The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi

Why Philosophers Should Care About Computational Complexity

One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed case that one would be wrong. In particular, I argue that computational complexity theory---the field that studies the resources (such as time, space, and randomness) needed to solve computational problems---leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume's problem of induction, Goodman's grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest. I end by discussing aspects of complexity theory itself that could benefit from philosophical analysis.Comment: 58 pages, to appear in "Computability: G\"odel, Turing, Church, and beyond," MIT Press, 2012. Some minor clarifications and corrections; new references adde

Open versus closed surgical exposure of canine teeth that are displaced in the roof of the mouth

Background: Palatal canines are upper permanent canine (eye) teeth that have become displaced in the roof of the mouth. They are a frequently occurring anomaly, present in 2% to 3% of the population. Management of this problem is both time consuming and expensive and involves surgical exposure (uncovering) followed by fixed braces for 2 to 3 years to bring the canine into alignment within the dental arch. Two techniques for exposing palatal canines are routinely used in the UK: one method (the closed technique) involves orthodontically moving the canine into its correct position beneath the palatal mucosa and the second method (the open technique) involves orthodontically moving the canine into its correct position above the palatal mucosa. Objectives: To establish if clinical, patient centred and economic outcomes are different according to whether an ’open’ or ’closed’ technique is employed for uncovering palatal canines. Search strategy: MEDLINE, EMBASE, the Cochrane Central Register of Controlled Trials (CENTRAL) and the Cochrane Oral Health Group’s Trials Register were searched (to 29th February 2008). There were no restrictions with regard to publication status or language. Selection criteria: Patients receiving surgical treatment to correct upper palatally impacted canines.Therewas no restriction for age, presenting malocclusion or the type of active orthodontic treatment undertaken. Unilateral and bilaterally displaced canines were included. Trials including participants with craniofacial deformity/syndrome were excluded. Data collection and analysis: Two review authors independently and in duplicate assessed studies for inclusion. The Cochrane Collaboration statistical guidelines were to be followed for data synthesis

Information structure and the accessibility of clausally introduced referents

This paper will examine the role of various factors in affecting the salience, and hence the accessibility to pronominal reference, of entities introduced into a discourse by a full clause. We begin with the premise that the possibility of pronominal reference with it versus that depends on the cognitive status of the referent, in the sense of Gundel, Hedberg and Zacharski (1993). This formulation of the problem provides grounds for an explanation of the data presented above, and provides a framework within which we examine the role of various other factors in promoting the salience of a clausally introduced entity, including the information structure of the utterance in which the entity is introduced. For entities introduced by clausal complements to bridge verbs, we show that the information structure of the utterance introducing the entity has a partial, or one-sided, effect on the salience of the entity. When the complement clause is focal, the salience of the entity depends only on its referential givenness-newness (in the sense of Gundel 1988, 1999b), as we would expect. But when the complement clause is ground material, the salience of an entity introduced by the clause is enhanced. Other factors, including the presuppositionality of factive and interrogative complements, also serve to enhance the salience of entities introduced by complement clauses

Stative sentences in Japanese and the role of the nominative marker "ga" : a thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts in Japanese at Massey University, Palmerston North, New Zealand

The Japanese nominative particle ga is normally associated with the marking of subjects. However, there are several constructions involving stative predicates, where it has been claimed, notably by those working within a generative framework, that a ga-marked NP can be an object and that such sentences are transitive. Such an analysis has particularly arisen in the case of sentences with more than one ga-marked NP, exhibiting so-called double ga marking. The following study makes two claims. Firstly, that one of the functions of ga in such sentences is to provide a discourse frame akin to the topic marking function of the postpositional particle wa. Secondly it argues that stative sentences associated with double ga-marking are in fact intransitive and that the ga-marked NP's that have been claimed to be objects are in fact subjects

An Atypical Survey of Typical-Case Heuristic Algorithms

Heuristic approaches often do so well that they seem to pretty much always give the right answer. How close can heuristic algorithms get to always giving the right answer, without inducing seismic complexity-theoretic consequences? This article first discusses how a series of results by Berman, Buhrman, Hartmanis, Homer, Longpr\'{e}, Ogiwara, Sch\"{o}ening, and Watanabe, from the early 1970s through the early 1990s, explicitly or implicitly limited how well heuristic algorithms can do on NP-hard problems. In particular, many desirable levels of heuristic success cannot be obtained unless severe, highly unlikely complexity class collapses occur. Second, we survey work initiated by Goldreich and Wigderson, who showed how under plausible assumptions deterministic heuristics for randomized computation can achieve a very high frequency of correctness. Finally, we consider formal ways in which theory can help explain the effectiveness of heuristics that solve NP-hard problems in practice.Comment: This article is currently scheduled to appear in the December 2012 issue of SIGACT New

Computation Environments, An Interactive Semantics for Turing Machines (which P is not equal to NP considering it)

To scrutinize notions of computation and time complexity, we introduce and formally define an interactive model for computation that we call it the \emph{computation environment}. A computation environment consists of two main parts: i) a universal processor and ii) a computist who uses the computability power of the universal processor to perform effective procedures. The notion of computation finds it meaning, for the computist, through his \underline{interaction} with the universal processor. We are interested in those computation environments which can be considered as alternative for the real computation environment that the human being is its computist. These computation environments must have two properties: 1- being physically plausible, and 2- being enough powerful. Based on Copeland' criteria for effective procedures, we define what a \emph{physically plausible} computation environment is. We construct two \emph{physically plausible} and \emph{enough powerful} computation environments: 1- the Turing computation environment, denoted by $E_T$, and 2- a persistently evolutionary computation environment, denoted by $E_e$, which persistently evolve in the course of executing the computations. We prove that the equality of complexity classes $\mathrm{P}$ and $\mathrm{NP}$ in the computation environment $E_e$ conflicts with the \underline{free will} of the computist. We provide an axiomatic system $\mathcal{T}$ for Turing computability and prove that ignoring just one of the axiom of $\mathcal{T}$, it would not be possible to derive $\mathrm{P=NP}$ from the rest of axioms. We prove that the computist who lives inside the environment $E_T$, can never be confident that whether he lives in a static environment or a persistently evolutionary one.Comment: 33 pages, interactive computation, P vs N