CNF Encodings of Parity

The minimum number of clauses in a CNF representation of the parity function $x_1 \oplus x_2 \oplus \dotsb \oplus x_n$ is $2^{n-1}$. One can obtain a more compact CNF encoding by using non-deterministic variables (also known as guess or auxiliary variables). In this paper, we prove the following lower bounds, that almost match known upper bounds, on the number $m$ of clauses and the maximum width $k$ of clauses: 1) if there are at most $s$ auxiliary variables, then $m \ge \Omega\left(2^{n/(s+1)}/n\right)$ and $k \ge n/(s+1)$; 2) the minimum number of clauses is at least $3n$. We derive the first two bounds from the Satisfiability Coding Lemma due to Paturi, Pudlak, and Zane

Peculiarities of the paradoxes of Oscar Wilde

The aim of study - on the basis of perused aphorisms of O. Wilde to prove that Oscar Wilde's paradoxes due to their peculiarities can be regarded as a special category of paradoxesЦель исследования - на основе сбора и анализа афоризмов Оскара Уйальда показать особенности парадоксов автора, позволяющие выделить их в определенную категорию

Assessment of allergen tolerance breakdown levels in local and classical allergic rhinitis

Local allergic rhinitis, a new endotype of allergic rhinitis discerned by researchers of the Spanish Allergy School, is now in the focus of interest of international allergological community. A special feature of local allergic rhinitis, which, being similar to conventional signs of allergic rhinitis, is, however, characterized by absence of systemic atopy manifestations, e.g., an increased total serum IgE content and positive allergic skin tests. In order to assess the level of tolerance breakdown to allergens in local and classical allergic rhinitis, we have studyed concentrations of IL-4, IL-22, and IFNγ in three biological fluids, blood, nasal secretions, and skin exudate. The whole study cohort consisted of 82 patients aged 18 to 60 years with established allergic rhinitis. The diagnosis was based on counseling by allergologist/immunologist, including clinical case history and possible inheritance of atopy as well as videorhinoscopy performed by an ENT specialist. The procedure of videorhinoscopy allowed to specify allergic origin of rhinitis and exclude the patients with non-allergic forms of the disease, but it did not enable us to differentiate between the endotypes of classic and local allergic rhinitis. Subsequently, all patients have been divided into two subgroups based on the criteria of systemic atopy: (1) with a high content of serum total IgE and positive skin allergy tests (n = 41) and (2) with a significantly lower concentration of IgE and negative allergy tests (n = 41). It was concluded that the patients with classic allergic rhinitis prevailed in the 1st subgroup, whereas local rhinitis predominated in the 2nd group. The study of IL-4, IL-22 and IFNγ concentrations in the three biological fluids allowed us to presume that the 1st subgroup was characterized by increased content of IL-4 and IL-22 in blood and skin exudate in comparison with controls, and the 2nd subgroup showed a decrease in IFNγ to control values. The cytokine concentrations in nasal secretions were not representative for the subgroups studied. The result has been interpreted as the absence of tolerance breakdown to causal allergens in the patients with local allergic rhinitis at the systemic level. The obtained data could be used in development of a diagnostic biomarker system for this specific endotype of allergic rhinitis, thus avoiding potential diagnostic errors which occurred in the past, when this endotype was classified as non-allergic form of the disease, thus administering non-adequate treatment, e.g., allergen-specific immunotherapy, which could be prescribed in these cases

SAT-Based Circuit Local Improvement

Finding exact circuit size is notoriously hard. Whereas modern computers and algorithmic techniques allow to find a circuit of size seven in the blink of an eye, it may take more than a week to search for a circuit of size thirteen. One of the reasons of this behavior is that the search space is enormous: the number of circuits of size s is s^?(s), the number of Boolean functions on n variables is 2^(2?). In this paper, we explore the following natural heuristic idea for decreasing the size of a given circuit: go through all its subcircuits of moderate size and check whether any of them can be improved by reducing to SAT. This may be viewed as a local search approach: we search for a smaller circuit in a ball around a given circuit. Through this approach, we prove new upper bounds on the circuit size of various symmetric functions. We also demonstrate that some upper bounds that were proved by hand decades ago, can nowadays be found automatically in a few seconds

Fantasmagorii︠a︡ : povestʹ /

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