arXiv subjects

Mikolaj Bojanczyk

Publications and source records attributed to Mikolaj Bojanczyk.

9 recordsLinked to original sources

arXiv2021

Separator logic and star-free expressions for graphs

We describe two formalisms for defining graph languages, and prove that they are equivalent: 1. Separator logic. This is first-order logic on graphs which is allowed to use the edge relation, and for every $n \in \{0,1,\ldots \}$ a relation of arity $n+2$ which says that "vertex $s$ can be connected to vertex $t$ by a path that avoids vertices $v_1,\ldots,v_n$". 2. Star-free graph expressions. These are expressions that describe graphs with distinguished vertices called ports, and which are built from finite languages via Boolean combinations and the operations on graphs with ports used to construct tree decompositions. Furthermore, we prove a variant of Schützenberger's theorem (about star-free languages being those recognized by a periodic monoids) for graphs of bounded pathwidth. A corollary is that, given $k$ and a graph language represented by an \mso formula, one can decide if the language can be defined in either of two equivalent formalisms on graphs of pathwidth at most $k$.

cs.LO↗

arXiv2018

Two monads for graphs

An introduction to algebras for graphs, based on Courcelle's algebras of hyperedge replacement and vertex replacement. The paper uses monad notation.

cs.LO↗

arXiv2018

Regular and First Order List Functions

We define two classes of functions, called regular (respectively, first-order) list functions, which manipulate objects such as lists, lists of lists, pairs of lists, lists of pairs of lists, etc. The definition is in the style of regular expressions: the functions are constructed by starting with some basic functions (e.g. projections from pairs, or head and tail operations on lists) and putting them together using four combinators (most importantly, composition of functions). Our main results are that first-order list functions are exactly the same as first-order transductions, under a suitable encoding of the inputs; and the regular list functions are exactly the same as MSO-transductions.

cs.FL↗

arXiv2017

Star Height via Games

This paper proposes a new algorithm deciding the star height problem. As shown by Kirsten, the star height problem reduces to a problem concerning automata with counters, called limitedness. The new contribution is a different algorithm for the limitedness problem, which reduces it to solving a Gale-Stewart game with an ω-regular winning condition.

cs.LO↗

arXiv2014

Decomposition Theorems and Model-Checking for the Modal $μ$-Calculus

We prove a general decomposition theorem for the modal $μ$-calculus $L_μ$ in the spirit of Feferman and Vaught's theorem for disjoint unions. In particular, we show that if a structure (i.e., transition system) is composed of two substructures $M_1$ and $M_2$ plus edges from $M_1$ to $M_2$, then the formulas true at a node in $M$ only depend on the formulas true in the respective substructures in a sense made precise below. As a consequence we show that the model-checking problem for $L_μ$ is fixed-parameter tractable (fpt) on classes of structures of bounded Kelly-width or bounded DAG-width. As far as we are aware, these are the first fpt results for $L_μ$ which do not follow from embedding into monadic second-order logic.

math.LO↗

arXiv2012

Wreath Products of Forest Algebras, with Applications to Tree Logics

We use the recently developed theory of forest algebras to find algebraic characterizations of the languages of unranked trees and forests definable in various logics. These include the temporal logics CTL and EF, and first-order logic over the ancestor relation. While the characterizations are in general non-effective, we are able to use them to formulate necessary conditions for definability and provide new proofs that a number of languages are not definable in these logics.

cs.LO↗

arXiv2010

Tree Languages Defined in First-Order Logic with One Quantifier Alternation

We study tree languages that can be defined in Δ_2 . These are tree languages definable by a first-order formula whose quantifier prefix is forall exists, and simultaneously by a first-order formula whose quantifier prefix is . For the quantifier free part we consider two signatures, either the descendant relation alone or together with the lexicographical order relation on nodes. We provide an effective characterization of tree and forest languages definable in Δ_2 . This characterization is in terms of algebraic equations. Over words, the class of word languages definable in Δ_2 forms a robust class, which was given an effective algebraic characterization by Pin and Weil.

cs.FL↗

arXiv2009

Two-Way Unary Temporal Logic over Trees

We consider a temporal logic EF+F^-1 for unranked, unordered finite trees. The logic has two operators: EFϕ, which says "in some proper descendant ϕholds", and F^-1ϕ, which says "in some proper ancestor ϕholds". We present an algorithm for deciding if a regular language of unranked finite trees can be expressed in EF+F^-1. The algorithm uses a characterization expressed in terms of forest algebras.

cs.LO↗

arXiv2009

Weak Mso with the Unbounding Quantifier

A new class of languages of infinite words is introduced, called the max-regular languages, extending the class of $ω$-regular languages. The class has two equivalent descriptions: in terms of automata (a type of deterministic counter automaton), and in terms of logic (weak monadic second-order logic with a bounding quantifier). Effective translations between the logic and automata are given.

cs.FL↗