FO2(<,+1,~) on data trees, data tree automata and branching vector addition systems

TL;DR

This paper explores the logical and automata-theoretic properties of data trees, introducing a new automata model DAD# and a counter automata model EBVASS, establishing their equivalence in decision problems.

Contribution

It introduces DAD# automata and EBVASS models, demonstrating their expressive power and the equivalence of key decision problems for data trees.

Findings

DAD# is more expressive than FO2(<,+1,~)

Emptiness of DAD# and satisfiability of FO2(<,+1,~) are inter-reducible

Reachability in EBVASS is equivalent to these decision problems.

Abstract

A data tree is an unranked ordered tree where each node carries a label from a finite alphabet and a datum from some infinite domain. We consider the two variable first order logic FO2(<,+1,~) over data trees. Here +1 refers to the child and the next sibling relations while < refers to the descendant and following sibling relations. Moreover, ~ is a binary predicate testing data equality. We exhibit an automata model, denoted DAD# that is more expressive than FO2(<,+1,~) but such that emptiness of DAD# and satisfiability of FO2(<,+1,~) are inter-reducible. This is proved via a model of counter tree automata, denoted EBVASS, that extends Branching Vector Addition Systems with States (BVASS) with extra features for merging counters. We show that, as decision problems, reachability for EBVASS, satisfiability of FO2(<,+1,~) and emptiness of DAD# are equivalent.