Theoretical Computer Science Research Articles

Advanced image analysis for medical diagnostics: a system for segmentation and classification using level set methods and AI algorithms

This work aims to implement and utilize an advanced computer system for image analysis and processing through artificial intelligence. The system will evaluate images from multiple sources. As a result, a comprehensive e-Medicus system will be developed to capture and analyze X-ray data and classify cancer cells. This innovative tool, with its unique features tailored for medical facilities, will assist them in capturing and analyzing X-ray images and CT scan results. The e-Medicus system offers several benefits for medical facilities, including efficient automation of photo analysis, tracking changes in a patient's condition over time, and facilitating the identification of medical changes and data classification. A multimedia presentation of the change process will use the contour set function, allowing for topological changes in solution properties. The system integrates novel procedures and algorithms from theoretical computer science and numerical mathematics, leveraging neural networks, genetic algorithms, semantic networks, image ontology, rough set theory, contour set methods, and hybrid algorithms. These developed algorithms enhance methods and concepts in image segmentation, gathering, transmitting, storing, extracting, and effectively presenting information.

Model for Designing Gamified Experiences Mediated by a Virtual Teaching and Learning Environment

Higher Education Institutions (HEIs) face new challenges in regard to technological development in light of necessary pedagogical and didactic innovations in educational action. This article proposes a Technological–Pedagogical Gamification Model (MGTP) that guides the design of gamified educational practices in Virtual Teaching and Learning Environments (EVEAs). The MGTP proposal is based on theoretical cores of Pedagogy and Computer Science theories, as well as works related to gamified experiences in EVEA where the social, cognitive, and teaching presences were analyzed. This work also presents an initial validation of the MGTP through expert judgment, and its results are analyzed from both a qualitative (content analysis and comments) and quantitative (using the Content Validity Coefficient method) perspective. These results reveal a high level of acceptance of the model by experts that is corroborated by reliability tests (Cronbach’s alpha and split-half reliability test). The results facilitated the development of a final version of the model for its subsequent application and evaluation in university practice.

Evaluating the explanatory power of the Conscious Turing Machine

The recent “Conscious Turing Machine” (CTM) proposal offered by Manuel and Lenore Blum aims to define and explore consciousness, contribute to the solution of the hard problem, and demonstrate the value of theoretical computer science with respect to the study of consciousness. Surprisingly, given the ambitiousness and novelty of the proposal (and the prominence of its creators), CTM has received relatively little attention. We here seek to remedy this by offering an exhaustive evaluation of CTM. Our evaluation considers the explanatory power of CTM in three different domains of interdisciplinary consciousness studies: the philosophy of mind, cognitive neuroscience, and computation. Based on our evaluation in each of the target domains, at present, any claim that CTM constitutes progress is premature. Nevertheless, the model has potential, and we highlight several possible avenues of future research which proponents of the model may pursue in its development.

The ultrametric backbone is the union of all minimum spanning forests

Minimum spanning trees and forests are powerful sparsification techniques that remove cycles from weighted graphs to minimize total edge weight while preserving node reachability, with applications in computer science, network science, and graph theory. Despite their utility and ubiquity, they have several limitations, including that they are only defined for undirected networks, they significantly alter dynamics on networks, and they do not generally preserve important network features such as shortest distances, shortest path distribution, and community structure. In contrast, distance backbones, which are subgraphs formed by all edges that obey a generalized triangle inequality, are well defined in directed and undirected graphs and preserve those and other important network features. The backbone of a graph is defined with respect to a specified path-length operator that aggregates weights along a path to define its length, thereby associating a cost to indirect connections. The backbone is the union of all shortest paths between each pair of nodes according to the specified operator. One such operator, the max function, computes the length of a path as the largest weight of the edges that compose it (a weakest link criterion). It is the only operator that yields an algebraic structure for computing shortest paths that is consistent with De Morgan's laws. Applying this operator yields the ultrametric backbone of a graph in that (semi-triangular) edges whose weights are larger than the length of an indirect path connecting the same nodes (i.e. those that break the generalized triangle inequality based on max as a path-length operator) are removed. We show that the ultrametric backbone is the union of minimum spanning forests in undirected graphs and provides a new generalization of minimum spanning trees to directed graphs that, unlike minimum equivalent graphs and minimum spanning arborescences, preserves all shortest paths and De Morgan's law consistency.

Fast Algorithms for \(\ell_p\)-Regression

The ℓ p -norm regression problem is a classic problem in optimization with wide ranging applications in machine learning and theoretical computer science. The goal is to compute \(\boldsymbol {\mathit {x}}^{\star } =\arg \min _{\boldsymbol {\mathit {A}}\boldsymbol {\mathit {x}}=\boldsymbol {\mathit {b}}}\Vert \boldsymbol {\mathit {x}}\Vert _p^p \) , where \(\boldsymbol {\mathit {x}}^{\star }\in \mathbb {R}^n,\boldsymbol {\mathit {A}}\in \mathbb {R}^{d\times n},\boldsymbol {\mathit {b}} \in \mathbb {R}^d \) and d ≤ n . Efficient high-accuracy algorithms for the problem have been challenging both in theory and practice and the state-of-the-art algorithms require \(poly(p)\cdot n^{\frac{1}{2}-\frac{1}{p}} \) linear system solves for p ≥ 2. In this paper, we provide new algorithms for ℓ p -regression (and a more general formulation of the problem) that obtain a high-accuracy solution in O ( pn ( p − 2)/(3 p − 2) ) linear system solves. We further propose a new inverse maintenance procedure that speeds-up our algorithm to \(\widetilde{O}(n^{\omega }) \) total runtime, where O ( n ω ) denotes the running time for multiplying n × n matrices. Additionally, we give the first Iteratively Reweighted Least Squares (IRLS) algorithm that is guaranteed to converge to an optimum in a few iterations. Our IRLS algorithm has shown exceptional practical performance, beating the currently available implementations in MATLAB/CVX by 10-50x.

New Type of Extended Soft Set Operation: Complementary Extended Union Operation

Soft set theory was proposed by Molodtsov in 1999 to model some problems involving uncertainty. It has a wide range of theoretical and practical applications. Soft set operations constitute the basic building blocks of soft set theory. Many kinds of soft set operations have been described and applied in various ways since the inception of the theory. In this paper, to contribute to the theory, a new soft set operation, called complementary extended union operation, is defined, its properties are discussed in detail to obtain the relationship of each operation with other soft set operations, and the distributions of these operations over other soft set operations are examined. We obtain that the complementary extended union operation along with other certain types of soft set operations construct some well-known algebraic structure such as Boolean Algebra, De Morgan Algebra, semiring, and hemiring in the set of soft sets with a fixed parameter set. Since Boolean Algebra is fundamental in digital logic design, computer science, information retrieval, set theory and probability; De Morgan Algebra in logic and set theory, computer science, artificial intelligence, circuit design; semirings in theoretical computer science, optimization problems, economics, cryptography and coding theory, and hemirings in combinatorics, mathematical economics, theoretical computer science, these algebraic structures provide essential tools for various applications, facilitating the analysis, design, and optimization of systems across many disciplines, and thus this study is expected to contribute to decision-making methods and cryptography based on soft sets.

Computationally Hard Problems for Logic Programs under Answer Set Semantics

Showing that a problem is hard for a model of computation is one of the most challenging tasks in theoretical computer science, logic and mathematics. For example, it remains beyond reach to find an explicit problem that cannot be computed by polynomial size propositional formulas (PF). As a model of computation, logic programs (LP) under answer set semantics are as expressive as PF, and also \(\mathtt{NP}\) -complete for satisfiability checking. In this paper, we show that the PAR problem is hard for LP, i.e., deciding whether a binary string contains an odd number of \(1\) ’s requires exponential size logic programs. The proof idea is first to transform logic programs into equivalent boolean circuits, and then apply a probabilistic method known as random restriction to obtain an exponential lower bound. Based on the main result, we generalize a sufficient condition for identifying hard problems for LP, and give a separation map for a logic program family from a computational point of view, whose members are all equally expressive and share the same reasoning complexity.

From the Editor-in-Chief

Welcome everybody to the third issue of 2024! Some of us are already in holiday, some are still at work; surely, many of us will meet in the forthcoming conferences, mostly our flagship conference (LICS). In the meanwhile, I invite you to read the present issue of SIGLOG News, whose contents are the following: -The SIGLOG Chair introduces this year edition of LICS and reports on the current situation of SIGLOG. -We present the outcome of the 2024 Alonzo Church Award. This prize was established in 2015 by SIGLOG, by the European Association for Theoretical Computer Science (EATCS), by the European Association for Computer Science Logic (EACSL), and by the Kurt Goedel Society (KGS). In 2022 the KGS was dissolved and, since then, the award has been sponsored by EACSL, EATCS and SIGLOG. The award is for an outstanding contribution represented by a paper or by a small group of papers published within the past 25 years. This year's recipients are Thomas Ehrhard and Laurent Regnier. -Then, you can read a new semantics column, edited by Prakash Panangaden and written by Samson Abramsky and Luca Reggio, on Game comonads ; these offer a categorical view of a number of model-comparison games central to model theory and provide a new bridge between categorical methods developed for semantics, and the combinatorial and algorithmic methods of resource-sensitive model theory. -To conclude, we wrap up this number with the latest issue of SIGLOG Monthly, prepared by Elli Anastasiadi.

On the maximum value of the stairs2 index

Measures of tree balance play an important role in different research areas such as mathematical phylogenetics or theoretical computer science. The balance of a tree is usually quantified in a single number, called a balance or imbalance index, and several such indices exist in the literature. Here, we focus on the stairs2 balance index for rooted binary trees, which was first introduced in the context of viral phylogenetics but has not been fully analyzed from a mathematical viewpoint yet. While it is known that the caterpillar tree uniquely minimizes the stairs2 index for all leaf numbers and the fully balanced tree uniquely maximizes the stairs2 index for leaf numbers that are powers of two, understanding the maximum value and maximal trees for arbitrary leaf numbers has been an open problem in the literature. In this note, we fill this gap by showing that for all leaf numbers, there is a unique rooted binary tree maximizing the stairs2 index. Additionally, we obtain recursive and closed expressions for the maximum value of the stairs2 index of a rooted binary tree with n leaves.

Tight General Bounds for the Extremal Numbers of 0–1 Matrices

Abstract A zero-one matrix $M$ is said to contain another zero-one matrix $A$ if we can delete some rows and columns of $M$ and replace some $1$-entries with $0$-entries such that the resulting matrix is $A$. The extremal number of $A$, denoted $\operatorname{ex}(n,A)$, is the maximum number of $1$-entries that an $n\times n$ zero-one matrix can have without containing $A$. The systematic study of this function for various patterns $A$ goes back to the work of Füredi and Hajnal from 1992, and the field has many connections to other areas of mathematics and theoretical computer science. The problem has been particularly extensively studied for so-called acyclic matrices, but very little is known about the general case (i.e., the case where $A$ is not necessarily acyclic). We prove the first asymptotically tight general result by showing that if $A$ has at most $t$ $1$-entries in every row, then $\operatorname{ex}(n,A)\leq n^{2-1/t+o(1)}$. This verifies a conjecture of Methuku and Tomon. Our result also provides the first tight general bound for the extremal number of vertex-ordered graphs with interval chromatic number $2$, generalizing a celebrated result of Füredi and Alon, Krivelevich, and Sudakov about the (unordered) extremal number of bipartite graphs with maximum degree $t$ in one of the vertex classes.

Formal definitions and proofs for partial (co)recursive functions

Partial functions are a key concept in programming. Without partiality a programming language has limited expressiveness – it is not Turing-complete, hence, it excludes some constructs such as while-loops. In functional programming languages, partiality mostly originates from the non-termination of recursive functions. Corecursive functions are another source of partiality: here, the issue is not termination, but the inability to produce arbitrary large, finite approximations of a theoretically infinite output.Partial functions have been formally studied in the branch of theoretical computer science called domain theory. In this paper we propose to step up the level of formality by using the Coq proof assistant. The main difficulty is that Coq requires all functions to be total, since partiality would break the soundness of its underlying logic. We propose practical solutions for this issue, and others, which appear when one attempts to define and reason about partial (co)recursive functions in a total functional language.

Exploring Clique Transversal Problems for d-degenerate Graphs with Fixed d: From Polynomial-Time Solvability to Parameterized Complexity

This paper explores the computational challenges of clique transversal problems in d-degenerate graphs, which are commonly encountered across theoretical computer science and various network applications. We examine d-degenerate graphs to highlight their utility in representing sparse structures and assess several variations of clique transversal problems, including the b-fold and {b}-clique transversal problems, focusing on their computational complexities for different graph categories. Our analysis identifies that certain instances of these problems are polynomial-time solvable in specific graph classes, such as 1-degenerate or 2-degenerate graphs. However, for d-degenerate graphs where d≥2, these problems generally escalate to NP-completeness. We also explore the parameterized complexity, pinpointing specific conditions that render these problems fixed-parameter tractable.

Mathematical aspects of digital technologies: The problem of the absent-minded passenger

<p>The example of the absent-minded passenger problem illustrates the fundamental difficulty of estimating practical values of probabilistic parameters on the basis of general theorizing and allows us to give more accurate and subtle estimates of various parameters, as well as their practical computable approximation. This article is important for building systems of artificial intelligence and artificial consciousness, as well as for modern didactics of higher education, especially for the disciplines of “applied mathematics” and “theoretical computer science”. It is shown that a detailed study of a random process provides more significant applied materials than formally correct theoretical calculations.</p>

Glassy Word Problems: Ultraslow Relaxation, Hilbert Space Jamming, and Computational Complexity

We introduce a family of local models of dynamics based on “word problems” from computer science and group theory, for which we can place rigorous lower bounds on relaxation timescales. These models can be regarded either as random circuit or local Hamiltonian dynamics and include many familiar examples of constrained dynamics as special cases. The configuration space of these models splits into dynamically disconnected sectors, and for initial states to relax, they must “work out” the other states in the sector to which they belong. When this problem has a high time complexity, relaxation is slow. In some of the cases we study, this problem also has high space complexity. When the space complexity is larger than the system size, an unconventional type of jamming transition can occur, whereby a system of a fixed size is not ergodic but can be made ergodic by appending a large reservoir of sites in a trivial product state. This finding manifests itself in a new type of Hilbert space fragmentation that we call fragile fragmentation. We present explicit examples where slow relaxation and jamming strongly modify the hydrodynamics of conserved densities. In one example, density modulations of wave vector q exhibit almost no relaxation until times O(exp(1/q)), at which point they abruptly collapse. We also comment on extensions of our results to higher dimensions. Published by the American Physical Society 2024

Specifying the Power and Limitations of Randomness

Avi Wigderson on the impact, and limitations, of randomness on complexity.

Thermodynamics of Computations with Absolute Irreversibility, Unidirectional Transitions, and Stochastic Computation Times

Developing a thermodynamic theory of computation is a challenging task at the interface of nonequilibrium thermodynamics and computer science. In particular, this task requires dealing with difficulties such as stochastic halting times, unidirectional (possibly deterministic) transitions, and restricted initial conditions, features common in real-world computers. Here, we present a framework which tackles all such difficulties by extending the martingale theory of nonequilibrium thermodynamics to generic nonstationary Markovian processes, including those with broken detailed balance and/or absolute irreversibility. We derive several universal fluctuation relations and second-law-like inequalities that provide both lower and upper bounds on the intrinsic dissipation (mismatch cost) associated with any periodic process—in particular, the periodic processes underlying all current digital computation. Crucially, these bounds apply even if the process has stochastic stopping times, as it does in many computational machines. We illustrate our results with exhaustive numerical simulations of deterministic finite automata processing bit strings, one of the fundamental models of computation from theoretical computer science. We also provide universal equalities and inequalities for the acceptance probability of words of a given length by a deterministic finite automaton in terms of thermodynamic quantities, and outline connections between computer science and stochastic resetting. Our results, while motivated from the computational context, are applicable far more broadly. Published by the American Physical Society 2024

SEMI GROUP IDENTITIES WITH APPLICATIONS TO SEMI GROUPS

This research paper explores the rich and intriguing world of semi-group identities, their properties, and their applications to various types of semi-groups. Semi-groups are algebraic structures that generalize the notion of groups, allowing for non-invertible elements. Despite their broader scope, semi-groups retain many important features found in group theory. This study investigates the identities that hold true in the context of semi-group algebra and sheds light on the underlying mathematical structures and relationships among these identities. By delving into specific applications, we illustrate the significance of these findings to various types of semi-groups, such as monoids, semigroups with zero, and cancellative semi-groups. Ultimately, our results not only deepen our understanding of the fundamental properties of semi-groups. But also provide valuable insights for researchers in the areas of algebraic structures, combinatorics, and theoretical computer science.

Improved Real‐Time Post‐Processing for Quantum Random Number Generators

AbstractRandomness extraction is a key problem in cryptography and theoretical computer science. With the recent rapid development of quantum cryptography, quantum‐proof randomness extraction has also been widely studied, addressing the security issues in the presence of a quantum adversary. In contrast with conventional quantum‐proof randomness extractors characterizing the input raw data as min‐entropy sources, it is found that the input raw data generated by a large class of trusted‐device quantum random number generators can be characterized as the so‐called reverse block source. This fact enables us to design improved extractors. Two novel quantum‐proof randomness extractors for reverse block sources that realize real‐time block‐wise extraction are proposed specifically. In comparison with the general min‐entropy randomness extractors, the designs achieve a significantly higher extraction speed and a longer output data length with the same seed length. In addition, they enjoy the property of online algorithms, which process the raw data on the fly without waiting for the entire input raw data to be available. These features make the design an adequate choice for the real‐time post‐processing of practical quantum random number generators. Applying the extractors to the raw data generated by a widely used quantum random number generator, a simulated extraction speed as high as 300 Gbps is achieved.

Optimal Layout of (Kp − Cp)n into Wheel-Like Networks

The problem of finding the induced subgraph with the maximum number of edges among all subsets of a fixed cardinality is known as the maximum subgraph problem (MSP). The lexicographic order has been proven optimal for Cartesian products of specific regular graphs [B. Sergei, B. Pavle and K. Nikola, New infinite family of regular edge isoperimetric graphs, Theoretical Computer Science 721 (2018) 42–53]. This paper is dedicated to the precise calculation of edge values within the resulting subgraphs, enhancing our comprehension of their structural properties and implications. In this paper, we focus on solving the MSP of [Formula: see text] for any [Formula: see text] and for odd [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is obtained by the Cartesian product of a complete graph with [Formula: see text] vertices with a removal of a cycle on [Formula: see text] vertices. It has been observed that the graph obtained by deleting a cycle of length [Formula: see text] from a complete graph [Formula: see text] is isomorphic to a circulant graph [Formula: see text], while the particular case [Formula: see text] has been previously studied in [A. Syeda and M. Rajesh, MinLA of [Formula: see text] and its optimal layout into certain trees, The Journal of Supercomputing (2023), https://doi.org/10.1007/s11227-023-05140-3]. Our work extends the understanding of the MSP to a broader class of graphs with similar properties and applications.

Common fuzzy fixed point results for f-contractive mappings with applications

The aim of this article is to establish some common fixed point theorems for -fuzzy mappings under F-contraction in the framework of complete metric spaces. To extend and improve some well-known results of literature, new results for multivalued mappings are obtained as application of established results. We have illustrated an appropriate example to rationalize the notions and outcomes. Also we investigated the solution of the domain of words as application of our results to theoretical computer science.