Jacek Cichoń

Decomposing Baire functions

We discuss in the paper the following problem: Given a function in a given Baire class, into “how many” (in terms of cardinal numbers) functions of lower classes can it be decomposed? The decomposition is understood here in the sense of the set-theoretical union.

On Ideals of Subsets of the Plane and on Cohen Reals

Let be any proper ideal of subsets of the real line R which contains all finite subsets of R . We define an ideal * ∣ as follows: X ∈ * ∣ if there exists a Borel set B ⊂ R × R such that X ⊂ B and for any x ∈ R we have { y ∈ R : 〈 x, y 〉 ∈ B } ∈ . We show that there exists a family ⊂ * ∣ of power ω 1 such that ⋃ ∉ * ∣ . In the last section we investigate properties of ideals of Lebesgue measure zero sets and meager sets in Cohen extensions of models of set theory.

Two-phase cardinality estimation protocols for sensor networks with provable precision

Efficient cardinality estimation is a common requirement for many wireless sensor network (WSN) applications. The task must be accomplished at extremely low overhead due to severe sensor resource limitation. This poses an interesting challenge for large-scale WSNs. In this paper we present a two-phase probabilistic algorithm based on order statistics and Bernoulli scheme, which effectively estimates the cardinality of WSNs. We thoroughly examine properties of estimators used in each phase as well as the precision of the whole procedure. The algorithm discussed in this paper is a modification of a recently published idea - the modification enables us to obtain a provable precision.

On cardinality estimation protocols for wireless sensor networks

In this article we address the problem of estimating a size of wireless sensor networks (WSNs). We restrict our attention to sensors with very limited storage capabilities. The problem arises when sensors have to quickly obtain approximate size of the network to use algorithms which require such information. Another application area is the problem of counting the number of different objects (e.g. people in public bus transportation) and use of such information to optimize the routes and frequency of buses. In this paper we present two-phase probabilistic algorithm based on order statistics and balls-bins model which effectively solves the presented problem.

Privacy Protection for RFID with Hidden Subset Identifiers

We propose very simple and cheap but nevertheless effective protection against privacy threats for RFID-tags. For the hidden subsetRFID-tags proposed in this paper, the ID string presented by an RFID-tag evolves rapidly. It is is not the bit value that enables one to recognize a tag. Instead, a reader detects some invariant properties that are hard to be recognized by a curious illegitimate reader. The solution is not based on any cryptographic primitive, it relies only on properties of random sets and on linear mappings between vector spaces. The solution proposed is well suited for low-end devices, since all mechanisms can be easily implemented by circuits of a small size.

Minimal Büchi Automata for Certain Classes of LTL Formulas

In this paper we calculate the minimal number of states of Buchi automata which encode some classes of linear temporal logic (LTL) formulas that are frequently used in model checking. Our results may be used for verification of the quality of algorithms which automatically translate LTL formulas into Buchi automata and for improving the quality and speed of such translators. In the last section of this paper we compare our lower-bound estimations to Buchi automata generated by two currently used translators: LTL2BA and SPOT.

Generating continuous mappings with Lipschitz mappings

If X is a metric space, then C X and £ X denote the semigroups of continuous and Lipschitz mappings, respectively, from X to itself. The relative rank of C X modulo £ X is the least cardinality of any set U £ X where U generates C X . For a large class of separable metric spaces X we prove that the relative rank of C X modulo £ X is uncountable. When X is the Baire space N N , this rank is 8 1 . A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.

How to Improve the Reliability of Chord

In this paper we focus on Chord P2P protocol and we study the process of unexpected departures of nodes from this system. Each of such departures may effect in losing any information and in classical versions of this protocol the probability of losing some information is proportional to the quantity of information put into this system. This effect can be partially solved by gathering in the protocol multiple copies (replicas) of information. The replication mechanism was proposed by many authors. We present a detailed analysis of one variant of blind replication and show that this solution only partially solves the problem. Next we propose two less obvious modifications of the Chord protocol. We call the first construction a direct sums of Chords and the second - a folded Chord. We discuss the recovery mechanisms of partially lost information in each of these systems and investigate their reliability. We show that our modification increases essentially the expected lifetime of information put into the system. Our modifications of the Chord protocol are very soft and require only a small interference in the programming code of the original Chord protocol.

On two-cardinal properties of ideals

We investigate two-cardinal properties of ideals. These properties involve notions such as Luzin sets, special coverings, etc. We apply our results to the ideals of meagre sets and of negligible sets in the real line. In case of the negligible sets, we relate these properties to caliber and precalibers of the measure algebra.

On dense subsets of the measure algebra

We show that the minimal cardinality of a dense subset of the measure algebra is the same as the minimal cardinality of a base of the ideal of Lebesgue measure zero subsets of the real line.