A pseudo-natural algorithm for the word problem of a finitely presented group is an algorithm which not only tells us whether or not a word w equals 1 in the group but also gives a derivation of 1 from w when w equals 1. In [13], [14] Madlener and Otto show that, if we measure complexity of a primitive recursive algorithm by its level in the Grzegorczyk hierarchy, there are groups in which a pseudo-natural algorithm is (...) arbitrarily more complicated than an algorithm which simply solves the word problem. In a given group the lowest degree of complexity that can be realised by a pseudo-natural algorithm is essentially the derivational complexity of that group. Thus the result separates the derivational complexity of the word problem of a finitely presented group from its intrinsic complexity. The proof given in [13] involves the construction of a finitely presented group G from a Turing machine T such that the intrinsic complexity of the word problem for G reflects the complexity of the halting problem of T, while the derivational complexity of the word problem for G reflects the runtime complexity of T. The proof of one of the crucial lemmas in [13] is only sketched, and part of the purpose of this paper is to give the full details of this proof. We will also obtain a variant of their proof, using modular machines rather than Turing machines. As for several other results, this simplifies the proofs considerably. MSC: 03D40, 20F10. (shrink)

By theword problemfor some class of algebraic structures we mean the problem of determining, given a finite setEof equations between words and an additional equationx=y, whetherx=ymust hold in all structures satisfying each member ofE. In 1947 Post [P] showed the word problem for semigroups to be undecidable. This result was strengthened in 1950 by Turing, who showed the word problem to be undecidable forcancellation semigroups,i.e. semigroups satisfying thecancellation propertyNovikov [N] eventually showed the word (...) class='Hi'>problem for groups to be undecidable.In 1966 Gurevich [G] showed the word problem to be undecidable forfinitesemigroups. However, this result on finite structures has not been extended to cancellation semigroups or groups; indeed it is easy to see that a finite cancellation semigroup is a group, so both questions are the same. We do not here settle the word problem for finite groups, but we do show that the word problem is undecidable for finite semigroups with zero satisfying an approximation to the cancellation property. (shrink)

This note addresses the issue as to which ceers can be realized by word problems of computably enumerable (or, simply, c.e.) structures (such as c.e. semigroups, groups, and rings), where being realized means to fall in the same reducibility degree (under the notion of reducibility for equivalence relations usually called “computable reducibility”), or in the same isomorphism type (with the isomorphism induced by a computable function), or in the same strong isomorphism type (with the isomorphism induced by a (...) computable permutation of the natural numbers). We observe, e.g., that every ceer is isomorphic to the word problem of some c.e. semigroup, but (answering a question of Gao and Gerdes) not every ceer is in the same reducibility degree of the word problem of some finitely presented semigroup, nor is it in the same reducibility degree of some non‐periodic semigroup. We also show that the ceer provided by provable equivalence of Peano Arithmetic is in the same strong isomorphism type as the word problem of some non‐commutative and non‐Boolean c.e. ring. (shrink)

Let G be a finite group, T denote the theory of Z[G]-lattices . It is shown that T is undecidable when there are a prime p and a p-subgroup S of G such that S is cyclic of order p4, or p is odd and S is non-cyclic of order p2, or p = 2 and S is a non-cyclic abelian group of order 8 . More precisely, first we prove that T is undecidable because it interprets the word (...) problem for finite groups; then we lift undecidability from T to T. (shrink)

Today, the earth is an interconnected globe humming with electronic transmissions-a chattering planet nestled in the provident silence of space. The ethical question is whether this is contributing to authentic human development and helping individuals and peoples to be true to their transcendent destiny. The new media are powerful tools for education, cultural enrichment, commercial activity, political participation, intercultural dialogue and understanding. They also can serve the cause of religion. Yet the new information technology needs to be informed and guided (...) by solidarity in the service of the common good bridging the digital divide within and among nations. The value-laden message of Western secular culture to people and traditional societies in many cases ill-prepared to evaluate it leads to widespread crisis, for example, in regard to marriage and family life. This technology can be a means for solving human problems, promoting the integral development of persons, creating a world governed by justice, peace, and love. It also can help men and women in their age-old search for self-understanding. Like today's world itself, the world of media, including the Internet, has been brought by Christ, inchoately yet truly, within the boundaries of the kingdom of God and placed in service to the word of salvation. (shrink)

Many individuals with developmental language disorder (DLD) and individuals who are deaf or hard of hearing (D/HH) have social–emotional problems, such as social difficulties, and show signs of aggression, depression, and anxiety. These problems can be partly associated with their executive functions (EFs) and theory of mind (ToM). The difficulties of both groups in EF and ToM may in turn be related to self-directed speech (i.e., overt or covert speech that is directed at the self). Self-directed speech is thought (...) to allow for the construction of non-sensory representations (i.e., representations that do not coincide with direct observation). Such non-sensory representations allow individuals to overcome the limits set upon them by the senses. This ability is constrained by the development of word meaning structure (i.e., the way words are understood). We argue that the greater ability to construct non-sensory representations may result in more enhanced forms of EF and ToM. We conclude that difficulties in EF, ToM, and social–emotional functioning in those with hearing and language problems may be accounted for in terms of word meaning impairments. We propose that word meaning structure and self-directed speech should be considered in assigning EF and ToM treatments to individuals with DLD and those who are D/HH. (shrink)

Earlier research by the author brought about findings suggesting that people in a special way process words related to demands of a problem they previously solved, even when they do not consciously notice this relationship. The findings concerned interference in the task in which the words appeared, a shift in affective responses to them that depended on sex of the participants, and impaired memory of the words. The aim of this study was to replicate these effects and to find (...) out whether they are related to working memory span of the participants, taken as a measure of the individual’s ability to control attention. Participants in the experimental group solved a divergent problem, then performed an ostensibly unrelated speeded affective classification task concerning each of a series of nouns, and then performed an unexpected cued recall task for the nouns. Afterwards, a task measuring WM span was administered. In the control group there was no problem-solving phase. Response latencies for words immediately following problem-related words in the classification task were longer in the experimental than in the control group, but there was no relationship between this effect and WM span. Solving the problem, in interaction with sex of the participants and, independently, with their WM span, influenced affective responses to problem-related words. Recall of these words, however, was not impaired in the experimental group. (shrink)

We prove that the conjugacy problem for the automorphism group of the random graph is Borel complete, and discuss the analogous problem for some other countably categorical structures.

The solution of the outer multiplicity problem in the tensor product of U(3) irreducible representations (irreps) developed by Biedenharn et al.(1–7) and realized through the well-known Draayer-Akiyama (DA) computer code(8) is extended to the quantum algebra Uq(3). An analytic formula for special stretched Uq(3) Wigner coefficients, $$\left\langle {(\lambda _1 \mu _1 ) H_1 , (\lambda _2 \mu _2 ) \varepsilon _2 \Lambda _2 m_2 \left| { (\lambda _3 \mu _3 ) H_3 } \right.} \right\rangle _{\max }^q $$ is derived (...) using a projection operator method.(9–10) In this expression Hi denotes the highest weight vector of the (λiμi) irrep; the subscript “max” means coefficients corresponding to a unit tensor operator with a maximal characteristic null space; and q is the usual quantum label so the standard U(3) Wigner coefficient, which is required in the DA code, can be obtained in the q→1 limit of the theory. To illustrate the theory, some Uq(3) Wigner coefficients for the tensor product (22)×(22) are calculated. The procedure for evaluating nonhighest weight Wigner Uq(3) coefficients follow the DA prescription. (shrink)

The explicit history of the “hidden variables” problem is well-known and established. The main events of its chronology are traced. An implicit context of that history is suggested. It links the problem with the “conservation of energy conservation” in quantum mechanics. Bohr, Kramers, and Slaters (1924) admitted its violation being due to the “fourth Heisenberg uncertainty”, that of energy in relation to time. Wolfgang Pauli rejected the conjecture and even forecast the existence of a new and unknown then (...) elementary particle, neutrino, on the ground of energy conservation in quantum mechanics, afterwards confirmed experimentally. Bohr recognized his defeat and Pauli’s truth: the paradigm of elementary particles (furthermore underlying the Standard model) dominates nowadays. However, the reason of energy conservation in quantum mechanics is quite different from that in classical mechanics (the Lie group of all translations in time). Even more, if the reason was the latter, Bohr, Cramers, and Slatters’s argument would be valid. The link between the “conservation of energy conservation” and the problem of hidden variables is the following: the former is equivalent to their absence. The same can be verified historically by the unification of Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics in the contemporary quantum mechanics by means of the separable complex Hilbert space. The Heisenberg version relies on the vector interpretation of Hilbert space, and the Schrödinger one, on the wave-function interpretation. However the both are equivalent to each other only under the additional condition that a certain well-ordering is equivalent to the corresponding ordinal number (as in Neumann’s definition of “ordinal number”). The same condition interpreted in the proper terms of quantum mechanics means its “unitarity”, therefore the “conservation of energy conservation”. In other words, the “conservation of energy conservation” is postulated in the foundations of quantum mechanics by means of the concept of the separable complex Hilbert space, which furthermore is equivalent to postulating the absence of hidden variables in quantum mechanics (directly deducible from the properties of that Hilbert space). Further, the lesson of that unification (of Heisenberg’s approach and Schrödinger’s version) can be directly interpreted in terms of the unification of general relativity and quantum mechanics in the cherished “quantum gravity” as well as a “manual” of how one can do this considering them as isomorphic to each other in a new mathematical structure corresponding to quantum information. Even more, the condition of the unification is analogical to that in the historical precedent of the unifying mathematical structure (namely the separable complex Hilbert space of quantum mechanics) and consists in the class of equivalence of any smooth deformations of the pseudo-Riemannian space of general relativity: each element of that class is a wave function and vice versa as well. Thus, quantum mechanics can be considered as a “thermodynamic version” of general relativity, after which the universe is observed as if “outside” (similarly to a phenomenological thermodynamic system observable only “outside” as a whole). The statistical approach to that “phenomenological thermodynamics” of quantum mechanics implies Gibbs classes of equivalence of all states of the universe, furthermore re-presentable in Boltzmann’s manner implying general relativity properly … The meta-lesson is that the historical lesson can serve for future discoveries. (shrink)

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out a sequence of examples. We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non-classifiable. (...) In this paper, we calculate the degree of the isomorphism problem for Abelian p-groups of bounded Ulm length. The result is a sequence of classes whose isomorphism problems are cofinal in the hyperarithmetical hierarchy. In the process, new back-and-forth relations on such groups are calculated. (shrink)

We prove that if p ≠ q are distinct primes, then the classification problems for p-local and q-local torsion-free abelian groups of rank two are incomparable with respect to Borel reducibility.

The Sunnah has an important role in the correct understanding and the protection from wrong interpretations of the Qurʾān. Accordingly, ḥadīth played a crucial role in shaping the opinions of the mufassirs. In this article, the tafsir titled Kur’an Yolu written by a group of scholars and first published by the Presidency of the Religious Affairs in 2003 which is widely read in Tur-key is examined from ḥadīth usage point of view. The problems in ḥadīth usage are classified under headings (...) such as insufficiency/incorrectness in the explanation of verses due to insuffi-cient usage of ḥadīth, not including ḥadīth which corroborate the verse, wrong conclusions derived from ḥadīth, problems with citations, ḥadīth translation problems, wrong explanation of ḥadīth, problems with some issues related to Sunnah. Over fifty examples where there were wrong citations of ḥadīth, wrong conclusions were derived, there was mixing up of ḥadīth and translation errors leading to wrong meanings were determined and the most important ones were studied. Due to these problems regarding ḥadīth usage, it was seen that sometimes the unity of Qurʾān and Sunnah was violated and some wrong explanations were given. The article is important due to determination of these mistakes and attracting attention to care and pre-caution in ḥadīth usage in this kind of works. This study indicates the importance of Qurʾān-Sunnah unity and shows the important role that ḥadīths play in understanding the verses. It is observed that in case hadiths related to a verse are not given or given in an insufficient or in-correct manner the verse in question is understood or interpreted incorrectly. Therefore, the correct translation and explanation of the ḥadīth, avoiding citation mistakes and making use of primary sources are very important. The factors that lead to wrong conclusions are not evalu-ating a ḥadīth in its entirety (taking into account only the first part of the ḥadīth), not evaluat-ing the ḥadīths related to a subject in entirety (taking into account only certain ḥadīths), not making use of commentaries in order to understand a ḥadīth correctly and wrong citation of a ḥadīth. The fact that the ḥadīth which is about the gladness of Allah due to the repentance of a person has been confused with another ḥadīth and an additional error has been made in the text of this ḥadīth indicates that the authors have remembered this ḥadīth incorrectly and they have not verified this ḥadīth in its source. Given the material mistakes in many other examples under the citation errors title, it has been concluded that the wrong citation of ḥadīths in a way that affects the meaning constitutes a serious problem for the work under examination. As to the translation problems, it is observed that they arise from wrong citation of the ḥadīth from its sources (for instance omitting a word or adding expressions that are not found in the ḥadīth) or not making use of commentaries in understanding the ḥadīth. It is necessary that for the analysis of the ḥadīth the authors make use of the tafsir of Ibn Kathīr who is a ḥadīth schol-ar along with the works of other scholars who are not specialized in ḥadīth in order to correct-ly determine the accuracy level and meaning of the ḥadīth. It seems necessary that a commis-sion that includes ḥadīth scholars review this work in order to correct the mistakes that are found and to make use of the suggestions that are put forward. This step is important for cor-recting the mistakes found in this prestigious and widely used work. Furthermore, since many readers gain knowledge about Qurʾān using this work, consideration of issues raised in this study will contribute to the forthcoming edition of the work. It is surprising that despite pro-posed corrections since its initial publication, so many material mistakes still exist in this valu-able work which is a product of great effort. It is hoped that it will continue to become useful for people by making necessary corrections. (shrink)

Epistemic paternalism occurs when a decision is made for an agent which helps them arrive at the truth, though they didn’t consent to that decision (and sometimes weren’t even aware of it). Common defenses of epistemic paternalism claim that it can help promote positive veritistic results. In other words, epistemic paternalism is often good for inquiry. I argue that there is often a better alternative available to epistemic paternalism in the form of what I call Open Group Inquiry. I then (...) examine how Open Group Inquiry can be applied to cases of religious inquiry, while noting that epistemic paternalism is impermissible in cases of general religious inquiry. I argue that in the case of religious inquiry, there are serious questions about what constitutes evidence along with how to evaluate it. Rather than posing a particular worry for Open Group Inquiry, I suggest these questions pose a problem for religious inquiry in general. I conclude that while it very much matters how concepts like religious knowledge, religious faith, scepticism, etc., are defined, these considerations may well pave the way for a novel argument for religious scepticism. (shrink)

It is proved that for a variety of groups in which the relatively free groups are solvable, the relatively free groups of distinct finite rank are not elementarily equivalent.

We have two polynomial time results for the uniform word problem for a quasivariety Q: The uniform word problem for Q can be solved in polynomial time iff one can find a certain congruence on finite partial algebras in polynomial time. Let Q* be the relational class determined by Q. If any universal Horn class between the universal closure S and the weak embedding closure S̄ of Q* is finitely axiomatizable then the uniform word (...) class='Hi'>problem for Q is solvable in polynomial time. This covers Skolem's 1920 solution to the uniform word problem for lattices and Evans' 1953 applications of the weak embeddability property for finite partial V algebras. (shrink)

In this paper, we will prove the inevitable non-uniformity of two constructions from combinatorial group theory related to the word problem for finitely generated groups and the Higman—Neumann—Neumann Embedding Theorem.

For my present purposes, and in terms of my immediate concerns, the problem of ideology in American literary history has three different though closely related aspects: first, the multivolume American literary history I have begun to edit; then, the concept of ideology as a constituent part of literary study, and, finally, the current revaluation of the American Renaissance. I select this period because it has been widely regarded as both the source and the epitome of our literary tradition; because (...) it has become, accordingly, the focal point of the critical revision now under way in American studies; and because, from either of these perspectives, literary or critical, it seems to me a particularly fitting subject for the occasion. For one thing, we owe the idea of an American Renaissance to F. O. Matthiessen, who was a prime mover of the Salzburg Seminar, and a member of its first faculty in 1947. Moreover, American Renaissance was a classic work of revisionist criticism. It reset the terms for the study of American literary history; it gave us a new canon of classic texts; and it inspired the growth of American studies in the United States and abroad. It is not too much to say that Matthiessen, American Renaissance, and the Salzburg Seminar brought American literature to postwar Europe. What followed, from the late forties through the sixties, was the flowering of a new academic field, complete with programs of study, periodicals, theses, conferences, and distinguished procession of scholarly authorities, including many graduates of the Salzburg Seminar.Matthiessen figures as a watershed in this development. For if American Renaissance marked the seeding-time of a new academic field, it was also the harvest of some three decades of literary study. I refer, first of all, to the dual legacy that Matthiessen acknowledges of T. S. Eliot and Vernon Parrington—which is to say, the partnership in American Renaissance between the terms “literary” and “history”; or, in the words of Matthiessen’s subtitle, between Art and Expression in the Age of Emerson and Whitman: “art,” meaning a small group of aesthetic masterpieces, and “expression,” meaning representative works, reflecting and illuminating the culture at large. It was the remarkable achievement of Matthiessen that his book yokes these concepts gracefully together. Somehow, one concept seems to support the other. The historical designation American seems richer for its association with an aesthetic renaissance; Emerson’s and Whitman’s art gains substance by its capacity to express the age. Matthiessen himself did not feel it necessary to explain the connection. But we can see in retrospect that what made it work—what made it, indeed, unnecessary for Matthiessen to explain the connection—was an established consensus, or rather a consensus long in the making, which American Renaissance helped establish. I mean a consensus about the term “literary” that involved the legitimation of a certain canon, and a consensus about the term “history” that was legitimated by a certain concept of America. Sacvan Bercovitch is Carswell Professor of English and American Literature at Harvard University. He is the Author of The Puritan Origins of the American Self and The American Jeremiad, among other works. He has also edited several collections of essays, most recently Reconstructing American Literary History. (shrink)

Every science has its technical vocabulary, consisting in part of terms coined for explicit purposes and in part of words borrowed from ordinary discourse and used with greater or lesser degrees of precision. Words of the latter sort pose curious problems, some of them familiar to those historians of science concerned with, for example, what Galileo meant by forza and Newton by attraction. Indeed, analogous problems face any historian seeking to understand the older meanings of terms still in use today.

In this paper, we shall discuss some recent contributions to the project [15, 14, 2, 18, 22, 23] of explaining why no satisfactory system of complete invariants has yet been found for the torsion-free abelian groups of finite rank n ≥ 2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n-dimensional vector space ℚn which contain n linearly independent elements. Thus the collection of torsion-free abelian groups (...) of rank at most n can be naturally identified with the set S of all nontrivial additive subgroups of ℚn. In 1937, Baer [4] solved the classification problem for the class Sof rank 1 groups as follows.Let ℙ be the set of primes. If G is a torsion-free abelian group and 0 ≠ x ϵ G, then the p-height of x is defined to behx = sup{n ϵ ℕ ∣ There exists y ϵ G such that pny = x} ϵ ℕ ∪{∞}; and the characteristic χ of x is defined to be the function. (shrink)

Generic generalisations like ‘Opioids are highly addictive’ are very useful in scientific communication, but they can often be interpreted in many different ways. Although this is not a problem when all interpretations provide the same answer to the question under discussion, a problem arises when a generic generalisation is used to answer a question other than that originally intended. In such cases, some interpretations of the generalisation might answer the question in a way that the original speaker would (...) not endorse. Rather than excising generic generalisations from scientific communication, I recommend that scientific communicators carefully consider the kinds of questions their words might be taken to answer and try to avoid phrasing that might be taken to provide unintended answers. (shrink)