Foundations Of Computer Science Research Articles

Artificial Intelligence Curriculum Development for Intelligent System Experts in University

Artificial intelligence (AI) has emerged as a pivotal technology for enhancing national and industrial competitiveness in the digital transformation era. Consequently, the cultivation of specialized talent in AI has garnered significant attention. This study analyzed AI-related department curricula at major universities worldwide, identifying critical courses for each academic semester. The data we collected included course titles, syllabi, and learning objectives, which were refined and analyzed afterward. Furthermore, we comparatively examined university AI education programs based on the content of Computer Science Curricula 2023, a widely recognized framework for computer science education. The insights gleaned from our analysis revealed that AI curricula are built upon a foundation of computer science, emphasizing the importance of a deep understanding of various related domains within the field of computer science. Based on these findings, we proposed a curriculum for AI departments, considering the need for a comprehensive understanding of computer science alongside specialized AI courses. This study aims to provide foundational data for advancing AI education and guide educational program improvements. Ultimately, it aspires to contribute to developing specialized professionals in the AI field, thereby bolstering national and industrial competitiveness in the rapidly evolving digital landscape.

Special Section on the Fifty-Ninth Annual IEEE Symposium on Foundations of Computer Science (2018)

Special Section on the Fifty-Ninth Annual IEEE Symposium on Foundations of Computer Science (2018)

Construction and Teaching Reform of “Research foundation of Computer Science” Course based on OBE Concept

The research foundation of computer science is a compulsory professional foundation course for master's students of computer science and technology and related disciplines and majors. Due to the extensive content of this course, covering four topics: advanced computer architecture, Internet of Things technology, network and information security, and algorithm design and analysis, students feel that it is difficult to master scientific learning methods and lack the goal and initiative of learning. This article is guided by the OBE concept and explores and practices the teaching reform of computer graduate courses in the context of the construction of new engineering disciplines, starting from five key links: setting teaching objectives, designing teaching content, reforming teaching modes, establishing teaching evaluation mechanisms.

“Intelligent Heuristics Are the Future of Computing”

Back in 1988, the partial game trees explored by computer chess programs were among the largest search structures in real-world computing. Because the game tree is too large to be fully evaluated, chess programs must make heuristic strategic decisions based on partial information, making it an illustrative subject for teaching AI search. In one of his lectures that year on AI search for games and puzzles, Professor Hans Berliner—a pioneer of computer chess programs 1 —stated: As a student in the field of the theory of computation, I was naturally perplexed but fascinated by this perspective. I had been trained to believe that “Algorithms and computational complexity theory are the foundation of computer science.” However, as it happens, my attempts to understand heuristics in computing have subsequently played a significant role in my career as a theoretical computer scientist. I have come to realize that Berliner’s postulation is a far-reaching worldview, particularly in the age of big, rich, complex, and multifaceted data and models, when computing has ubiquitous interactions with science, engineering, humanity, and society. In this article, 2 I will share some of my experiences on the subject of heuristics in computing, presenting examples of theoretical attempts to understand the behavior of heuristics on real data, as well as efforts to design practical heuristics with desirable theoretical characterizations. My hope is that these theoretical insights from past heuristics—such as spectral partitioning, multilevel methods, evolutionary algorithms, and simplex methods—can shed light on and further inspire a deeper understanding of the current and future techniques in AI and data mining.

Effective Projections on Group Shifts to Decide Properties of Group Cellular Automata

Many decision problems concerning cellular automata are known to be decidable in the case of algebraic cellular automata, that is, when the state set has an algebraic structure and the automaton acts as a morphism. The most studied cases include finite fields, finite commutative rings and finite commutative groups. In this paper, we provide methods to generalize these results to the broader case of group cellular automata, that is, the case where the state set is a finite (possibly non-commutative) finite group. The configuration space is not even necessarily the full shift but a subshift — called a group shift — that is a subgroup of the full shift on [Formula: see text], for any number [Formula: see text] of dimensions. We show, in particular, that injectivity, surjectivity, equicontinuity, sensitivity and nilpotency are decidable for group cellular automata, and non-transitivity is semi-decidable. Injectivity always implies surjectivity, and jointly periodic points are dense in the limit set. The Moore direction of the Garden-of-Eden theorem holds for all group cellular automata, while the Myhill direction fails in some cases. The proofs are based on effective projection operations on group shifts that are, in particular, applied on the set of valid space-time diagrams of group cellular automata. This allows one to effectively construct the traces and the limit sets of group cellular automata. A preliminary version of this work was presented at the conference Mathematical Foundations of Computer Science 2020.

Blocking Trails for f-factors of Multigraphs

Blocking flows were introduced by Dinic (Soviet Math Doklady 11: 1277–1280, 1970) to speed up the computation of maximum network flows. They have been used in algorithms for problems such as maximum cardinality matching of bipartite graphs Hopcroft and Karp (SIAM J Comput 2(4), 225–231, 1973) and general graphs Micali and Vazirani (in: Proceedings of the 21st Annual Symposium on Foundations of Computer Science, 17–27, 1980), maximum weight matching of general graphs Gabow and Tarjan (J ACM 38(4), 815–853, 1991), and many others. The blocking algorithm of Gabow and Tarjan (1991) for matching is based on depth-first search. We extend the depth-first search approach to find f-factors of general multigraphs. Here f is an arbitrary integral-valued function on vertices, an f-matching is a subgraph where every vertex x has degree $$\le f(x)$$ , an f-factor has equality in every degree bound. A set of blocking trails for an f-matching M is a maximal collection $$\mathcal{A}$$ of edge-disjoint augmenting trails such that $$M\bigoplus _{A\in {\mathcal{A}}} A $$ is a valid f-matching. Blocking trails are needed in efficient algorithms for maximum cardinality f-matching Huang and Pettie (Algorithmica 84(7): 1952–1992, 2022), maximum weight f-factors/matchings by scaling Duan et al. (In: Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), Vol. 168 of LIPIcs, 41:1-41:17, 2020; Gabow (A weight-scaling algorithm for f-factors of multigraphs. arXiv:2010.01102 , 2020), and approximate maximum weight f-factors and f-edge covers Huang and Pettie (2022). Since these algorithms find many sets of blocking trails, the time to find blocking trails is a dominant factor in the running time. Our blocking trail algorithm runs in linear time O(m). In independent work and using a different approach, Huang and Pettie (2022) present a blocking trails algorithm using time $$O(m\alpha (m,n))$$ . As examples of the time bounds for the above applications, an approximate maximum weight f-factor is found in time $$O(m\,\alpha (m,n))$$ using Huang and Pettie (2022), and our algorithm eliminates the factor $$\alpha (m,n)$$ . Similarly a maximum weight f-factor is found in time $$O(\sqrt{\Phi \log \Phi }\, m\,\alpha (m,n)\, \log (\Phi W))\,$$ using Huang and Pettie (2022) , ( $$\Phi =\sum _{v\in V} f(v)$$ , W the maximum edge weight) and our algorithm eliminates the $$\alpha (m,n)$$ factor, making the time within a factor $$\sqrt{\log {\Phi }}$$ of the bound for bipartite multigraphs. The technical difficulty for this work stems from the fact that a fixed vertex can occur many times in a given search. This does not occur in ordinary matching or in algorithms for maximum cardinality or maximum weight f-matching. These multiple occurrences can create a new variant of blossom, the “skew blossom”. Also they can make blossoms become “incomplete”, i.e., partially processed yet still relevant in future searches.

An Optimal Separation of Randomized and Quantum Query Complexity

We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order sum to at most , where is the number of variables, is the tree depth, and is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal [Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]. The bounds prior to our work degraded rapidly with , becoming trivial already at . As an application, we obtain, for every integer , a partial Boolean function on bits that has bounded-error quantum query complexity at most and randomized query complexity . This separation of bounded-error quantum versus randomized query complexity is best possible, by the results of Aaronson and Ambainis [SIAM J. Comput., 47 (2018), pp. 982–1038] and Bravyi et al. [Classical Algorithms for Forrelation, arXiv preprint, 2021]. Prior to our work, the best known separation was polynomially weaker: versus for any [A. Tal, Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]. As another application, we obtain an essentially optimal separation of versus for bounded-error quantum versus randomized communication complexity for any . The best previous separation was polynomially weaker: versus (this is implicit in [A. Tal, Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]).

Rapid Mixing of Glauber Dynamics up to Uniqueness via Contraction

For general antiferromagnetic 2-spin systems, including the hardcore model on weighted independent sets and the antiferromagnetic Ising model, there is an for the partition function on graphs of maximum degree when the infinite regular tree lies in the uniqueness region by Li, Lu, and Yin [Correlation Decay up to Uniqueness in Spin Systems, preprint, https://arxiv.org/abs/1111.7064, 2021]. Moreover, in the tree nonuniqueness region, Sly in [Computational transition at the uniqueness threshold, in Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, 2010, pp. 287–296] showed that there is no to estimate the partition function unless . The algorithmic results follow from the correlation decay approach due to Weitz [Counting independent sets up to the tree threshold, in Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 140–149] or the polynomial interpolation approach developed by Barvinok [Combinatorics and Complexity of Partition Functions, Springer, 2016]. However, the running time is only polynomial for constant . For the hardcore model, recent work of Anari, Liu, and Oveis Gharan [Spectral independence in high-dimensional expanders and applications to the hardcore model, in Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science, 2020, pp. 1319–1330] establishes rapid mixing of the simple single-site Markov chain, known as the Glauber dynamics, in the tree uniqueness region. Our work simplifies their analysis of the Glauber dynamics by considering the total pairwise influence of a fixed vertex on other vertices, as opposed to the total influence of other vertices on , thereby extending their work to all 2-spin models and improving the mixing time. More important, our proof ties together the three disparate algorithmic approaches: we show that contraction of the so-called tree recursions with a suitable potential function, which is the primary technique for establishing efficiency of Weitz’s correlation decay approach and Barvinok’s polynomial interpolation approach, also establishes rapid mixing of the Glauber dynamics. We emphasize that this connection holds for all 2-spin models (both antiferromagnetic and ferromagnetic), and existing proofs for the correlation decay and polynomial interpolation approaches immediately imply rapid mixing of the Glauber dynamics. Our proof utilizes the fact that the graph partition function is a divisor of the partition function for Weitz’s self-avoiding walk tree. This fact leads to new tools for the analysis of the influence of vertices and may be of independent interest for the study of complex zeros.

Hop-Constrained Oblivious Routing

We prove the existence of an oblivious routing scheme that is -competitive in terms of , thus resolving a well-known question in oblivious routing. Concretely, consider an undirected network and a set of packets each with its own source and destination. The objective is to choose a path for each packet, from its source to its destination, so as to minimize , defined as follows: The dilation is the maximum path hop length, and the congestion is the maximum number of paths that include any single edge. The routing scheme obliviously and randomly selects a path for each packet independent of (the existence of) the other packets. Despite this obliviousness, the selected paths have within a factor of the best possible value. More precisely, for any integer hop constraint , this oblivious routing scheme selects paths of length at most and is -competitive in terms of congestion in comparison to the best possible congestion achievable via paths of length at most hops. These paths can be sampled in polynomial time. This result can be viewed as an analogue of the celebrated oblivious routing results of Räcke [Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002; Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 2008], which are -competitive in terms of congestion but are not competitive in terms of dilation.

Applications of Mathematical in Computer Science

Abstract: Mathematics (The QUEEN’s mother of all Sciences), is the foundation of Computer Science. Mathematics can be perceived in our garden or park from symmetry of leaves, flowers, fruits etc. and by so many examples of Geometry and symmetry can be seen in nature. Scientists and researchers cannot ideally accomplish their work without the inclusion of mathematics. Mathematics is sociable for analytical skills needed in Computer Disciplines like; Concepts of binary number system, Boolean algebra, Calculus, Discrete mathematics, linear algebra, number theory, and graph theory are the most applicable to the subject of computer science with the accessional emergence of new concepts like machine learning, artificial intelligence, virtual reality and augmented reality make the future of mathematics grow endless. Mathematics has been an important intellectual preoccupation of man for a long time. Computer Science as a formal discipline is about seven decades young. Is the almost spontaneous use of computing? In this article, this paper convey to the frontage the many close connections and parallels between the Mother and daughter sciences. The paper underscores the strong interplay and interactions by looking at some exciting contemporary results from number theory and combinatorial mathematics and algorithms of computer science.

Distributed Shor's algorithm

Shor's algorithm is one of the most important quantum algorithm proposed by Peter Shor [Proceedings of the 35th Annual Symposium on Foundations of Computer Science, 1994, pp. 124--134]. Shor's algorithm can factor a large integer with certain probability and costs polynomial time in the length of the input integer. The key step of Shor's algorithm is the order-finding algorithm, the quantum part of which is to estimate $s/r$, where $r$ is the ``order" and $s$ is some natural number that less than $r$. {{Shor's algorithm requires lots of qubits and a deep circuit depth, which is unaffordable for current physical devices.}} In this paper, to reduce the number of qubits required and circuit depth, we propose a quantum-classical hybrid distributed order-finding algorithm for Shor's algorithm, which combines the advantages of both quantum processing and classical processing. {{ In our distributed order-finding algorithm, we use two quantum computers with the ability of quantum teleportation separately to estimate partial bits of $s/r$.}} The measuring results will be processed through a classical algorithm to ensure the accuracy of the results. Compared with the traditional Shor's algorithm that uses multiple control qubits, our algorithm reduces nearly $L/2$ qubits for factoring an $L$-bit integer and reduces the circuit depth of each computer.

The "never-proved" triangle inequality: A GeoGebra & CAS approach

<abstract><p>We use a quite simple, yet challenging, elementary geometry statement, the so-called "never proved" (by a mathematician) theorem, introduced by Prof. Jiawei Hong in his communication to the IEEE 1986 Symposium on Foundations of Computer Science, to exemplify and analyze the current situation of achievements, ongoing improvements and limitations of GeoGebra's automated reasoning tools, as well as other computer algebra systems, in dealing with geometric inequalities. We present a large collection of facts describing the curious (and confusing) history behind the statement and its connection to automated deduction. An easy proof of the "never proved" theorem, relying on some previous (but not trivial) human work is included. Moreover, as part of our strategy to address this challenging result with automated tools, we formulate a large list of variants of the "never proved" statement (generalizations, special cases, etc.). Addressing such variants with GeoGebra Discovery, ${\texttt{Maple}}$, ${\texttt{REDUCE/Redlog}}$ or ${\texttt{Mathematica}}$ leads us to introduce and reflect on some new approaches (e.g., partial elimination of quantifiers, consideration of symmetries, relevance of discovery vs. proving, etc.) that could be relevant to consider for future improvements of automated reasoning in geometry algorithms. As a byproduct, we obtain an original result (to our knowledge) concerning the family of triangles inscribable in a given triangle.</p></abstract>

A construction of combinatorial NLTS

The NLTS (No Low-Energy Trivial State) conjecture [M. H. Freedman and M. B. Hastings, Quantum Inf. Comput. 14, 144 (2014)] posits that there exist families of Hamiltonians with all low energy states of high complexity (with complexity measured by the quantum circuit depth preparing the state). Here, we prove a weaker version called the combinatorial no low error trivial states (NLETS), where a quantum circuit lower bound is shown against states that violate a (small) constant fraction of local terms. This generalizes the prior NLETS results [L. Eldar and A. W. Harrow, in 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2017), pp. 427–438] and [Nirkhe et al., in 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018), Leibniz International Proceedings in Informatics (LIPIcs), edited by Chatzigiannakis et al. (Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2018), Vol. 107, pp. 1–11]. Our construction is obtained by combining tensor networks with expander codes [M. Sipser and D. Spielman, IEEE Trans. Inf. Theory 42, 1710 (1996)]. The Hamiltonian is the parent Hamiltonian of a perturbed tensor network, inspired by the “uncle Hamiltonian” of Fernández-González et al. [Commun. Math. Phys. 333, 299 (2015)]. Thus, we deviate from the quantum Calderbank-Shor-Steane (CSS) code Hamiltonians considered in most prior works.

Contributions of Quantum Factoring on Quantum Research

This installment of Computer’s series highlights work published in IEEE Symposium on Foundations of Computer Science. Although we typically highlight articles from work published in IEEE Computer Society journals, this month’s exception is due to the prevalence and significance of quantum computing research progress in recent years.

Caching with Time Windows and Delays

We consider two generalizations of the classical weighted paging problem that incorporate the notion of delayed service of page requests. The first is the (weighted) paging with time windows (\sf PageTW) problem, which is like the classical weighted paging problem except that each page request only needs to be served before a given deadline. This problem arises in many practical applications of online caching, such as the “deadline” I/O scheduler in the Linux kernel and video-on-demand streaming. The second, and more general, problem is the (weighted) paging with delay (\sf PageD) problem, where the delay in serving a page request results in a penalty being added to the objective. This problem generalizes the caching problem to allow delayed service, a line of work that has recently gained traction in online algorithms (e.g., [Y. Emek, S. Kutten, and R. Wattenhofer, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, 2016, pp. 333--344; Y. Azar et al., Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, 2017, pp. 551--563; Y. Azar and N. Touitou, Proceedings of the 60th IEEE Annual Symposium on Foundations of Computer Science, 2019, pp. 60--71]). We give $O(\log k\log n)$-competitive algorithms for both the \sf PageTW and \sf PageD problems on $n$ pages with a cache of size $k$. This significantly improves on the previous best bounds of $O(k)$ for both problems [Y. Azar et al., Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, 2017, pp. 551--563]. We also consider the offline \sf PageTW and \sf PageD problems, for which we give $O(1)$-approximation algorithms and prove APX-hardness. These are the first results for the offline problems; even NP-hardness was not known before our work. At the heart of our algorithms is a novel “hitting-set” LP relaxation of the \sf PageTW problem that overcomes the $\Omega(k)$ integrality gap of the natural LP for the problem. To the best of our knowledge, this is the first example of an LP-based algorithm for an online problem with delays/deadlines.

Teaching Mathematics with the Software Engineering Body of Knowledge

It is a fact that employability of the graduates and post-graduates in engineering disciplines does not explicitly require any specific mathematical knowledge. It is unfortunate but true that even though Mathematics is the foundation of computer science it is now being considered as a totally separate subject. The inclusion of topics from Mathematics in the Computing Curricula all over the world has often times been very tough to justify. Mathematical foundations any beyond the first year in a typical Undergraduate Engineering programme are rarely found in the computing curricula today. The author is associated with working on teaching computer science to K-12 schools. Integrating Computer Science and Mathematics has a huge potential but so are the risks. The focus is the IEEE Software Engineering Body of Knowledge [SWEBOK].

Rolling backwards can move you forward: On embedding problems in sparse expanders

We develop a general embedding method based on the Friedman-Pippenger tree embedding technique and its algorithmic version, enhanced with a roll-back idea allowing a sequential retracing of previously performed embedding steps. We use this method to obtain the following results. We show that the size-Ramsey number of logarithmically long subdivisions of bounded degree graphs is linear in their number of vertices, settling a conjecture of Pak [Proceedings of the thirteenth annual ACMSIAM symposium on discrete algorithms (SODA’02), 2002, pp. 321-328]. We give a deterministic, polynomial time online algorithm for finding vertex-disjoint paths of a prescribed length between given pairs of vertices in an expander graph. Our result answers a question of Alon and Capalbo [48th annual IEEE symposium on foundations of computer science (FOCS’07), 2007, pp. 518-524]. We show that relatively weak bounds on the spectral ratio λ / d \lambda /d of d d -regular graphs force the existence of a topological minor of K t K_t where t = ( 1 − o ( 1 ) ) d t=(1-o(1))d . We also exhibit a construction which shows that the theoretical maximum t = d + 1 t=d+1 cannot be attained even if λ = O ( d ) \lambda =O(\sqrt {d}) . This answers a question of Fountoulakis, Kühn and Osthus [Random Structures Algorithms 35 (2009), pp. 444-463].

Application of Mathematics in Computer Science

No one escape the learning of mathematics in one way or other, ranging from our kitchen to our journey from earth to Moon or Mars. Mathematics persists everywhere around us. It can be perceived in our garden or park from symmetry of leaves, flowers, fruits etc. and by so many examples of Geometry and symmetry can be seen in nature. God used mathematics in creation of the universe in one form or the other. Likewise, Mathematics is the queen of all sciences. Scientists and researchers can not perfectly accomplish their work without including mathematics. Mathematics is the foundation of Computer Science. If one is eager to learn any arena of Computer Science, first he/she has to imbibe a love of Mathematics that will be supportive for progressive learning of the said subject. Mathematics is friendly for analytical skills needed in Computer Science. Concepts of binary number system, Boolean algebra, Calculus, Discrete mathematics, linear algebra, number theory, and graph theory are the most applicable to the subject of computer science with the emergence of new concepts like machine learning, artificial intelligence, virtual reality and augmented reality.

On Information Orders on Metric Spaces

Information orders play a central role in the mathematical foundations of Computer Science. Concretely, they are a suitable tool to describe processes in which the information increases successively in each step of the computation. In order to provide numerical quantifications of the amount of information in the aforementioned processes, S.G. Matthews introduced the notions of partial metric and Scott-like topology. The success of partial metrics is given mainly by two facts. On the one hand, they can induce the so-called specialization partial order, which is able to encode the existing order structure in many examples of spaces that arise in a natural way in Computer Science. On the other hand, their associated topology is Scott-like when the partial metric space is complete and, thus, it is able to describe the aforementioned increasing information processes in such a way that the supremum of the sequence always exists and captures the amount of information, measured by the partial metric; it also contains no information other than that which may be derived from the members of the sequence. R. Heckmann showed that the method to induce the partial order associated with a partial metric could be retrieved as a particular case of a celebrated method for generating partial orders through metrics and non-negative real-valued functions. Motivated by this fact, we explore this general method from an information orders theory viewpoint. Specifically, we show that such a method captures the essence of information orders in such a way that the function under consideration is able to quantify the amount of information and, in addition, its measurement can be used to distinguish maximal elements. Moreover, we show that this method for endowing a metric space with a partial order can also be applied to partial metric spaces in order to generate new partial orders different from the specialization one. Furthermore, we show that given a complete metric space and an inf-continuous function, the partially ordered set induced by this general method enjoys rich properties. Concretely, we will show not only its order-completeness but the directed-completeness and, in addition, that the topology induced by the metric is Scott-like. Therefore, such a mathematical structure could be used for developing metric-based tools for modeling increasing information processes in Computer Science. As a particular case of our new results, we retrieve, for a complete partial metric space, the above-explained celebrated fact about the Scott-like character of the associated topology and, in addition, that the induced partial ordered set is directed-complete and not only order-complete.

The Work of Writing Programs: Logic and Inscriptive Practice in the History of Computing

This article explores the entanglement of logic and computing by focusing on the activity of writing. Although mathematical logic is sometimes cast as the immaterial spirit of the computer’s material body, the study of logic also takes place in the physical world through the manipulation of symbols on paper. Already in the 19th century, mathematical logic was understood to be related to mechanization, though not as the science behind an as-yet-uninvented technology. Rather, symbolic notations were seen as tools that opened possibilities but required new kinds of work. Turning to early electronic computing in the 1950s, I observe that researchers similarly relied on novel inscriptive techniques to mitigate labor. Finally, considering Charles Hamblin’s Reverse Polish Notation, I show how logic was a source of notational invention, emerging as a practical resource for the work of writing programs independently of its role as a plausible theoretical foundation for computer science.