Lectures on Semantics: The Initial Algebra and Final Coalgebra Perspectives

1997 ◽  
pp. 1-33 ◽  
Author(s):  
Peter Aczel
Keyword(s):  
Initial Algebra

The initial algebra of maximal minors of a generalized hankel matrix

1999 ◽  
Vol 27 (1) ◽  
pp. 429-450
Author(s):  
Paulo F. Machado
Keyword(s):  
Hankel Matrix ◽  
Initial Algebra

Initial Algebra Semantics and Continuous Algebras

1977 ◽  
Vol 24 (1) ◽  
pp. 68-95 ◽  
Author(s):  
J. A. Goguen ◽  
J. W. Thatcher ◽  
E. G. Wagner ◽  
J. B. Wright
Keyword(s):  
Initial Algebra ◽  
Initial Algebra Semantics

scholarly journals Crisp-determinization of weighted tree automata over strong bimonoids

2021 ◽  
Vol vol. 23 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
Zoltán Fülöp ◽  
Dávid Kószó ◽  
Heiko Vogler
Keyword(s):  
Finite Order ◽  
Positive Answer ◽  
Order Property ◽  
Sufficient Condition ◽  
Tree Automata ◽  
Weighted Tree ◽  
Bottom Up ◽  
Initial Algebra ◽  
Initial Algebra Semantics ◽  
Weighted Tree Automata

We consider weighted tree automata (wta) over strong bimonoids and their initial algebra semantics and their run semantics. There are wta for which these semantics are different; however, for bottom-up deterministic wta and for wta over semirings, the difference vanishes. A wta is crisp-deterministic if it is bottom-up deterministic and each transition is weighted by one of the unit elements of the strong bimonoid. We prove that the class of weighted tree languages recognized by crisp-deterministic wta is the same as the class of recognizable step mappings. Moreover, we investigate the following two crisp-determinization problems: for a given wta ${\cal A}$, (a) does there exist a crisp-deterministic wta which computes the initial algebra semantics of ${\cal A}$ and (b) does there exist a crisp-deterministic wta which computes the run semantics of ${\cal A}$? We show that the finiteness of the Nerode algebra ${\cal N}({\cal A})$ of ${\cal A}$ implies a positive answer for (a), and that the finite order property of ${\cal A}$ implies a positive answer for (b). We show a sufficient condition which guarantees the finiteness of ${\cal N}({\cal A})$ and a sufficient condition which guarantees the finite order property of ${\cal A}$. Also, we provide an algorithm for the construction of the crisp-deterministic wta according to (a) if ${\cal N}({\cal A})$ is finite, and similarly for (b) if ${\cal A}$ has finite order property. We prove that it is undecidable whether an arbitrary wta ${\cal A}$ is crisp-determinizable. We also prove that both, the finiteness of ${\cal N}({\cal A})$ and the finite order property of ${\cal A}$ are undecidable.

scholarly journals Difficulties in initial algebra learning in Indonesia

2014 ◽  
Vol 26 (4) ◽  
pp. 683-710 ◽  
Author(s):  
Al Jupri ◽  
Paul Drijvers ◽  
Marja van den Heuvel-Panhuizen
Keyword(s):  
Initial Algebra ◽  
Algebra Learning

scholarly journals On Well-Founded and Recursive Coalgebras

2020 ◽  
pp. 17-36
Author(s):  
Jiří Adámek ◽  
Stefan Milius ◽  
Lawrence S. Moss
Keyword(s):  
General Proof ◽  
Initial Algebra ◽  
Equivalent Characterization ◽  
Algebra Morphism ◽  
Recursion Theorem ◽  
The Relationship ◽  
Well Foundedness

AbstractThis paper studies fundamental questions concerning category-theoretic models of induction and recursion. We are concerned with the relationship between well-founded and recursive coalgebras for an endofunctor. For monomorphism preserving endofunctors on complete and well-powered categories every coalgebra has a well-founded part, and we provide a new, shorter proof that this is the coreflection in the category of all well-founded coalgebras. We present a new more general proof of Taylor’s General Recursion Theorem that every well-founded coalgebra is recursive, and we study conditions which imply the converse. In addition, we present a new equivalent characterization of well-foundedness: a coalgebra is well-founded iff it admits a coalgebra-to-algebra morphism to the initial algebra.

scholarly journals Dual Number Meadows

2019 ◽  
Author(s):  
Jan Aldert Bergstra
Keyword(s):  
Characteristic Zero ◽  
Equational Theory ◽  
Rational Numbers ◽  
Zero Element ◽  
Open Problems ◽  
Dual Number ◽  
Initial Algebra

The class of dual number meadows is introduced. By definition this class is a quasivariety. Dual number meadows contain a non-zero element the square of which is zero. These structures are non-involutive and coregular. Some properties of the equational theory of dual number meadows are discussed and an initial algebra specification is given for the minimal dual number meadow of characteristic zero which contains the dual rational numbers. Several open problems are stated.

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