Action Logic is Undecidable

Action logic is the algebraic logic (inequational theory) of residuated Kleene lattices. One of the operations of this logic is the Kleene star, which is axiomatized by an induction scheme. For a stronger system that uses an -rule instead (infinitary action logic), Buszkowski and Palka (2007) proved -completeness (thus, undecidability). Decidability of action logic itself was an open question, raised by Kozen in 1994. In this article, we show that it is undecidable, more precisely, -complete. We also prove the same undecidability results for all recursively enumerable logics between action logic and infinitary action logic, for fragments of these logics with only one of the two lattice (additive) connectives, and for action logic extended with the law of distributivity.

Non-well-founded Deduction for Induction and Coinduction

Induction and coinduction are both used extensively within mathematics and computer science. Algebraic formulations of these principles make the duality between them apparent, but do not account well for the way they are commonly used in deduction. Generally, the formalization of these reasoning methods employs inference rules that express a general explicit (co)induction scheme. Non-well-founded proof theory provides an alternative, more robust approach for formalizing implicit (co)inductive reasoning. This approach has been extremely successful in recent years in supporting implicit inductive reasoning, but is not as well-developed in the context of coinductive reasoning. This paper reviews the general method of non-well-founded proofs, and puts forward a concrete natural framework for (co)inductive reasoning, based on (co)closure operators, that offers a concise framework in which inductive and coinductive reasoning are captured as we intuitively understand and use them. Through this framework we demonstrate the enormous potential of non-well-founded deduction, both in the foundational theoretical exploration of (co)inductive reasoning and in the provision of proof support for (co)inductive reasoning within (semi-)automated proof tools.

Towards automated reasoning in Herbrand structures

Herbrand structures have the advantage, computationally speaking, of being guided by the definability of all elements in them. A salient feature of the logics induced by them is that they internally exhibit the induction scheme, thus providing a congenial, computationally oriented framework for formal inductive reasoning. Nonetheless, their enhanced expressivity renders any effective proof system for them incomplete. Furthermore, the fact that they are not compact poses yet another proof-theoretic challenge. This paper offers several layers for coping with the inherent incompleteness and non-compactness of these logics. First, two types of infinitary proof system are introduced—one of infinite width and one of infinite height—which manipulate infinite sequents and are sound and complete for the intended semantics. The restriction of these systems to finite sequents induces a completeness result for finite entailments. Then, in search of effectiveness, two finite approximations of these systems are presented and explored. Interestingly, the approximation of the infinite-width system via an explicit induction scheme turns out to be weaker than the effective cyclic fragment of the infinite-height system.

NOTES ON BOUNDED INDUCTION FOR THE COMPOSITIONAL TRUTH PREDICATE

We prove that the theory of the extensional compositional truth predicate for the language of arithmetic with Δ0-induction scheme for the truth predicate and the full arithmetical induction scheme is not conservative over Peano Arithmetic. In addition, we show that a slightly modified theory of truth actually proves the global reflection principle over the base theory.

THE TEACHER QUALIFICATION SCHEME: A CASE OF ALBANIA

The study aims to analyse the results of “Teacher Qualification Exam” in Albania, and to link these results with teacher preparation curricula taught in public universities. The methodology of research includes desk research on the literature about curricula and teacher continuous professional development, elaboration of the results of the testing of 3064 teachers, analysis of the university teacher preparation curricula in terms of skills development, the elaboration of the results of questionnaires developed by novice teachers, and in depth interviews with students graduating from the education departments. As a result, the research shows the relationships amongst the curricula developed in the teacher preparation faculties in Albania, the poor results of novice teachers involved in the induction scheme, and the comparatively low results of teachers pertaining to the third category of the Qualification Scheme (novice teachers having no more than 5 years of teaching experience) in Albania. It is advisable to adopt a better professional development scheme. Key words: desk research, novice teachers, teacher qualification.

Early Cytokine Modulation after the Rapid Induction Phase of Sublingual Immunotherapy with Mite Monomeric Allergoids

The influence of different treatment schedules of sublingual immunotherapy (SLIT) in activating IL-10-producing T-cells, crucial in inducing allergen-specific tolerance, is not completely understood. The present work was designed to evaluate allergen driven interleukin release by mononuclear cells in the early phase of SLIT, after application of different induction schemes. Twenty mite-allergic patients were enrolled, 10 (group A) treated with a traditional 98 day induction scheme and 10 (group B) with a 16 day scheme with monomeric allergoid vaccine. At the end of the induction phase, the cumulative doses taken by group A and group B patients were equivalent to 50.5 and 50.3 μg of mite group 1 allergens, respectively. The release of Th1-, Th2- and Treg-related interleukins was assessed in culture supernatants of 5 μg/ml Der-p1-stimulated mononuclear cells, isolated before and after the induction phases. No relevant treatment-related side effects were observed. Interleukin release was similar in the two groups at the enrolment. Non-stimulated and Der p 1 stimulated release of studied cytokines was similar in the two groups at enrolment. Der p 1 stimulation significantly increased IL-10 release (p<0.0002) after treatment in group B patients, and this effect was higher (p=0.05) compared to group A patients. Furthermore, at the end of SLIT induction TNF-alpha, IL-4 and IFN-gamma production were reduced in group B patients (p<0.05, p=0.062 and p=0.060, respectively). The rapid induction scheme of sublingual immunotherapy induces an early immune suppression more effectively than the slower one. The rapid induction scheme should be the preferential way to start sublingual immunotherapy, particularly when monomeric allergoids are utilized.