Future-Looking Logics on Data Words and Trees

Abstract

Abstract. In a data word or a data tree each position carries a label from a finite alphabet and a data value from an infinite domain. Over data words we consider the logic LTL ↓ 1 (F), that extends LTL(F) with one register for storing data values for later comparisons. We show that satisfiability over data words of LTL ↓ 1 (F) is already non primitive recursive. We also show that the extension of LTL ↓ 1 (F) with either the backward modality F −1 or with one extra register is undecidable. All these lower bounds were already known for LTL ↓ 1 (X, F) and our results essentially show that the X modality was not necessary. Moreover we show that over data trees similar lower bounds hold for certain fragments of XPath.

Forward-XPath and extended register automata on data-trees

“…This settles a natural question left from the work in [13], also mentioned in [8,9]. As a consequence this also answers positively the open question raised in [1] on whether the downward fragment of XPath in the presence of DTDs is decidable.…”

“…Notably, none of these works considers horizontal axes to navigate between siblings: By exploiting the bisimulation invariance property enjoyed by these logics, the complexity of the satisfiability problem is kept relatively low (at most ExpTime) in the presence of data values. However, when horizontal axes are present, most of the problems have a non-primitive recursive complexity (including the fragment of [13], or even much simpler ones without the one-step '→' axis [9]). In [11], several fragments with horizontal axes are treated.…”

“…This is absurd, as the ATRA model is decidable. We refer the reader to [6,9] for more details on these kind of codings.…”

Witness Runs for Counter Machines

“…For instance, the statement that the program variable x never decreases below its initial value can be expressed by the formula below that uses a form of freeze operator: ∃y (x = y) ∧ G(x ≥ y). Recent results on satisfiability and model-checking problems can be found in [FS09,DLS10].…”

Lossy Counter Machines Decidability Cheat Sheet

“…Secondly, some problems that are decidable for lossy counters machines are still Ackermann-hard, i.e., they require nonprimitive-recursive time and space [43,44]. This can be used to show Ackermann-hardness of problems that are decidable but rich enough to encode lossy counters, see [18,19,32,24,46] for examples.…”