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1

Dwa spojrzenia na nieskończoność

Samulewicz Alicja, Starosolski Andrzej

MINUT Matematyka i Informatyka na Uczelniach Technicznych

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2021

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Nr 3

52-60

PL

Z nieskończonością spotykamy się bardzo często w wielu działach matematyki. Celem tego artykułu jest przybliżenie Czytelnikowi, czym właściwie jest ta nieskończoność, a raczej czym są te nieskończoności.

2

Verification of equivalence with reference method for measurements of PM10 concentrations using low-cost devices

Owczarek Tomasz, Rogulski Mariusz, Czechowski Piotr O.

Zeszyty Naukowe Akademii Morskiej w Szczecinie

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2019

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nr 60 (132)

84--89

EN

This study presents an assessment of the equivalence of measurements of particulate matter PM10 concentrations using a low-cost electronic device as compared to the reference method. Data for the study were collected in accordance with the guidelines for research equivalence of the two devices operating in parallel. On this basis, a model correcting raw measurement results was developed. The best results were obtained for the model having the form of a second degree polynomial and taking into account air temperature. Corrected measurement results were used in the equivalence testing procedure. As a result, confirmation of equivalence was obtained for the vast majority of data sets generated from original measurements. This confirms the usefulness of the device as a tool for monitoring air quality.

3

From Box Algebra to Interval Temporal Logic

Klaudel Hanna, Koutny Maciej, Duan Zhenhua, Moszkowski Ben

Fundamenta Informaticae

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2019

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Vol. 167, nr 4

323--354

EN

In this paper, we further develop a recently introduced semantic link between temporal logics and Petri nets. We focus on two specific formalisms, Interval Temporal Logic (ITL) and Box Algebra (BA), which are closely related by their compositional approach to constructing system descriptions. The overall goal of our investigation is to translate Petri nets into behaviourally equivalent logical formulas. As a result, the analysis of system properties can be carried out using either of the two formalisms, exploiting their respective strengths and powerful tool support. The contribution of this paper is twofold. First, we extend the existing translation from BA to ITL, by removing restrictions concerning the way control flow of concurrent system is modelled, and by allowing a fully general synchronisation operator. Second, we strengthen the notion of equivalence between a Petri net and the corresponding logical formula by proving such an equivalence at the level of transition-based executions of Petri nets rather than just by looking at their labels. We also show that the complexity of the proposed translation compares favourably with the complexity of the translation from BA expressions to Petri nets.

4

A Note on Nonemptiness of Cognitive Semantics for Linguistic Representations of Modal Equivalence

Katarzyniak R., Więcek D.

Computational Methods in Science and Technology

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2018

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Vol. 24, No. 4

301--315

EN

In this paper we deal with an original model of technically oriented cognitive semantics of modal equivalences, stated in semi-controlled natural languages and used as surface linguistic representations of internal knowledge states of artificial autonomous systems. The cognitive semantics is defined to support generation of natural language statements extended with auto-epistemic operators of modality, representing subjectively experienced certainty, strong belief and week belief of the autonomous system that the relationship between occurrence of two chosen properties in a real object is the logical equivalence relation. An internal architecture of the autonomous system is briefly presented with particular emphasis on computational modules responsible for verification of compliance of current knowledge processing states with states described by the cognitive semantics as proper for application of a particular case of modal equivalence. The major issue discussed in the paper concerns an original theory of grounding of modal equivalences in empirical knowledge resources, encapsulated in the autonomous system. The whole theory consists of theorems describing basic and advanced properties of the cognitive semantics. In the paper, the attention is limited to presentation and proof of four theorems about nonemptiness of the proposed cognitive semantics. The research results cover the next stage of some research and development efforts, concerning the technical implementation of cognitive computing models, describing the artificial generation of semi-natural languages by the interactive autonomous system to describe their encapsulated knowledge resources.

5

Rozwiązania równoważne umów o zamówienie publiczne

Dzierżanowski W.

Budownictwo i Prawo

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2018

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R. 21, nr 3

33--36

PL

Przedmiot zamówienia opisuje się w sposób jednoznaczny i wyczerpujący za pomocą dostatecznie dokładnych i zrozumiałych określeń. Zakazuje się dokonywania opisu przedmiotu zamówienia przez wskazanie znaków towarowych, patentów lub pochodzenia produktu, a wyjątkiem jest dopuszczalność takiego opisu, gdy wymaga tego specyfika przedmiotu zamówienia. Wówczas konieczne jest dopuszczenie rozwiązań równoważnych wskazanemu z nazwy i określenie jasnych parametrów równoważności.

EN

The object of the order is described in an unambiguous and exhaustive way with sufficiently precise and understandable terms. It is forbidden to describe the subject of the order by indication of trademarks, patents or origin of the product, and the exception is the admissibility of such a description when the specificity of the subject matter of the contract requires it. Then, it is necessary to allow equivalent solutions to the name indicated and to define clear equivalence parameters.

6

Opisanie wyrobu w specyfikacji istotnych warunków zamówienia publicznego na roboty budowlane w sposób mogący utrudnić uczciwą konkurencję

Więcław-Bator L.

Budownictwo i Prawo

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2018

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R. 21, nr 1

26--29

PL

W artykule wskazano najczęściej popełniane nieprawidłowości w zakresie opisywania wyrobów w dokumentacji projektowej i specyfikacjach technicznych wykonania oraz odbioru robót budowlanych realizowanych w formule "zbuduj". Szczegółowej analizie poddano ugruntowane orzecznictwo Krajowej Izby Odwoławczej i sądów powszechnych. Autorka sformułuje zalecenia dla zamawiających, których wdrożenie może zminimalizować ilość zapytań do treści specyfikacji istotnych warunków zamówienia, postępowań odwoławczych, sądowych i dyscyplinarnych, a także korekt finansowych nakładanych na wydatki ponoszone z funduszy strukturalnych i Funduszu Spójności.

EN

The article identifies the most common irregularities in the description of products in project documentation and technical specifications for the execution and acceptance of construction works carried out in the "build" formula. The well-established case law of the National Chamber of Appeals and common courts has been thoroughly analyzed. The author formulates recommendations for ordering entities, implementation of which may minimize the number of inquiries to the contents of the tender specifications, appeal proceedings, court and disciplinary proceedings, as well as financial corrections imposed on expenses incurred from structural funds and the Cohesion Fund.

7

Tinput-Driven Pushdown, Counter, and Stack Automata

Kutrib M., Malcher A., Wendlandt M.

Fundamenta Informaticae

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2017

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Vol. 155, nr 1/2

59--88

EN

In input-driven automata the input alphabet is divided into distinct classes and different actions on the storage medium are solely governed by the input symbols. For example, in inputdriven pushdown automata (IDPDA) there are three distinct classes of input symbols determining the action of pushing, popping, or doing nothing on the pushdown store. Here, input-driven automata are extended in such a way that the input is preprocessed by a deterministic sequential transducer. IDPDAs extended in this way are called tinput-driven pushdown automata (TDPDA) and it turns out that TDPDAs are more powerful than IDPDAs but still not as powerful as real-time deterministic pushdown automata. Nevertheless, even this stronger model has still good closure and decidability properties. In detail, it is shown that TDPDAs are closed under the Boolean operations union, intersection, and complementation. Furthermore, decidability procedures for the inclusion problem as well as for the questions of whether a given automaton is a TDPDA or an IDPDA are developed. Additionally, representation theorems for the context-free languages using IDPDAs and TDPDAs are established. Two other classes investigated are on the one hand TDPDAs restricted to tinput-driven counter automata and on the other hand TDPDAs generalized to tinput-driven stack automata. In both cases, it is possible to preserve the good closure and decidability properties of TDPDAs, namely, the closure under the Boolean operations as well as the decidability of the inclusion problem.

8

The Simulation Powers and Limitations of Higher Temperature Hierarchical Self-Assembly Systems

Hendricks J., Patitz M., Rogers T. A.

Fundamenta Informaticae

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2017

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Vol. 155, nr 1/2

131--162

EN

In this paper, we extend existing results about simulation and intrinsic universality in a model of tile-based self-assembly. Namely, we work within the 2-Handed Assembly Model (2HAM), which is a model of self-assembly in which assemblies are formed by square tiles that are allowed to combine, using glues along their edges, individually or as pairs of arbitrarily large assemblies in a hierarchical manner, and we explore the abilities of these systems to simulate each other when the simulating systems have a higher "temperature" parameter, which is a system wide threshold dictating how many glue bonds must be formed between two assemblies to allow them to combine. It has previously been shown that systems with lower temperatures cannot simulate arbitrary systems with higher temperatures, and also that systems at some higher temperatures can simulate those at particular lower temperatures, creating an infinite set of infinite hierarchies of 2HAM systems with strictly increasing simulation power within each hierarchy. These previous results relied on two different definitions of simulation, one (strong simulation) seemingly more restrictive than the other (standard simulation), but which have previously not been proven to be distinct. Here we prove distinctions between them by first fully characterizing the set of pairs of temperatures such that the high temperature systems are intrinsically universal for the lower temperature systems (i.e. one tile set at the higher temperature can simulate any at the lower) using strong simulation. This includes the first impossibility result for simulation downward in temperature. We then show that lower temperature systems which cannot be simulated by higher temperature systems using the strong definition, can in fact be simulated using the standard definition, proving the distinction between the types of simulation.

9

Some Notes on Extracting Linguistic Summaries Built with Epistemic Modalities and Natural Language Connectives of Equivalence

Katarzyniak R., Lorkiewicz W., Więcek D.

Computational Methods in Science and Technology

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2017

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Vol. 23, No. 1

19--41

EN

In this paper we deal with an original technically oriented model for cognitive semantics. As the expected area of application we focus on the process of extraction of modal linguistic summaries from data managed by autonomous components of ambient systems and intelligent environments. As such, the cognitive semantics is defined for a particular case of modal natural language statements with epistemic modalities. The statements of interest are built with natural language operators, representing epistemic modalities (related to the main cognitive states of knowledge certainty: full certainty, strong belief and epistemic possibility), and natural language connectives of equivalence. Furthermore, an approach to their effective processing by autonomous computational systems is designed. An internal architecture of the autonomous computational component is designed with respect to modular model for natural language processing with separate modules for epistemic and semantic memory storage and processing. An original theoretical concept underlying the model of semantic memory is a holon defined as a collection of complementary linguistic protoforms. Finally, we provide several illustrative computational examples of linguistic summaries’ extraction, based on artificial and real data.

10

Completeness Results for Generalized Communication-free Petri Nets with Arbitrary Arc Multiplicities

Mayr E. W., Weihmann J.

Fundamenta Informaticae

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2016

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Vol. 143, nr 3/4

355--391

EN

We investigate gcf-Petri nets, a generalization of communication-free Petri nets allowing arbitrary arc multiplicities, and characterized by the sole restriction that each transition has at most one incoming arc. We use canonical firing sequences with nice properties for gcf-PNs to show that the RecLFS, (zero-)reachability, covering, and boundedness problems of gcf-PNs are in PSPACE. By simulating PSPACE-Turing machines by gss-PNs, a subclass of gcf-PNs where additionally all transitions have at most one outgoing arc, we ultimately obtain PSPACE-completess for these problems in case of gss-PNs or gcf-PNs. Additionally, we prove PSPACE-completeness for the liveness problem of gcf-PNs. Last, we show PSPACE-hardness as well as a doubly exponential space upper bound for the containment and equivalence problems of gss-PNs or gcf-PNs.

11

A Generalization of a Theorem of Móricz and Rhoades on Weighted Means

Natarajan P.N.

Commentationes Mathematicae

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2012

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Vol. 52, [Z] 1

29-37

EN

In this paper, we prove a theorem which gives an equivalent formulation of summability by weighted mean methods. The result of Hardy [1] and that of Móricz and Rhoades [2] are special cases of this theorem. In this context, it is important to note that the result of Móricz and Rhoades is valid even without the assumption Pn/Pn→ 0 as n→∞.

12

Equivalence, Indicators, Quasi-indicators and Optimal Steiner Topologies on Four Points in Space

Weng J.F., Smith J.MacGregor, Brazil M., Thomas D.A.

Fundamenta Informaticae

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2008

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Vol. 84, nr 1

135-149

EN

The Steiner tree problem is an intractable optimization problem, which asks for a network, in fact a tree, interconnecting a given point set V in a metric space and minimizing the total length of the network. The tree topology t of the network is called a Steiner topology and a tree T with minimum length with respect to its Steiner topology is called a Steiner tree. As a combinatorial optimization problem, the Steiner tree problem asks for a Steiner tree T with minimum length over all possible topologies t on V. It has been proved that if T is in E3 then the length of T cannot be expressed by radicals even when T spans just 4 points. For such optimization problems in which the objective functions do not have closed form solutions the traditional approach is approximation. In this paper we propose a new approach by introducing some new concepts: equivalence, indicators and quasi-indicators, and then we apply these concepts to the Steiner tree problem. Roughly speaking, a quasi-indicator is a function that is simple to compute but indicates with high probability the optimal solution to the original optimisation problem. For a specific optimisation problem - finding the optimal Steiner topologies on 4 points in space, we demonstrate how to find good quasi-indicators. The extensive computational experiments over 5000 random 4-point sets show that the best quasi-indicator for finding optimal Steiner topologies on 4 points in space is not only easy to compute but also extremely successful with less than 1.5% failures in indicating optimal topologies even if degeneracy of Steiner minimal trees exists. Moreover, within the 1.5% cases of failure, the maximum and the average relative error are 1.5% and 0.2% respectively. Therefore, the performance of the proposed Q-indicator is very good and could be applied to the four vertices surrounding any pair of adjacent Steiner points in a Steiner tree on n ( > 4) points in space to make local improvements to the topology of the Steiner minimal tree in space.

13

Structure of Taylor coefficients by equivalence of Tauberian conditions

Canak I.

Demonstratio Mathematica

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2008

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Vol. 41, nr 2

309-315

EN

From the equivalent statement of a sequence (un) whose general control modulo of the oscillatory behavior of integer order m is (C, 1) slowly oscillating, we obtain some conclusions regarding the structure of the general control modulo of the oscillatory behavior of integer order k, k < m of (un) and investigate subsequential convergence of some sequences related to (un).

14

Implications and equivalences in orthomodular lattices

Hyčko M.

Demonstratio Mathematica

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2005

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Vol. 38, nr 4

777--792

EN

The present article describes a method for checking the validity of implications or equivalences in the free orthomodular lattice on two generators and in the F(a, b, c1,..., cn), which is the free orthomodular lattice generated by the elements a, 6, ci,... Cn, where the elements ci, i = 1,..., n are central in it. The structure of the previous lattices is described in [3] and [1]. The method presented is based on comparing the elements that are assigned to each expression on both sides of an implication or an equivalence. It gives a necessary condition for the implication or equivalence of arbitrary positive statements (a combination of identities and logical connectives AND and OR) to hold. When the conclusion part is an identity or a conjunction of identities, these conditions become also sufficient.

15

Algebraic Structures Related to Many Valued Logical Systems. Part 2, Equivalence Among some Widespread Structures

Cattaneo G., Ciucci D., Giuntini R., Konig M.

Fundamenta Informaticae

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2004

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Vol. 63, nr 4

357--373

EN

Several algebraic structures (namely HW, BZMVdM, Stonean MV and MVΔ algebras) related to many valued logical systems are considered and their equivalence is proved. Four propositional calculi whose Lindenbaum-Tarski algebra corresponds to the four equivalent algebraic structures are axiomatized and their semantical completeness is given.

16

Infinite Elementary Divisor Structure-Preserving Transformations for Polynomial Matrices

Karampetakis N. P., Vologiannidis S.

International Journal of Applied Mathematics and Computer Science

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2003

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Vol. 13, no 4

493-503

EN

The main purpose of this work is to propose new notions of equivalence between polynomial matrices that preserve both the finite and infinite elementary divisor structures. The approach we use is twofold: (a) the 'homogeneous polynomial matrix approach', where in place of the polynomial matrices we study their homogeneous polynomial matrix forms and use 2-D equivalence transformations in order to preserve their elementary divisor structure, and (b) the `polynomial matrix approach', where some conditions between the 1-D polynomial matrices and their transforming matrices are proposed.

17

Komori identities in algebraic logic

Blok W., La Falce S.

Reports on Mathematical Logic

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2000

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No. 34

79-106

EN

A variety generated by a class K of BCK-algebras consists of BCK-algebras if and only if it satisfies a certain kind of identity, first discovered by Komori. A similar phenomenon is shown to hold more generally in a certain class of quasivarieties of logic that includes not only the class of BCK-algebras but also such classes as the quasivariety of biresiduation algebras and quasivarieties of algebras with an equivalence operation. We describe a set of identities (which we call Komori identities), and show that the variety generated by a class K of algebras in one of the quasivarieties considered is contained in the quasivariety if and only it it satisfies a Komori identity. We use the result to establish (i) that the subvarieties of any of the quasivarieties studied are congruence 3-permutable and (ii) that the varietal join of two subvarieties of any of the quasivarieties studied is contained in the quasivariety.

18

An algebra of data flows

Winkowski J.

Fundamenta Informaticae

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2000

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Vol. 42, Nr 1

75-104

EN

An algebra of processes is proposed in which processes are modelled in terms of data rather than events. Structures representing processes are similar to the labelled event structures of Winskel (1982), but they denote flows of data. For processes represented in such a way operations similar to those of CCS (the Calculus of Communicating Systems as developed by Milner (1980)) are defined and their fundamental properties are stated.

19

On the decidability of some problems concerning morphisms, regular sets, and post correspondence problem

Turakainen P.

Fundamenta Informaticae

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1999

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Vol. 38, Nr 1,2

209-221

EN

Regular languages are divided into equivalence classes according to the lengths of the words and both the universal and the existential equivalence of rational transductions on the set of these classes is studied. It is shown that the cardinality equivalence problem is undecidable for e-free finite substitutions. The morphic replication equivalence problem is arithmetized and an application to word equations is presented. Finally, the generalized Post correspondence problem is modified by using a single inverse morphism or a single finite substitution or its inverse instead of two morphisms.

20

Equivalence of nD Singular Roesser and Fornasini-Marchesini Models

Kaczorek T.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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1999

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Vol. 47, nr 3

235-246

EN

A new nD Roesser model with extended inputs is introduced. nD Roesser models with square matrices and extended inputs equivalent to the singular nD Fornasini-Marchesini models and nD Fornasini-Marchesini models with square matrices equivalent to the singular nD Roesser models are derived. A formula determination the solution of standard 2D Roesser model with extended inputs is also derived.