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An algebraic view of structural induction [chapter]

Claudio Hermida, Bart Jacobs
1995 Lecture Notes in Computer Science  
B over B , we consider a logical predicate lifting 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  ... 
doi:10.1007/bfb0022272 fatcat:mffbyj4ohrcafind7abxlrpbda

Reasoning about modular datatypes with Mendler induction

Paolo Torrini, Tom Schrijvers
2015 Electronic Proceedings in Theoretical Computer Science  
In functional programming, datatypes a la carte provide a convenient modular representation 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.  ... 
doi:10.4204/eptcs.191.13 fatcat:hjx4q5zskrf35or3j7xx5rjt2q

Equivariant Syntax and Semantics [chapter]

Andrew M. Pitts
2002 Lecture Notes in Computer Science  
The notion 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.  ... 
doi:10.1007/3-540-45465-9_3 fatcat:j5yhzgv4sbhxndrjnfqfltbbim

Deductively Definable Logics of Induction

John D. Norton
2010 Journal of Philosophical Logic  
A broad class 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  ... 
doi:10.1007/s10992-010-9146-2 fatcat:5zz2mlhcpre4jpx4w7x5c3ou7y

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

Claudio Hermida, Bart Jacobs
1998 Information and Computation  
After giving 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  ... 
doi:10.1006/inco.1998.2725 fatcat:qwvx6yx3j5evdn7zrtdb6rox3i

Initial Algebra Semantics for Cyclic Sharing Structures [chapter]

Makoto Hamana
2009 Lecture Notes in Computer Science  
Since they can be naturally defined by 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.  ... 
doi:10.1007/978-3-642-02273-9_11 fatcat:ktvhen52arccxbxt7ytyspjjyq

What inductive explanations could not be

John Dougherty
2017 Synthese  
Thus, neither is 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.  ... 
doi:10.1007/s11229-017-1457-1 fatcat:7yzcb5hefzgkbfjrys3mkfuqty

Homotopy theory of the master equation package applied to algebra and geometry: a sketch of two interlocking programs [article]

Dennis Sullivan
2008 arXiv   pre-print
to describe the 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.  ... 
arXiv:0803.0588v1 fatcat:pklswzt54fetld5wohkqsw3y2a

On Sahlqvist theory for hybrid logics

Willem Conradie, Claudette Robinson
2015 Journal of Logic and Computation  
SQEMA based on order-theoretic and 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  ... 
doi:10.1093/logcom/exv045 fatcat:3vknyc43znd4zp7ohuo5uu2fuu

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  ... 

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.  ... 

Algebraically closed structures in Positive Logic [article]

Mohammed Belkasmi
2019 arXiv   pre-print
In this paper we extend 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).  ... 
arXiv:1911.02997v1 fatcat:hwolrlt75jaerpiofeuvdkbtnq

An Axiomatic Theory of Inductive Inference

Luciano Pomatto, Alvaro Sandroni
2018 Philosophy of Science  
This paper develops 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.  ... 
doi:10.1086/696386 fatcat:mn4axo3rcjhdvmxsx2opeauygy

Polarized Process Algebra and Program Equivalence [chapter]

Jan A. Bergstra, Inge Bethke
2003 Lecture Notes in Computer Science  
It becomes possible to find 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.  ... 
doi:10.1007/3-540-45061-0_1 fatcat:tkqu5qhoqrdbzmrgrouz3bku5m
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