62 research outputs found
Linear Hashing is Awesome
We consider the hash function where
are chosen uniformly at random from . We prove that when we
use in hashing with chaining to insert elements into a table of size
the expected length of the longest chain is
. The proof also generalises to give the same
bound when we use the multiply-shift hash function by Dietzfelbinger et al.
[Journal of Algorithms 1997].Comment: A preliminary version appeared at FOCS'1
Univalent Double Categories
Category theory is a branch of mathematics that provides a formal framework
for understanding the relationship between mathematical structures. To this
end, a category not only incorporates the data of the desired objects, but also
"morphisms", which capture how different objects interact with each other.
Category theory has found many applications in mathematics and in computer
science, for example in functional programming. Double categories are a natural
generalization of categories which incorporate the data of two separate classes
of morphisms, allowing a more nuanced representation of relationships and
interactions between objects. Similar to category theory, double categories
have been successfully applied to various situations in mathematics and
computer science, in which objects naturally exhibit two types of morphisms.
Examples include categories themselves, but also lenses, petri nets, and spans.
While categories have already been formalized in a variety of proof assistants,
double categories have received far less attention. In this paper we remedy
this situation by presenting a formalization of double categories via the proof
assistant Coq, relying on the Coq UniMath library. As part of this work we
present two equivalent formalizations of the definition of a double category,
an unfolded explicit definition and a second definition which exhibits
excellent formal properties via 2-sided displayed categories. As an application
of the formal approach we establish a notion of univalent double category along
with a univalence principle: equivalences of univalent double categories
coincide with their identitie
On the Termination Problem for Probabilistic Higher-Order Recursive Programs
In the last two decades, there has been much progress on model checking of
both probabilistic systems and higher-order programs. In spite of the emergence
of higher-order probabilistic programming languages, not much has been done to
combine those two approaches. In this paper, we initiate a study on the
probabilistic higher-order model checking problem, by giving some first
theoretical and experimental results. As a first step towards our goal, we
introduce PHORS, a probabilistic extension of higher-order recursion schemes
(HORS), as a model of probabilistic higher-order programs. The model of PHORS
may alternatively be viewed as a higher-order extension of recursive Markov
chains. We then investigate the probabilistic termination problem -- or,
equivalently, the probabilistic reachability problem. We prove that almost sure
termination of order-2 PHORS is undecidable. We also provide a fixpoint
characterization of the termination probability of PHORS, and develop a sound
(but possibly incomplete) procedure for approximately computing the termination
probability. We have implemented the procedure for order-2 PHORSs, and
confirmed that the procedure works well through preliminary experiments that
are reported at the end of the article
Efficient computer-aided verification of parallel and distributed software systems
The society is becoming increasingly dependent on applications of distributed software systems, such as controller systems and wireless telecommunications. It is very difficult to guarantee the correct operation of this kind of systems with traditional software quality assurance methods, such as code reviews and testing. Formal methods, which are based on mathematical theories, have been suggested as a solution. Unfortunately, the vast complexity of the systems and the lack of competent personnel have prevented the adoption of sophisticated methods, such as theorem proving.
Computerised tools for verifying finite state asynchronous systems exist, and they been successful on locating errors in relatively small software systems. However, a direct translation of software to low-level formal models may lead to unmanageably large models or complex behaviour. Abstract models and algorithms that operate on compact high-level designs are needed to analyse larger systems.
This work introduces modelling formalisms and verification methods of distributed systems, presents efficient algorithms for verifying high-level models of large software systems, including an automated method for abstracting unneeded details from systems consisting of loosely connected components, and shows how the methods can be applied in the software development industry.reviewe
Confusion Control in Generalized Petri Nets Using Synchronized Events
The loss of conflicting information in a Petri net (PN), usually called confusions, leads to incomplete and faulty system behavior. Confusions, as an unfortunate phenomenon in discrete event systems modeled with Petri nets, are caused by the frequent interlacement of conflicting and concurrent transitions. In this paper, confusions are defined and investigated in bounded generalized PNs. A reasonable control strategy for conflicts and confusions in a PN is formulated by proposing elementary conflict resolution sequences (ECRSs) and a class of local synchronized Petri nets (LSPNs). Two control algorithms are reported to control the appeared confusions by generating a series of external events. Finally, an example of confusion analysis and control in an automated manufacturing system is presented
Informational Paradigm, management of uncertainty and theoretical formalisms in the clustering framework: A review
Fifty years have gone by since the publication of the first paper on clustering based on fuzzy sets theory. In 1965, L.A. Zadeh had published “Fuzzy Sets” [335]. After only one year, the first effects of this seminal paper began to emerge, with the pioneering paper on clustering by Bellman, Kalaba, Zadeh [33], in which they proposed a prototypal of clustering algorithm based on the fuzzy sets theory
- …