Computer supported mathematics with Ωmega

Classical automated theorem proving of today is based on ingenious search techniques to find a proof for a given theorem in very large search spaces—often in the range of several billion clauses. But in spite of many successful attempts to prove even open mathematical problems automatically, their use in everyday mathematical practice is still limited.The shift from search based methods to more abstract planning techniques however opened up a paradigm for mathematical reasoning on a computer and several systems of that kind now employ a mix of interactive, search based as well as proof planning techniques.The Ωmega system is at the core of several related and well-integrated research projects of the Ωmega research group, whose aim is to develop system support for a working mathematician as well as a software engineer when employing formal methods for quality assurance. In particular, Ωmega supports proof development at a human-oriented abstract level of proof granularity. It is a modular system with a central proof data structure and several supplementary subsystems including automated deduction and computer algebra systems. Ωmega has many characteristics in common with systems like NuPrL, CoQ, Hol, Pvs, and Isabelle. However, it differs from these systems with respect to its focus on proof planning and in that respect it is more similar to the proof planning systems Clam and λClam at Edinburgh.     

An example of formalizing recent mathematical results in Mizar

This paper describes an example of the successful formalization of quite advanced and new mathematics using the Mizar system. It shows that although much effort is required to formalize nontrivial facts in a formal computer deduction system, still it is possible to obtain the level of full logical correctness of all inference steps. We also discuss some problems encountered during the formalization, and try to point out some of the features of the Mizar system responsible for that. The formalization described in this paper allows also for contrasting the linguistic capability of the Mizar language and some of the phrases commonly used in “informal” mathematical papers that the Mizar system lacks, and consequently presents the methods of how to cope with it during the formalization. Yet, apart from the problems, this paper shows some definite benefits from using a formal computer system in the work of a mathematician.     

Purple sweet potato color repairs d-galactose-induced spatial learning and memory impairment by regulating the expression of synaptic proteins

Purple sweet potato color (PSPC), a class of naturally occurring anthocyanins used to color food (E163), has been reported to possess a variety of biological activities, including anti-oxidant, anti-tumor, and anti-inflammatory. The effect of PSPC on the spatial learning and memory of mice treated with d-galactose (d-gal) was evaluated by the Morris water maze; d-gal-treated mice had decreased performance compared with mice in the vehicle and PSPC groups, while the PSPC + d-gal group showed significantly shortened escape latency to platform, increased swimming speed, more target quadrant search time and more platform crossings as compared with the d-gal group. Brain functions, such as memory formation and recovery of function after injury, depend on proper regulation of the expression levels of the pre- and post-synaptic proteins. We investigated the expression of four pre-synaptic proteins (growth-associated protein-43, synapsin-I, synaptophysin, and synaptotagmin) and two post-synaptic proteins (post-synaptic density protein-95 and Ca²⁺/calmodulin-dependent protein kinase II) in the hippocampus and cerebral cortex, respectively, in response to different treatments. Western blotting analysis showed that there were significant decreases in the expression of these representative synaptic proteins in the hippocampus and cerebral cortex of d-gal-treated mice. Interestingly, these decreased expression levels of synaptic proteins could be reversed by PSPC. The levels of expression of these representative synaptic proteins in mice treated with PSPC alone were not significantly different from those in untreated mice. The results of this study suggested that memory impairment and synaptic protein loss in d-gal-treated mice may be improved by treatment with PSPC.     

Completeness in Hybrid Type Theory

We show that basic hybridization (adding nominals and @ operators) makes it possible to give straightforward Henkin-style completeness proofs even when the modal logic being hybridized is higher-order. The key ideas are to add nominals as expressions of type t, and to extend to arbitrary types the way we interpret \(@_i\) in propositional and first-order hybrid logic. This means: interpret \(@_i\alpha _a\) , where \(\alpha _a\) is an expression of any type \(a\) , as an expression of type \(a\) that rigidly returns the value that \(\alpha_a\) receives at the i-world. The axiomatization and completeness proofs are generalizations of those found in propositional and first-order hybrid logic, and (as is usual inhybrid logic) we automatically obtain a wide range of completeness results for stronger logics and languages. Our approach is deliberately low-tech. We don’t, for example, make use of Montague’s intensional type s, or Fitting-style intensional models; we build, as simply as we can, hybrid logicover Henkin’s logic.     

Importing Logics

The novel notion of importing logics is introduced, subsuming as special cases several kinds of asymmetric combination mechanisms, like temporalization [8, 9], modalization [7] and exogenous enrichment [13, 5, 12, 4, 1]. The graph-theoretic approach proposed in [15] is used, but formulas are identified with irreducible paths in the signature multi-graph instead of equivalence classes of such paths, facilitating proofs involving inductions on formulas. Importing is proved to be strongly conservative. Conservative results follow as corollaries for temporalization, modalization and exogenous enrichment.     

d-Serine facilitates the effects of extinction to reduce cocaine-primed reinstatement of drug-seeking behavior

Male Sprague Dawley rats were allowed to self-administer cocaine (0.5 mg/kg) during 90 min sessions for a period of 15 days. On day 16, rats were either held abstinent in their home cage environment or experienced an extinction session in which the active lever had no programmed consequences. Facilitating N-methyl-d-aspartate (NMDA) receptor activity with the coagonist d-serine (100 mg/kg i.p.) before or following the extinction session significantly reduced the subsequent cocaine-primed reinstatement of drug-seeking behavior tested on day 17. d-Serine significantly reduced drug-primed reinstatement only when combined with extinction, and its effectiveness when administered following the training session suggested that an enhancement of consolidation of extinction learning had occurred. In contrast, d-serine treatment did not reduce sucrose-primed reinstatement, indicating that the beneficial effects of this adjunct pharmacotherapy with extinction training were specific to an addictive substance (cocaine) and did not generalize to a natural reward (sucrose).     

Actuality in Propositional Modal Logic

We show that the actuality operator A is redundant in any propositional modal logic characterized by a class of Kripke models (respectively, neighborhood models). Specifically, we prove that for every formula ${\phi}$ in the propositional modal language with A, there is a formula ${\psi}$ not containing A such that ${\phi}$ and ${\psi}$ are materially equivalent at the actual world in every Kripke model (respectively, neighborhood model). Inspection of the proofs leads to corresponding proof-theoretic results concerning the eliminability of the actuality operator in the actuality extension of any normal propositional modal logic and of any “classical” modal logic. As an application, we provide an alternative proof of a result of Williamson’s to the effect that the compound operator A□ behaves, in any normal logic between T and S5, like the simple necessity operator □ in S5.     

School-Based Mental Health Services: Definitions and Models of Effective Practice

ABSTRACT

School-based mental health services are those delivered by school-employed and community-employed providers in school buildings. With the implementation of provisions of the Patient Protection and Affordable Care Act (2010) that funds school-based health centers, school-based mental health services could become more broadly available in communities throughout the United States. Playing a pivotal role in promoting models for school-based services that maximize benefits to children and adolescents while making efficient use of communities' mental health resources are school psychologists. School psychologists, who recognize and respect the separated development of school and community mental health professions, can foster school–agency partnerships to coordinate children's mental health services that are comprehensive, child centered, family focused, and culturally competent. In this article, we provide three case examples using the Participatory Culture-Specific Intervention Model (B. K. Nastasi, R. B. Moore, & K. M. Varjas, 2004 Nastasi, B. K., Moore, R. B., & Varjas, K. M. (2004). School-based mental health services: Creating comprehensive and culturally specific programs. Washington, DC: American Psychological Association. [Google Scholar]) as exemplars for the implementation of community partnerships providing comprehensive culturally and contextually relevant school-based mental health services.     

On All Strong Kleene Generalizations of Classical Logic

By using the notions of exact truth (‘true and not false’) and exact falsity (‘false and not true’), one can give 16 distinct definitions of classical consequence. This paper studies the class of relations that results from these definitions in settings that are paracomplete, paraconsistent or both and that are governed by the (extended) Strong Kleene schema. Besides familiar logics such as Strong Kleene logic (K3), the Logic of Paradox (LP) and First Degree Entailment (FDE), the resulting class of all Strong Kleene generalizations of classical logic also contains a host of unfamiliar logics. We first study the members of our class semantically, after which we present a uniform sequent calculus (the SK calculus) that is sound and complete with respect to all of them. Two further sequent calculi (the \({{\bf SK}^\mathcal{P}}\) and \({\bf SK}^{\mathcal{N}}\) calculus) will be considered, which serve the same purpose and which are obtained by applying general methods (due to Baaz et al.) to construct sequent calculi for many-valued logics. Rules and proofs in the SK calculus are much simpler and shorter than those of the \({\bf SK}^{\mathcal{P}}\) and the \({\bf SK}^{\mathcal{N}}\) calculus, which is one of the reasons to prefer the SK calculus over the latter two. Besides favourably comparing the SK calculus to both the \({\bf SK}^{\mathcal{P}}\) and the \({\bf SK}^{\mathcal{N}}\) calculus, we also hint at its philosophical significance.     

Finding Missing Proofs with Automated Reasoning

This article features long-sought proofs with intriguing properties (such as the absence of double negation and the avoidance of lemmas that appeared to be indispensable), and it features the automated methods for finding them. The theorems of concern are taken from various areas of logic that include two-valued sentential (or propositional) calculus and infinite-valued sentential calculus. Many of the proofs (in effect) answer questions that had remained open for decades, questions focusing on axiomatic proofs. The approaches we take are of added interest in that all rely heavily on the use of a single program that offers logical reasoning, William McCune's automated reasoning program OTTER. The nature of the successes and approaches suggests that this program offers researchers a valuable automated assistant. This article has three main components. First, in view of the interdisciplinary nature of the audience, we discuss the means for using the program in question (OTTER), which flags, parameters, and lists have which effects, and how the proofs it finds are easily read. Second, because of the variety of proofs that we have found and their significance, we discuss them in a manner that permits comparison with the literature. Among those proofs, we offer a proof shorter than that given by Meredith and Prior in their treatment of ukasiewicz's shortest single axiom for the implicational fragment of two-valued sentential calculus, and we offer a proof for the ukasiewicz 23-letter single axiom for the full calculus. Third, with the intent of producing a fruitful dialogue, we pose questions concerning the properties of proofs and, even more pressing, invite questions similar to those this article answers.     

Determining transformation distance in similarity: Considerations for assessing representational changes a priori

The representational distortion (RD) approach to similarity (e.g., Hahn, Chater, & Richardson, 2003 Hahn, U., Chater, N. and Richardson, L. B. 2003. Similarity as transformation. Cognition, 87: 1–32.  [Google Scholar]) proposes that similarity is computed using the transformation distance between two entities. We argue that researchers who adopt this approach need to be concerned with how representational transformations can be determined a priori. We discuss several roadblocks to using this approach. Specifically we demonstrate the difficulties inherent in determining what transformations are psychologically salient and the importance of considering the directionality of transformations.     

Medial prefrontal cortex infusions of bupivacaine or AP-5 block extinction of amphetamine conditioned place preference

The present experiments used reversible lesion techniques and intra-mPFC infusions of the n-methyl d-aspartate (NMDA) receptor antagonist d,l-2-amino-5-phosphonovaleric acid (AP-5) to examine the role of the mPFC in extinction of an amphetamine conditioned place preference (CPP). Following initial training and testing for an amphetamine (2 mg/kg) CPP, adult male Long–Evans rats were given extinction trials that were identical to training, except in the absence of peripheral amphetamine injections. Immediately prior to each extinction trial, rats received intra-mPFC infusions of the anesthetic drug bupivacaine (0.75% solution/0.5 μl), AP-5 (1.25, 2.5, 5.0 μg/0.5 μl), or saline. Following extinction training, rats were given a second CPP test session. Rats receiving intra-mPFC infusions of saline displayed extinction of CPP behavior. In contrast, intra-mPFC infusions of bupivacaine or AP-5 (2.5, 5.0 μg) blocked CPP extinction. The findings indicate (1) the mPFC mediates extinction of approach behavior to drug-associated environmental contexts, and (2) NMDA receptor blockade within the mPFC is sufficient to block extinction of amphetamine CPP behavior.     

Penitentiary institutions: Transforming processes and well-being

Although penitentiaries represent places whose necessary function is to limit anomy and deviance, a formally managed and closed organizational system prohibits the ability to make changes. Anyway, without changes these organizations may become ‘inadequate or alienating’ (Jaques, 1955 Jaques, E. (1955). Sistemi sociali come difesa contro l'ansia persecutoria e depressiva [Social Systems as Defense against Persecution- and Depression-Related Anxiety]. In M. Klein, P. Heimann, R. Money-Kyrle (Eds.), Nuove vie della psicoanalisi [New ways in psychoanalysis] (pp. 609–633). Milan, Italy: Il Saggiatore. Italian Translation 1966. [Google Scholar]). In the present issue we discuss relevant changes that occurred in the Italian penitentiary system and we highlight how this transformation can only occur as a result of dynamism and evolution skills; characteristics that are not often identified as prisons' distinctive features. We propose that a ‘cultural reorganization’, developed in close collaboration with the human resources of the organization, is necessary. It follows that these changes require a profound restructuring of the organizational culture and several modifications at different levels are involved: a. institutional regulation, b. roles and identities of penitentiary professionals, and c. evaluation of the appropriateness of the punishment. For these reasons it's essential to concentrate our attention on the ongoing transformational processes involving penitentiary institutions, carried out with the goal to achieve higher suitability of the penitentiary institutions to serve the community given the present socio-cultural context and the different needs of both the operators and the inmates.     

The variance of d′ estimates obtained in yes—no and two-interval forced choice procedures

In studies of detection and discrimination, data are often obtained in the form of a 2 × 2 matrix and then converted to an estimate of d′, based on the assumptions that the underlying decision distributions are Gaussian and equal in variance. The statistical properties of the estimate of d′, $\hat d'$ , are well understood for data obtained using the yes—no procedure, but less effort has been devoted to the more commonly used two-interval forced choice (2IFC) procedure. The variance associated with $\hat d'$ is a function of trued′ in both procedures, but for small values of trued′, the variance of $\hat d'$ obtained using the 2IFC procedure is predicted to be less than the variance of $\hat d'$ obtained using yes—no; for large values of trued′, the variance of $\hat d'$ obtained using the 2IFC procedure is predicted to be greater than the variance of $\hat d'$ from yes—no. These results follow from standard assumptions about the relationship between the two procedures. The present paper reviews the statistical properties of $\hat d'$ obtained using the two standard procedures and compares estimates of the variance of $\hat d'$ as a function of trued′ with the variance observed in values of $\hat d'$ obtained with a 2IFC procedure.     

Steps Toward a Computational Metaphysics

In this paper, the authors describe their initial investigations in computational metaphysics. Our method is to implement axiomatic metaphysics in an automated reasoning system. In this paper, we describe what we have discovered when the theory of abstract objects is implemented in prover9 (a first-order automated reasoning system which is the successor to otter). After reviewing the second-order, axiomatic theory of abstract objects, we show (1) how to represent a fragment of that theory in prover9’s first-order syntax, and (2) how prover9 then finds proofs of interesting theorems of metaphysics, such as that every possible world is maximal. We conclude the paper by discussing some issues for further research.     

The sampling distribution ofd′

The distribution of sample $\hat d's$ , although mathematically intractable, can be tabulated readily by computer. Such tabulations reveal a number of interesting properties of this distribution, including: (1) sample $\hat d's$ are biased, with an expected value that can be higher or lower than the true value, depending on the sample size, the true value itself, and the convention adopted for handling cases in which the sample $\hat d'$ is undefined; (2) the variance of $\hat d'$ also depends on the convention adopted for handling cases in which the sample $\hat d'$ is undefined and is in some cases poorly approximated by the standard approximation formula, (3) the standard formula for a confidence interval for $\hat d'$ is quite accurate with at least 50–100 trials per condition, but more accurate intervals can be obtained by direct computation with smaller samples.     

Marc lange on essentialism

For scientific essentialists, the only logical possibilities of existence are the real (or metaphysical) ones, and such possibilities, they say, are relative to worlds. They are not a priori, and they cannot just be invented. Rather, they are discoverable only by the a posteriori methods of science. There are, however, many philosophers who think that real possibilities are knowable a priori, or that they can just be invented. Marc Lange [Lange 2004 Lange, Marc. 2004. A Note on Scientific Essentialism, Laws of Nature, and Counterfactual Conditionals. Australasian Journal of Philosophy, 82: 227–41.  ] thinks that they can be invented, and tries to use his inventions to argue that the essentialist theory of counterfactual conditionals developed in Scientific Essentialism [Ellis 2001 Ellis BD 2001 Scientific Essentialism Cambridge: Cambridge University Press  , hereafter SE] is flawed.     

Using a Multivariate Multilevel Polytomous Item Response Theory Model to Study Parallel Processes of Change: The Dynamic Association Between Adolescents' Social Isolation and Engagement With Delinquent Peers in the National Youth Survey

The application of multidimensional item response theory models to repeated observations has demonstrated great promise in developmental research. It allows researchers to take into consideration both the characteristics of item response and measurement error in longitudinal trajectory analysis, which improves the reliability and validity of the latent growth curve (LGC) model. The purpose of this study is to demonstrate the potential of Bayesian methods and the utility of a comprehensive modeling framework, the one combining a measurement model (e.g., a multidimensional graded response model, MGRM) with a structural model (e.g., an associative latent growth curve analysis, ALGC). All analyses are implemented in WinBUGS 1.4.3 (Spiegelhalter, Thomas, Best, &; Lunn, 2003 Spiegelhalter, D. J., Thomas, A., Best, N. G. and Lunn, D. 2003. WinBUGS user manual Cambridge, , UK: MRC Biostatistics Unit, Institute of Public Health. Retrieved October 24, 2006, from http://www.mrc-bsu.cam.ac.uk/bugs [Google Scholar]), which allows researchers to use Markov chain Monte Carlo simulation methods to fit complex statistical models and circumvent intractable analytic or numerical integrations. The utility of this MGRM-ALGC modeling framework was investigated with both simulated and empirical data, and promising results were obtained. As the results indicate, being a flexible multivariate multilevel model, this MGRM-ALGC model not only produces item parameter estimates that are readily estimable and interpretable but also estimates the corresponding covariation in the developmental dimensions. In terms of substantive interpretation, as adolescents perceived themselves more socially isolated, the chance that they are engaged with delinquent peers becomes profoundly larger. Generally, boys have a higher initial exposure extent than girls. However, there is no gender difference associated with other latent growth parameters.     

Advanced Brief Strategic Therapy: An overview of interventions with eating disorders to exemplify how theory and practice work

This paper aims at introducing some of the central aspects of the evolution that brief strategic therapy has undergone at the Centro di Terapia Strategica of Arezzo, Italy, towards advanced therapeutic strategies which differ from the original Palo Alto model. (Fisch, Weakland, &; Segal, 1982; Watzlawick, 1978 Watzlawick, P. 1978. The language of change: Elements of therapeutic communication, New York: Basic Books.  [Google Scholar]; Watzlawick, Beavin, &; Jackson, 1967; Watzlawick, Weakland, &; Fisch, 1974). We will focus on how the concept of self-deception is central to the formation and the persistence of psychological disorders; and how the usage of non-ordinary logics and the understanding of the individual's perceptive-reactive system are of key importance in unravelling such disorders, allowing the therapist to guide the patient towards an efficacious and efficient solution. Rather than attempting to describe and enlist the theoretical corpus that underlies brief strategic therapy, we have chosen to exemplify some focal concepts that connect theory to practice, and vice versa, by presenting the outline of some therapeutic protocols devised for solving eating disorders which can be specifically tailored for each individual patient.     

Public announcement logic with distributed knowledge: expressivity,completeness and complexity

While dynamic epistemic logics with common knowledge have been extensively studied, dynamic epistemic logics with distributed knowledge have so far received far less attention. In this paper we study extensions of public announcement logic (\(\mathcal{PAL }\)) with distributed knowledge, in particular their expressivity, axiomatisations and complexity. \(\mathcal{PAL }\) extended only with distributed knowledge is not more expressive than standard epistemic logic with distributed knowledge. Our focus is therefore on \(\mathcal{PACD }\), the result of adding both common and distributed knowledge to \(\mathcal{PAL }\), which is more expressive than each of its component logics. We introduce an axiomatisation of \(\mathcal{PACD }\), which is not surprising: it is the combination of well-known axioms. The completeness proof, however, is not trivial, and requires novel combinations and extensions of techniques for dealing with \(S5\) knowledge, distributed knowledge, common knowledge and public announcements at the same time. We furthermore show that \(\mathcal{PACD }\) is decidable, more precisely that it is \(\textsc {exptime}\)-complete. This result also carries over to \(\mathcal{S 5\mathcal CD }\) with common and distributed knowledge operators for all coalitions (and not only the grand coalition). Finally, we propose a notion of a trans-bisimulation to generalise certain results and give deeper insight into the proofs.