Alexander Kartzow

Strictness of the collapsible pushdown hierarchy

We present a pumping lemma for each level of the collapsible pushdown graph hierarchy in analogy to the second authors pumping lemma for higher-order pushdown graphs (without collapse). Using this lemma, we give the first known examples that separate the levels of the collapsible pushdown graph hierarchy and of the collapsible pushdown tree hierarchy, i.e., the hierarchy of trees generated by higher-order recursion schemes. This confirms the open conjecture that higher orders allow one to generate more graphs and more trees. Full proofs can be found in the arXiv version[10] of this paper.

Satisfiability of CTL* with constraints

We show that satisfiability for CTL* with equality-, order-, and modulo-constraints over ℤ is decidable. Previously, decidability was only known for certain fragments of CTL*, e.g., the existential and positive fragments and EF.

Satisfiability of ECTL* with constraints

Satisfiability for ECTL* with local constraints over the integers is decidable.Finite satisfiability for ECTL* with local constraints over the integers is decidable.Locality of the constraints is necessary for decidability. We show that satisfiability and finite satisfiability for ECTL * with equality-, order-, and modulo-constraints over Z are decidable. Since ECTL * is a proper extension of CTL * this greatly improves the previously known decidability results for certain fragments of CTL * , e.g., the existential and positive fragments and EF. We also show that our choice of local constraints is necessary for the result in the sense that, if we add the possibility to state non-local constraints over Z , the resulting logic becomes undecidable.

Tree-Automatic Well-Founded Trees

We investigate tree-automatic well-founded trees. Using Delhommes decomposition technique for tree-automatic structures, we show that the (ordinal) rank of a tree-automatic well-founded tree is strictly below omega^omega. Moreover, we make a step towards proving that the ranks of tree-automatic well-founded partial orders are bounded by omega^omega^omega: we prove this bound for what we call upwards linear partial orders. As an application of our result, we show that the isomorphism problem for tree-automatic well-founded trees is complete for level Delta^0_{omega^omega} of the hyperarithmetical hierarchy with respect to Turing-reductions.

Tree-Automatic well-founded trees

We investigate tree-automatic well-founded trees. For this, we introduce a new ordinal measure for well-founded trees, called ∞-rank. The ∞-rankof a well-founded tree is always bounded from above by the ordinary (ordinal) rank of a tree. We also show that the ordinal rank of a well-founded tree of ∞-rankα is smaller than ω·(α+1). For string-automatic well-founded trees, it follows from [16] that the ∞-rankis always finite. Here, using Delhommes decomposition technique for tree-automatic structures, we show that the ∞-rankof a tree-automatic well-founded tree is strictly below ωω. As a corollary, we obtain that the ordinal rank of a string-automatic (resp., tree-automatic) well-founded tree is strictly below ω2 (resp., ωω). The result for the string-automatic case nicely contrasts a result of Delhomme, saying that the ranks of string-automatic well-founded partial orders reach all ordinals below ωω. As a second application of the ∞-rankwe show that the isomorphism problem for tree-automatic well-founded trees is complete for level

Structures without Scattered-Automatic Presentation

\Delta^0_{\omega^\omega}

First-Order model checking on nested pushdown trees is complete for doubly exponential alternating time

of the hyperarithmetical hierarchy (under Turing-reductions). Full proofs can be found in the arXiv-version [11] of this paper.

Satisfiability of ECTL ∗ with Local Tree Constraints

Bruyere and Carton lifted the notion of finite automata reading infinite words to finite automata reading words with shape an arbitrary linear order \(\mathfrak{L}\). Automata on finite words can be used to represent infinite structures, the so-called word-automatic structures. Analogously, for a linear order \(\mathfrak{L}\) there is the class of \(\mathfrak{L}\)-automatic structures. In this paper we prove the following limitations on the class of \(\mathfrak{L}\)-automatic structures for a fixed \(\mathfrak{L}\) of finite condensation rank 1 + α.

Reachability in Higher-Order-Counters

Recently we proved that first-order model checking on nested pushdown trees can be done in doubly exponential alternating time with linearly many alternations. Using the interpretation method of Compton and Henson we give a matching lower bound, i.e., we prove that first-order model checking on nested pushdown trees is complete for ATIME(exp2(cn), cn) with respect to log-lin reductions.

Satisfiability of ECTL* with Tree Constraints

Recently, we have shown that satisfiability for the temporal logic ECTL∗ with local constraints over (ℤ, <, =) is decidable using a new technique (Carapelle et al., 2013). This approach reduces the satisfiability problem of ECTL∗ with constraints over some structure 𝓐