:10.1086/696386
fatcat:mn4axo3rcjhdvmxsx2opeauygy
*an*axiomatic theory*of**induction*that speaks to the recent debate on Bayesian orgulity. ... We identify two types*of*disbelief about*induction*: skepticism that the existence*of*universal laws*of*Nature can be determined empirically, and skepticism that the true law*of*Nature, if it exists, can ... The*Structure*Theorem is a full characterization result that delivers*an*axiomatic understanding*of**induction*. ...##
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Polarized Process Algebra and Program Equivalence
[chapter]

2003
*
Lecture Notes in Computer Science
*

It becomes possible to find

doi:10.1007/3-540-45061-0_1
fatcat:tkqu5qhoqrdbzmrgrouz3bku5m
*an*axiomatization*of*program inequality. Technically this axiomatization is*an*infinite final*algebra*specification using conditional equations and auxiliary objects. ... In particular, the fully abstract model*of*the program*algebra*axioms*of*[2] is considered which results by working modulo behavioral congruence. ... This suggests the existence*of*a partial ordering and*an*operator which finitely approximates every basic process. Proof. We employ*structural**induction*. ...
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