Extending the WMSO+U Logic With Quantification Over Tuples

We study a new extension of the weak MSO logic, talking about boundedness. Instead of a previously considered quantifier U, expressing the fact that there exist arbitrarily large finite sets satisfying a given property, we consider a generalized quantifier U, expressing the fact that there exist tuples of arbitrarily large finite sets satisfying a given property. First, we prove that the new logic WMSO+U_tup is strictly more expressive than WMSO+U. In particular, WMSO+U_tup is able to express the so-called simultaneous unboundedness property, for which we prove that it is not expressible in WMSO+U. Second, we prove that it is decidable whether the tree generated by a given higher-order recursion scheme satisfies a given sentence of WMSO+K_tup.Comment: This is an extended version of a paper published at the CSL 2024 conferenc

Praktyczna implementacja algorytmu Micali-Vazirani

Algorytm Micali-Vazirani, znajdujący skojarzenia maksymalne w dowolnych grafach, był pierwszym algorytmem, którego złożoność równała się tej osiąganej przez klasyczne algorytmy dla grafów dwudzielnych. Późniejsze publikacje uzupełniły początkowo nieprecyzyjne opis i dowód algorytmu. Niniejsza praca omawia zasady jego działania i szczegóły implementacji.The Micali-Vazirani matching algorithm for general graphs was the first algorithm to achieve the time complexity of classic solutions to the same problem for bipartite graphs. Despite the initial lack of precision in its description and proof, newer publications have clarified and simplified its structure. We present an overview and an implementation of the algorithm