179 research outputs found
Fixpoint Theorem for Continuous Functions on Chain-Complete Posets
This text includes the definition of chain-complete poset, fix-point theorem on it, and the definition of the function space of continuous functions on chain-complete posets [10].Ishida Kazuhisa - Neyagawa-shi, Osaka, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. Bounds in posets and relational substructures. Formalized Mathematics, 6(1):81-91, 1997.Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesĆaw ByliĆski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesĆaw ByliĆski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Adam Grabowski. On the category of posets. Formalized Mathematics, 5(4):501-505, 1996. http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=000258624500003&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3Piotr Rudnicki and Andrzej Trybulec. Abian's fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma. Formalized Mathematics, 1(2):387-393, 1990.Glynn Winskel. The Formal Semantics of Programming Languages. The MIT Press, 1993.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85-89, 1990.Mariusz Ć»ynel and CzesĆaw ByliĆski. Properties of relational structures, posets, lattices and maps. Formalized Mathematics, 6(1):123-130, 1997
A point on fixpoints in posets
Let be a {\em non-empty strictly inductive poset}, that is, a
non-empty partially ordered set such that every non-empty chain has a least
upper bound lub, a chain being a subset of totally ordered by
. We are interested in sufficient conditions such that, given an element
and a function f:X\a X, there is some ordinal such that
, where is the transfinite sequence of iterates of
starting from (implying that is a fixpoint of ):
\begin{itemize}\itemsep=0mm \item \item a_l=\lub\{a_k\mid k
\textless{} l\} if is a limit ordinal, i.e. \end{itemize}
This note summarizes known results about this problem and provides a slight
generalization of some of them
A Fibrational Approach to Automata Theory
For predual categories C and D we establish isomorphisms between opfibrations
representing local varieties of languages in C, local pseudovarieties of
D-monoids, and finitely generated profinite D-monoids. The global sections of
these opfibrations are shown to correspond to varieties of languages in C,
pseudovarieties of D-monoids, and profinite equational theories of D-monoids,
respectively. As an application, we obtain a new proof of Eilenberg's variety
theorem along with several related results, covering varieties of languages and
their coalgebraic modifications, Straubing's C-varieties, fully invariant local
varieties, etc., within a single framework
Generalizing the Paige-Tarjan Algorithm by Abstract Interpretation
The Paige and Tarjan algorithm (PT) for computing the coarsest refinement of
a state partition which is a bisimulation on some Kripke structure is well
known. It is also well known in model checking that bisimulation is equivalent
to strong preservation of CTL, or, equivalently, of Hennessy-Milner logic.
Drawing on these observations, we analyze the basic steps of the PT algorithm
from an abstract interpretation perspective, which allows us to reason on
strong preservation in the context of generic inductively defined (temporal)
languages and of possibly non-partitioning abstract models specified by
abstract interpretation. This leads us to design a generalized Paige-Tarjan
algorithm, called GPT, for computing the minimal refinement of an abstract
interpretation-based model that strongly preserves some given language. It
turns out that PT is a straight instance of GPT on the domain of state
partitions for the case of strong preservation of Hennessy-Milner logic. We
provide a number of examples showing that GPT is of general use. We first show
how a well-known efficient algorithm for computing stuttering equivalence can
be viewed as a simple instance of GPT. We then instantiate GPT in order to
design a new efficient algorithm for computing simulation equivalence that is
competitive with the best available algorithms. Finally, we show how GPT allows
to compute new strongly preserving abstract models by providing an efficient
algorithm that computes the coarsest refinement of a given partition that
strongly preserves the language generated by the reachability operator.Comment: Keywords: Abstract interpretation, abstract model checking, strong
preservation, Paige-Tarjan algorithm, refinement algorith
Data types
A Mathematical
interpretation is given to the notion of a data type.
The main novelty is in the generality of the mathematical treatment
which allows procedural data types and circularly defined data types.
What is meant by data type is pretty close to what any computer
scientist would understand by this term or by data structure, type,
mode, cluster, class. The mathematical treatment is the conjunction
of the ideas of D. Scott on the solution of domain equations (Scott
(71), (72) and (76)) and the initiality property noticed by the
ADJ group (ADJ (75), ADJ (77)). The present work adds operations
to the data types proposed by Scott and generalizes the data types
of ADJ to procedural types and arbitrary circular type definitions.
The advantages of a mathematical interpretation of data types are
those of mathematical semantics in general : throwing light on some
ill-understood constructs in high-level programming languages, easing
the task of writing correct programs and making possible proofs of
correctness for programs or implementations"
Approximating least fixpoints
I try to come up with general techniques for approximating least ïŹxpoints from below and greatest ïŹxpoints from above. In this, I try to place as few restrictions as possible on the underlying partial orders; in particular, I avoid the use of linear orders. The approach is intended to abstract and thus generalise approximation techniques used in [10,1]. I hope that I am not falling into Einsteinâs trap with this note
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