Initial algebra semantics for lambda calculi

2006 ◽  
pp. 418-439
Author(s):  
John W. Gray
Keyword(s):  
Initial Algebra ◽  
Initial Algebra Semantics

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 Initial Algebra Semantics Is Enough!

2007 ◽  
pp. 207-222 ◽  
Author(s):  
Patricia Johann ◽  
Neil Ghani
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 The Transrational Numbers as an Abstract Data Type

2020 ◽  
Author(s):  
Jan Aldert Bergstra ◽  
John V. Tucker
Keyword(s):  
Rational Numbers ◽  
Data Type ◽  
Abstract Data Type ◽  
Constant Symbol ◽  
Total Function ◽  
Initial Algebra ◽  
Abstract Data ◽  
Initial Algebra Semantics

In an arithmetical structure one can make division a total function by defining 1/0 to be an element of the structure, or by adding a new element, such as an error element also denoted with a new constant symbol, an unsigned infinity or one or both signed infinities, one positive and one negative. We define an enlargement of a field to a transfield, in which division is totalised by setting 1/0 equal to the positive infinite value and -1/0 equal to its opposite, and which also contains an error element to help control their effects. We construct the transrational numbers as a transfield of the field of rational numbers and consider it as an abstract data type. We give it an equational specification under initial algebra semantics.

scholarly journals Polytypic Functions Over Nested Datatypes

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Ralf Hinze
Keyword(s):  
Recent Work ◽  
Theory And Practice ◽  
Basic Knowledge ◽  
Central Idea ◽  
Underlying Theory ◽  
Initial Algebra ◽  
International Audience ◽  
Initial Algebra Semantics

International audience The theory and practice of polytypic programming is intimately connected with the initial algebra semantics of datatypes. This is both a blessing and a curse. It is a blessing because the underlying theory is beautiful and well developed. It is a curse because the initial algebra semantics is restricted to so-called regular datatypes. Recent work by R. Bird and L. Meertens [3] on the semantics of non-regular or nested datatypes suggests that an extension to general datatypes is not entirely straightforward. Here we propose an alternative that extends polytypism to arbitrary datatypes, including nested datatypes and mutually recursive datatypes. The central idea is to use rational trees over a suitable set of functor symbols as type arguments for polytypic functions. Besides covering a wider range of types the approach is also simpler and technically less involving than previous ones. We present several examples of polytypic functions, among others polytypic reduction and polytypic equality. The presentation assumes some background in functional and in polytypic programming. A basic knowledge of monads is required for some of the examples.

scholarly journals Abstract Semantic Algebras!

1982 ◽  
Vol 11 (145) ◽  
Author(s):  
Peter D. Mosses
Keyword(s):  
Programming Languages ◽  
Programming Language ◽  
Conceptual Analysis ◽  
Formal Description ◽  
Data Types ◽  
New Approach ◽  
Initial Algebra ◽  
Language Semantics ◽  
Abstract Data ◽  
Initial Algebra Semantics

A new approach to the formal description of programming language semantics is described and illustrated. ''Abstract semantic algebras'' are just algebraically-specified abstract data types whose operations correspond to fundamental concepts of programming languages. The values of abstract semantic algebras are taken as meanings of programs in Denotational (or Initial Algebra) Semantics, instead of using Scott domains. This leads to semantic descriptions that clearly exhibit the underlying conceptual analysis, and which are rather easy to modify and extend. Some basic abstract semantic algebras corresponding to fundamental concepts of programming languages are given; they could be used in the description of many different programming languages.

scholarly journals Factorising folds for faster functions

2010 ◽  
Vol 20 (3-4) ◽  
pp. 353-373 ◽  
Author(s):  
GRAHAM HUTTON ◽  
MAURO JASKELIOFF ◽  
ANDY GILL
Keyword(s):  
Fixed Point ◽  
New Technique ◽  
General Technique ◽  
Initial Algebra ◽  
Recursive Programs ◽  
Fixed Point Semantics ◽  
Initial Algebra Semantics ◽  
Structured Approach

AbstractThe worker/wrapper transformation is a general technique for improving the performance of recursive programs by changing their types. The previous formalisation (A. Gill & G. Hutton, J. Funct. Program., vol. 19, 2009, pp. 227–251) was based upon a simple fixed-point semantics of recursion. In this paper, we develop a more structured approach, based upon initial-algebra semantics. In particular, we show how the worker/wrapper transformation can be applied to programs defined using the structured pattern of recursion captured by fold operators, and illustrate our new technique with a number of examples.

Initial algebra semantics

1974 ◽  
Author(s):  
J. A. Goguen ◽  
J. W. Thatcher
Keyword(s):  
Initial Algebra ◽  
Initial Algebra Semantics
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