The Intersection Problem for Finite Semigroups

The intersection problem for finite semigroups asks, given a set of regular languages, represented by recognizing morphisms to finite semigroups, whether there exists a word contained in their intersection. In previous work, it was shown that is problem is [Formula: see text]-complete. We introduce compressibility measures as a useful tool to classify the complexity of the intersection problem for certain classes of finite semigroups. Using this framework, we obtain a new and simple proof that for groups and for commutative semigroups, the problem (as well as the variant where the languages are represented by finite automata) is contained in [Formula: see text]. We uncover certain structural and non-structural properties determining the complexity of the intersection problem for varieties of semigroups containing only trivial submonoids. More specifically, we prove [Formula: see text]-hardness for classes of semigroups having a property called unbounded order and for the class of all nilpotent semigroups of bounded order. On the contrary, we show that bounded order and commutativity imply decidability in poly-logarithmic time on alternating random-access Turing machines with a single alternation. We also establish connections to the monoid variant of the problem.

Strongly Possible Keys for SQL

Missing data value is an extensive problem in both research and industrial developers. Two general approaches are there to deal with the problem of missing values in databases; they could be either ignored (removed) or imputed (filled in) with new values (Farhangfar et al. in IEEE Trans Syst Man Cybern-Part A: Syst Hum 37(5):692–709, 2007). For some SQL tables, it is possible that some candidate key of the table is not null-free and this needs to be handled. Possible keys and certain keys to deal with this situation were introduced in Köhler et al. (VLDB J 25(4):571–596, 2016). In the present paper, we introduce an intermediate concept called strongly possible keys that is based on a data mining approach using only information already contained in the SQL table. A strongly possible key is a key that holds for some possible world which is obtained by replacing any occurrences of nulls with some values already appearing in the corresponding attributes. Implication among strongly possible keys is characterized, and Armstrong tables are constructed. An algorithm to verify a strongly possible key is given applying bipartite matching. Connection between matroid intersection problem and system of strongly possible keys is established. For the cases when no strongly possible keys hold, an approximation notion is introduced to calculate the closeness of any given set of attributes to be considered as a strongly possible key using the $$g_3$$ g 3 measure, and we derive its component version $$g_4$$ g 4 . Analytical comparisons are given between the two measures.

Intersection problem for Droms RAAGs

We solve the subgroup intersection problem (SIP) for any RAAG [Formula: see text] of Droms type (i.e. with defining graph not containing induced squares or paths of length [Formula: see text]): there is an algorithm which, given finite sets of generators for two subgroups [Formula: see text], decides whether [Formula: see text] is finitely generated or not, and, in the affirmative case, it computes a set of generators for [Formula: see text]. Taking advantage of the recursive characterization of Droms groups, the proof consists in separately showing that the solvability of SIP passes through free products, and through direct products with free-abelian groups. We note that most of RAAGs are not Howson, and many (e.g. [Formula: see text]) even have unsolvable SIP.