Donald E. Knuth has been making foundational contributions to the field of computer science for as long as computer science has been a field. His award-winning textbooks are often given credit for shaping the field, and his scientific papers are widely referenced and stand as milestones of development over a wide variety of topics. The present volume, the seventh in a series of his collected papers, is devoted to his work on the design of new algorithms. Nearly thirty of (...) Knuth’s classic papers are collected in this book and brought up to date with extensive revisions and notes on subsequent developments. The papers cover numerous discrete problems, such as assorting, searching, data compression, theorem proving, and cryptography, as well as methods for controlling errors in numerical computations. (shrink)

We present a criterion for the use of thought experiments as a guide to possibilia that bear on important arguments in philosophy of religion. We propose that the more successful thought experiments are closer to the world in terms of phenomenological realism and the values they are intended to track. This proposal is filled out by comparing thought experiments of life after death by Peter van Inwagen and Dean Zimmerman with an idealist thought experiment. In terms of realism and values (...) we contrast an exemplary thought experiment by Iris Murdoch with one we find problematic by William Irwin. (shrink)

This piece raises a new challenge for Christian materialist accounts of human persons. Revisiting one of the perennial challenges for Christian materialism, explaining the metaphysical compatibility of resurrection and the life everlasting with materialist metaphysics, I argue that resuscitation phenomena reported in scripture undermine van Inwagen’s and Zimmerman’s attempts to reconcile resurrection and materialism. Although this challenge to Christian materialism is not insurmountable, it provides good reason to reject several of the most serious Christian materialist projects and offers a reason (...) for Christians to consider alternatives to materialism. (shrink)

The critique of mission history often involves perpetuating the overestimated impact of the missionaries from opposite sides. It was not so much the missionaries who mattered, but what mattered more was whether the people were responding to the message or not. Today we see the translating function of the missionaries in a clearer way and the people’s reception as the crucial factor in the process of modern Christianization. The World Council of Churches in its declaration “Together Towards Life” separates mission (...) from its entanglement with colonialism as a mission from the margins by grounding it in the triune God, but it seems to overlook the contributing factor of the people as the human subject of the Christianization process. Instead of following a simple input-impact model, we have to acknowledge more those who were adapting themselves to the Christian faith from within their own context. (shrink)

We defend the guise of the good thesis in a tradition going back to Socrates and Plato, according to which persons act on the basis of what appears to them as good or the least bad or evil act available to them. This seems contrary to moral experience, but we defend the thesis against plausible counter-examples in life as well as fiction. We contend that the thesis makes wrong-doing and vice intelligible, but still wrong, dysfunctional and horrific.

Collectively these essays inform educators and researchers at different grade levels about the teaching and learning of proof at each level and, thus, help ...

The present article delves into the extension of Knuth’s fundamental Boolean logic table to accommodate the complexities of indeterminate truth values through the integration of neutrosophic logic (Smarandache & Christianto, 2008). Neutrosophic logic, rooted in Florentin Smarandache’s groundbreaking work on Neutrosophic Logic (cf. Smarandache, 2005, and his other works), introduces an additional truth value, ‘indeterminate,’ enabling a more comprehensive framework to analyze uncertainties inherent in computational systems. By bridging the gap between traditional boolean operations and the indeterminacy present in (...) various real-world scenarios, this extension redefines logic tables, introducing neutrosophic operators that capture nuances beyond the binary realm. Through a thorough exploration of neutrosophic logic's principles and its implications in computational paradigms, this study proposes a novel approach to logic design that accommodates uncertain, imprecise, and incomplete information. This paradigm shift in logic tables not only broadens the spectrum of computing methodologies but also holds promise in fields such as decision-making systems and data analytics. This article amalgamates insights from over twelve key references encompassing seminal works in boolean logic, neutrosophic logic, and their applications in diverse scientific and computational domains, aiming to pave the way for a more robust and adaptable logic framework in computation. (shrink)

In the children's book ``The Phantom Tollbooth'' by Norton Juster one can find the following passage:``Yes, please,'' said Milo. ``Can you show me the biggest number there is?''.

This paper consists primarily of a survey of results of Harvey Friedman about some proof-theoretic aspects of various forms of Kruskal's tree theorem, and in particular the connection with the ordinal Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Verlen hierarchies, some subsystems of second-order logic, slow-growing and fast-growing hierarchies including Girard's result, and Goodstein sequences. The central theme of this paper is a powerful theorem due to (...) Kruskal, the ‘tree theorem’, as well as a ‘finite miniaturization’ of Kruskal's theorem due to Harvey Friedman. These versions of Kruskal's theorem are remarkable from a proof-theoretic point of view because they are not provable in relatively strong logical systems. They are examples of so-called ‘natural independence phenomena’, which are considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Gödel. Kruskal's tree theorem also plays a fundamental role in computer science, because it is one of the main tools for showing that certain orderings on trees are well founded. These orderings play a crucial role in proving the termination of systems of rewrite rules and the correctness of Knuth-Bendix completion procedures. There is also a close connection between a certain infinite countable ordinal called Γo and Kruskal's theorem. Previous definitions of the function involved in this connection are known to be incorrect, in that, the function is not monotonic. We offer a repaired definition of this function, and explore briefly the consequences of its existence. (shrink)

Problem-solving software that is not-necessarily infallible is central to AI. Such software whose correctness and incorrectness properties are deducible by agents is an issue at the foundations of AI. The Comprehensibility Theorem, which appeared in a journal for specialists in formal mathematical logic, might provide a limitation concerning this issue and might be applicable to any agents, regardless of whether the agents are artificial or natural. The present article, aimed at researchers interested in the foundations of AI, addresses many questions (...) related to that theorem, including differences between it and results of Gödel and Turing that have sometimes played key roles in Minds and Machines articles. This study also suggests that—if one is willing to assume a thesis due to Donald Knuth—the Comprehensibility Theorem is the first mathematical theorem implying the impossibility of any AI agent or natural agent—including a not-necessarily infallible human agent—satisfying a rigorous and deductive interpretation of the self-comprehensibility challenge. Some have pointed out the difficulty of self-comprehensibility, even according to presumably a less rigorous interpretation. This includes Socrates, who considered it to be among the most important of intellectual tasks. Self-comprehensibility in some form might be essential for a kind of self-reflection useful for self-improvement that might enable some agents to increase their success. We use the methods of applied mathematics, rather than philosophy, although some topics considered could be of interest to philosophers. (shrink)

We re-express our theorem on the strong-normalisation of display calculi as a theorem about the well-foundedness of a certain ordering on first-order terms, thereby allowing us to prove the termination of systems of rewrite rules. We first show how to use our theorem to prove the well-foundedness of the lexicographic ordering, the multiset ordering and the recursive path ordering. Next, we give examples of systems of rewrite rules which cannot be handled by these methods but which can be handled by (...) ours. Finally, we show that our method can also prove the termination of the Knuth-Bendix ordering and of dependency pairs. Keywords: rewriting, termination, well-founded ordering, recursive path ordering.. (shrink)

A central concern of generative grammar is the relationship between hierarchy and word order, traditionally understood as two dimensions of a single syntactic representation. A related concern is directionality in the grammar. Traditional approaches posit process-neutral grammars, embodying knowledge of language, put to use with infinite facility both for production and comprehension. This has crystallized in the view of Merge as the central property of syntax, perhaps its only novel feature. A growing number of approaches explore grammars with different directionalities, (...) often with more direct connections to performance mechanisms. This paper describes a novel model of universal grammar as a one-directional, universal parser. Mismatch between word order and interpretation order is pervasive in comprehension; in the present model, word order is language-particular and interpretation order (i.e., hierarchy) is universal. These orders are not two dimensions of a unified abstract object (e.g., precedence and dominance in a single tree); rather, both are temporal sequences, and UG is an invariant real-time procedure (based on Knuth's stack-sorting algorithm) transforming word order into hierarchical order. This shift in perspective has several desirable consequences. It collapses linearization, displacement, and composition into a single performance process. The architecture provides a novel source of brackets (labeled unambiguously and without search), which are understood not as part-whole constituency relations, but as storage and retrieval routines in parsing. It also explains why neutral word order within single syntactic cycles avoids 213-like permutations. The model identifies cycles as extended projections of lexical heads, grounding the notion of phase. This is achieved with a universal processor, dispensing with parameters. The empirical focus is word order in noun phrases. This domain provides some of the clearest evidence for 213-avoidance as a cross-linguistic word order generalization. Importantly, recursive phrase structure “bottoms out” in noun phrases, which are typically a single cycle (though further cycles may be embedded, e.g., relative clauses). By contrast, a simple transitive clause plausibly involves two cycles (vP and CP), embedding further nominal cycles. In the present theory, recursion is fundamentally distinct from structure-building within a single cycle, and different word order restrictions might emerge in larger domains like clauses. (shrink)

Perceptions of justice in decision outcomes and processes increases trust, support for authorities, and satisfaction with unfavorable outcomes (Lind and Tyler 1988). However, very little research has been done to apply justice concepts to natural resource decision making contexts (an exception is Lauber and Knuth 1997; 1998). In the study presented here, two rounds of focus groups with public participants in the Northern Lower Michigan Ecosystem Management Project were conducted. Participants were asked to describe their experiences with and attitudes (...) toward different types of public participation. They were also asked what criteria they use when judging the fairness of a natural resource agency decision. Themes emerging in the answers included the importance of representation, voice, consideration, logic, and desired outcomes. These are very similar to justice principles documented in other fields. This suggests managers would be wise to focus less on public participation techniques and more on achieving fair decision-making processes. (shrink)

One shortcoming of the chain rule is that it does not iterate: it gives the derivative of f(g(x)), but not (directly) the second or higher-order derivatives. We present iterated differentials and a version of the multivariable chain rule which iterates to any desired level of derivative. We first present this material informally, and later discuss how to make it rigorous (a discussion which touches on formal foundations of calculus). We also suggest a finite calculus chain rule (contrary to Graham, (...) class='Hi'>Knuth and Patashnik's claim that "there's no corresponding chain rule of finite calculus"). (shrink)

The notion of a minimal form is defined as an extension of the notion of a normal form in λ-β-calculus and its meaning is discussed in a computational environment. The features of the Knuth-Gross reduction strategy are used to prove that to possess a minimal form, for a generic term, is a semidecidable predicate.

Propositional logic has been recognized throughout the centuries as one of the cornerstones of reasoning in philosophy and mathematics. Over time, its formalization into Boolean algebra was accompanied by the recognition that a wide range of combinatorial problems can be expressed as propositional satisfiability (SAT) problems. Because of this dual role, SAT developed into a mature, multi-faceted scientific discipline, and from the earliest days of computing a search was underway to discover how to solve SAT problems in an automated fashion. (...) This book, the Handbook of Satisfiability, is the second, updated and revised edition of the book first published in 2009 under the same name. The handbook aims to capture the full breadth and depth of SAT and to bring together significant progress and advances in automated solving. Topics covered span practical and theoretical research on SAT and its applications and include search algorithms, heuristics, analysis of algorithms, hard instances, randomized formulae, problem encodings, industrial applications, solvers, simplifiers, tools, case studies and empirical results. SAT is interpreted in a broad sense, so as well as propositional satisfiability, there are chapters covering the domain of quantified Boolean formulae (QBF), constraints programming techniques (CSP) for word-level problems and their propositional encoding, and satisfiability modulo theories (SMT). An extensive bibliography completes each chapter. This second edition of the handbook will be of interest to researchers, graduate students, final-year undergraduates, and practitioners using or contributing to SAT, and will provide both an inspiration and a rich resource for their work. Edmund Clarke, 2007 ACM Turing Award Recipient: "SAT solving is a key technology for 21st century computer science." Donald Knuth, 1974 ACM Turing Award Recipient: "SAT is evidently a killer app, because it is key to the solution of so many other problems." Stephen Cook, 1982 ACM Turing Award Recipient: "The SAT problem is at the core of arguably the most fundamental question in computer science: What makes a problem hard?". (shrink)

Given two strings, text t of length n, and pattern p = p1…pk of length k, and given a natural number w, the subsequence matching problem consists in finding the number of size w windows of text t which contain pattern p as a subsequence, i.e. the letters p1,…,pk occur in the window, in the same order as in p, but not necessarily consecutively . Subsequence matching is used for finding frequent patterns and association rules in databases. We generalize the (...) Knuth–Morris–Pratt pattern matching algorithm; we define a non-conventional kind of RAM, the MP-RAMs which model more closely the microprocessor operations; we design an O on-line algorithm for solving the subsequence matching problem on MP-RAMs. (shrink)

• alpha-beta pruning characterizes human play, but it ticed by early chess programmers—Turing, Shannon, Ulam, and Bernstein. We humans are not very good ing the heuristics we ourselves use. Approximations to used by Samuel, Newell and Simon, McCarthy. Proved lent to minimax by Hart and Levine, independently Knuth gives details.