arXiv subjects

Tomasz Kowalski

Publications and source records attributed to Tomasz Kowalski.

33 records · Page 2Linked to original sources

arXiv2022

Omitting Types Theorem in hybrid-dynamic first-order logic with rigid symbols

In the the present contribution, we prove an Omitting Types Theorem (OTT) for an arbitrary fragment of hybriddynamic first-order logic with rigid symbols (i.e. symbols with fixed interpretations across worlds) closed under negation and retrieve. The logical framework can be regarded as a parameter and it is instantiated by some well-known hybrid and/or dynamic logics from the literature. We develop a forcing technique and then we study a forcing property based on local satisfiability, which lead to a refined proof of the OTT. For uncountable signatures, the result requires compactness, while for countable signatures, compactness is not necessary. We apply the OTT to obtain upwards and downwards Löwenheim-Skolem theorems for our logic, as well as a completeness theorem for its constructor-based variant. The main result of this paper can easily be recast in the institutional model theory framework, giving it a higher level of generality.

math.LO↗

arXiv2022

Qualitative representations of chromatic algebras

Conventional Ramsey-theoretic investigations for edge-colourings of complete graphs are framed around avoidance of certain configurations. Motivated by considerations arising in the field of Qualitative Reasoning, we explore edge colourings that in addition to forbidding certain triangle configurations also require others to be present. These conditions have natural combinatorial interest in their own right, but also correspond to qualitative representability of certain nonassociative relation algebras, which we will call chromatic.

math.CO↗

arXiv2021

Normal extensions of KTB of codimension 3

It is known that in the lattice of normal extensions of the logic KTB there are unique logics of codimensions 1 and 2, namely, the logic of a single reflexive point, and the logic of the total relation on two points. A natural question arises about the cardinality of the set of normal extensions of KTB of codimension 3. Generalising two finite examples found by a computer search, we construct an uncountable family of (countable) graphs, and prove that certain frames based on these produce a continuum of normal extensions of KTB of codimension 3. We use algebraic methods, which in this case turn out to be better suited to the task than frame-theoretic ones.

math.LO↗

arXiv2021

Varieties of semiassociative relation algebras and tense algebras

It is well known that the subvariety lattice of the variety of relation algebras has exactly three atoms. The (join-irreducible) covers of two of these atoms are known, but a complete classification of the (join-irreducible) covers of the remaining atom has not yet been found. These statements are also true of a related subvariety lattice, namely the subvariety lattice of the variety of semiassociative relation algebras. The present article shows that this atom has continuum many covers in this subvariety lattice (and in some related subvariety lattices) using a previously established term equivalence between a variety of tense algebras and a variety of semiassociative r-algebras.

math.LO↗

arXiv2019

Coherence in Modal Logic

A variety is said to be coherent if the finitely generated subalgebras of its finitely presented members are also finitely presented. In a recent paper by the authors it was shown that coherence forms a key ingredient of the uniform deductive interpolation property for equational consequence in a variety, and a general criterion was given for the failure of coherence (and hence uniform deductive interpolation) in varieties of algebras with a term-definable semilattice reduct. In this paper, a more general criterion is obtained and used to prove the failure of coherence and uniform deductive interpolation for a broad family of modal logics, including K, KT, K4, and S4.

math.LO↗

arXiv2018

Two-sided wreath product done right

We investigate a semigroup construction related to the two-sided wreath product. It encompasses a range of known constructions and gives a slightly finer version of the decomposition in the Krohn-Rhodes Theorem, in which the three-element flip-flop is replaced by the two-element semilattice. We develop foundations of the theory of our construction, showing in the process that it naturally combines ideas from semigroup theory (wreath products), category theory (Grothendieck construction), and ordered structures (residuated lattices).

math.RA↗

arXiv2018

Uniform interpolation and coherence

A variety V is said to be coherent if any finitely generated subalgebra of a finitely presented member of V is finitely presented. It is shown here that V is coherent if and only if it satisfies a restricted form of uniform deductive interpolation: that is, any compact congruence on a finitely generated free algebra of V restricted to a free algebra over a subset of the generators is again compact. A general criterion is obtained for establishing failures of coherence, and hence also of uniform deductive interpolation. This criterion is then used in conjunction with properties of canonical extensions to prove that coherence and uniform deductive interpolation fail for certain varieties of Boolean algebras with operators (in particular, algebras of modal logic K and its standard non-transitive extensions), double-Heyting algebras, residuated lattices, and lattices.

math.LO↗

arXiv2017

Faster range minimum queries

Range Minimum Query (RMQ) is an important building brick of many compressed data structures and string matching algorithms. Although this problem is essentially solved in theory, with sophisticated data structures allowing for constant time queries, practical performance and construction time also matter. Additionally, there are offline scenarios in which the number of queries, $q$, is rather small and given beforehand, which encourages to use a simpler approach. In this work, we present a simple data structure, with very fast construction, which allows to handle queries in constant time on average. This algorithm, however, requires access to the input data during queries (which is not the case of sophisticated RMQ solutions). We subsequently refine our technique, combining it with one of the existing succinct solutions with $O(1)$ worst-case time queries and no access to the input array. The resulting hybrid is still a memory frugal data structure, spending usually up to about $3n$ bits, and providing competitive query times, especially for wide ranges. We also show how to make our baseline data structure more compact. Experimental results demonstrate that the proposed BbST (Block-based Sparse Table) variants are competitive to existing solutions, also in the offline scenario.

cs.DS↗

arXiv2017

Faster batched range minimum queries

Range Minimum Query (RMQ) is an important building brick of many compressed data structures and string matching algorithms. Although this problem is essentially solved in theory, with sophisticated data structures allowing for constant time queries, there are scenarios in which the number of queries, $q$, is rather small and given beforehand, which encourages to use a simpler approach. A recent work by Alzamel et al. starts with contracting the input array to a much shorter one, with its size proportional to $q$. In this work, we build upon their solution, speeding up handling small batches of queries by a factor of 3.8--7.8 (the gap grows with $q$). The key idea that helped us achieve this advantage is adapting the well-known Sparse Table technique to work on blocks, with speculative block minima comparisons. We also propose an even much faster (but possibly using more space) variant without the array contraction.

cs.DS↗

arXiv2017

Algebraic foundations for qualitative calculi and networks

A qualitative representation $ϕ$ is like an ordinary representation of a relation algebra, but instead of requiring $(a; b)^ϕ= a^ϕ| b^ϕ$, as we do for ordinary representations, we only require that $c^ϕ\supseteq a^ϕ| b^ϕ\iff c\geq a ; b$, for each $c$ in the algebra. A constraint network is qualitatively satisfiable if its nodes can be mapped to elements of a qualitative representation, preserving the constraints. If a constraint network is satisfiable then it is clearly qualitatively satisfiable, but the converse can fail. However, for a wide range of relation algebras including the point algebra, the Allen Interval Algebra, RCC8 and many others, a network is satisfiable if and only if it is qualitatively satisfiable. Unlike ordinary composition, the weak composition arising from qualitative representations need not be associative, so we can generalise by considering network satisfaction problems over non-associative algebras. We prove that computationally, qualitative representations have many advantages over ordinary representations: whereas many finite relation algebras have only infinite representations, every finite qualitatively representable algebra has a finite qualitative representation; the representability problem for (the atom structures of) finite non-associative algebras is NP-complete; the network satisfaction problem over a finite qualitatively representable algebra is always in NP; the validity of equations over qualitative representations is co-NP-complete. On the other hand we prove that there is no finite axiomatisation of the class of qualitatively representable algebras.

cs.AI↗

arXiv2016

Suffix arrays with a twist

The suffix array is a classic full-text index, combining effectiveness with simplicity. We discuss three approaches aiming to improve its efficiency even more: changes to the navigation, data layout and adding extra data. In short, we show that $(i)$ how we search for the right interval boundary impacts significantly the overall search speed, $(ii)$ a B-tree data layout easily wins over the standard one, $(iii)$ the well-known idea of a lookup table for the prefixes of the suffixes can be refined with using compression, $(iv)$ caching prefixes of the suffixes in a helper array can pose a(nother) practical space-time tradeoff.

cs.DS↗

arXiv2016

Complexity and polymorphisms for digraph constraint problems under some basic constructions

The role of polymorphisms in determining the complexity of constraint satisfaction problems is well established. In this context we study the stability of CSP complexity and polymorphism properties under some basic graph theoretic constructions. As applications we observe a collapse in the applicability of algorithms for CSPs over directed graphs with both a total source and a total sink: the corresponding CSP is solvable by the "few subpowers algorithm" if and only if it is solvable by a local consistency check algorithm. Moreover, we find that the property of "strict width" and solvability by few subpowers are unstable under first order reductions. The analysis also yields a complete characterisation of the main polymorphism properties for digraphs whose symmetric closure is a complete graph.

math.CO↗

arXiv2015

Indexing arbitrary-length $k$-mers in sequencing reads

We propose a lightweight data structure for indexing and querying collections of NGS reads data in main memory. The data structure supports the interface proposed in the pioneering work by Philippe et al. for counting and locating $k$-mers in sequencing reads. Our solution, PgSA (pseudogenome suffix array), based on finding overlapping reads, is competitive to the existing algorithms in the space use, query times, or both. The main applications of our index include variant calling, error correction and analysis of reads from RNA-seq experiments.

cs.DS↗

arXiv2012

Kites and Pseudo BL-algebras

We investigate a construction of a pseudo BL-algebra out of an $\ell$-group called a kite. We show that many well-known examples of algebras related to fuzzy logics can be obtained in that way. We describe subdirectly irreducible kites. As another application, we exhibit a new countably infinite family of varieties of pseudo BL-algebras covering the variety of Boolean algebras.

math.RA↗

arXiv2011

On Normal-Valued Basic Pseudo Hoops

We show that every pseudo hoop satisfies the Riesz Decomposition Property. We visualize basic pseudo hoops by functions on a linearly ordered set. Finally, we study normal-valued basic pseudo hoops giving a countable base of equations for them.

math.AC↗