In this paper, I provide a thorough discussion and reconstruction of Bernard Bolzano’s theory of grounding and a detailed investigation into the parallels between his concept of grounding and current notions of normal proofs. Grounding (Abfolge) is an objective ground-consequence relation among true propositions that is explanatory in nature. The grounding relation plays a crucial role in Bolzano’s proof-theory, and it is essential for his views on the ideal buildup of scientific theories. Occasionally, similarities have been pointed out between Bolzano’s ideas on grounding and cut-free proofs in Gentzen’s sequent calculus. My thesis is, however, that they bear an even stronger resemblance to the normal natural deduction proofs employed in proof-theoretic semantics in the tradition of Dummett and Prawitz.

A property of finite graphs is called non-deterministically testable if it has a ‘certificate’ such that once the certificate is specified, its correctness can be verified by random local testing. In this paper we study certificates that consist of one or more unary and/or binary relations on the nodes, in the case of dense graphs. Using the theory of graph limits, we prove that non-deterministically testable properties are also deterministically testable.

Balakrishnan and Zhao does an excellent job in this issue at reviewing the recent advances on stochastic comparison between order statistics from independent and heterogeneous observations with proportional hazard rates, gamma distribution, geometric distribution, and negative binomial distributions, the relation between various stochastic order and majorization order of concerned heterogeneous parameters is highlighted. Some examples are presented to illustrate main results while pointing out the potential direction for further discussion.

The traveling salesman problem (TSP) is one of the most fundamental optimizationproblems. We consider the β-metric traveling salesman problem(Δβ-TSP), i.e., the TSPrestricted to graphs satisfying the β-triangle inequalityc({v,w}) ≤ β(c({v,u}) + c({u,w})),for some cost function c and any three vertices u,v,w.The well-known path matching Christofides algorithm (PMCA) guarantees an approximationratio of 3β2/2 and is the best known algorithm for theΔβ-TSP, for 1 ≤ β ≤ 2. Weprovide a complete analysis of the algorithm. First, we correct an error in the originalimplementation that may produce an invalid solution. Using a worst-case example, we thenshow that the algorithm cannot guarantee a better approximation ratio. The example canalso be used for the PMCA variants for the Hamiltonian path problem with zero and oneprespecified endpoints. For two prespecified endpoints, we cannot reuse the example, butwe construct another worst-case example to show the optimality of the analysis also inthis case.

We provide an algorithm for listing all minimal 2-dominating sets of a tree of ordern in time 𝒪(1.3248n). This implies that every tree has at most1.3248n minimal 2-dominating sets. We also show that thisbound is tight.

We discuss how much space is sufficient to decide whether a unary given numbern is a prime. We show thatO(log log n) space is sufficient for a deterministicTuring machine, if it is equipped with an additional pebble movable along the input tape,and also for an alternating machine, if the space restriction applies only to itsaccepting computation subtrees. In other words, the language is a prime is inpebble–DSPACE(log log n) and also inaccept–ASPACE(log log n). Moreover, if the givenn is composite, such machines are able to find a divisor ofn. Since O(log log n) space is toosmall to write down a divisor, which might requireΩ(log n) bits, the witness divisor is indicated by theinput head position at the moment when the machine halts.

We give several new applications of the wreath product of forest algebras to the study oflogics on trees. These include new simplified proofs of necessary conditions fordefinability in CTL and first-order logic with the ancestor relation; asequence of identities satisfied by all forest languages definable inPDL; and new examples of languages outside CTL, alongwith an application to the question of what properties are definable in bothCTL and LTL.

This paper is a sequel to Beall (2011), in which I both give and discuss the philosophical import of a ‘classical collapse’ result for the propositional (multiple-conclusion) logic LP+. Feedback on such ideas prompted a spelling out of the first-order case. My aim in this paper is to do just that: namely, explicitly record the first-order result(s), including the collapse results for K3+ and FDE+.

The purpose of this paper is to investigate categoricity arguments conducted in second order logic and the philosophical conclusions that can be drawn from them. We provide a way of seeing this result, so to speak, through a first order lens divested of its second order garb. Our purpose is to draw into sharper relief exactly what is involved in this kind of categoricity proof and to highlight the fact that we should be reserved before drawing powerful philosophical conclusions from it.

This study designed and developed a web-based reading strategy training program and investigated students’ use of its features and EFL teachers’ and students’ perceptions of the program. The recent proliferation of online reading materials has made information easily available to L2 readers; however, L2 readers’ ability to deal with them requires the development of specific reading strategies. The researcher therefore constructed a web-based strategy training program on the basis of L2 reading strategy research and pedagogy. The program offers four types of reading strategy functions (Global, Problem-solving, Support, and Socio-affective) through 15 strategy buttons: Keyword, Preview, Prediction, Outline, Summary, Semantic Mapping, Pronunciation, Speed Reading, Dictionary, Translation, Grammar, Highlight, Notebook, Music Box, and My Questions. Forty college teachers and thirty-two EFL students in Taiwan were invited to use and evaluate this program. The researcher tracked students’ use of the functions, and teachers and students completed a survey and written reflections that documented their perceptions of the program. Both groups gave positive feedback on the program's user-friendly interface design and the effectiveness of its strategy function keys for enhancing reading comprehension and motivating learning. They also thought highly of the site's extensive offerings of reading opportunities supported by effective reading aids and a computerized classroom management system, features not available in large traditional classes. There was, however, a gap between what teachers thought and what students did. The teachers thought highly of Global strategies, whereas students regarded Support strategies as more useful. The low-proficiency group's heavy use of Support strategies explained this gap. The high-proficiency group's more frequent use of Global strategies echoed teachers’ preference for teaching Global strategies. This connection suggests that teachers should provide more explicit training to encourage all students to use Global strategies for overall textual understanding.