We consider recursion schemes (not assumed to be homogeneously typed, and hence not necessarily safe) and use them as generators of (possibly infinite) ranked trees. A recursion scheme is essentially a finite typed deterministic term rewriting system that generates, when one applies the rewriting rules ad infinitum, an infinite tree, called its value tree. A fundamental question is to provide an equivalent description of the trees generated by recursion schemes by a class of machines. In this article, we answer this open question by introducing collapsible pushdown automata (CPDA), which are an extension of deterministic (higher-order) pushdown automata. A CPDA generates a tree as follows. One considers its transition graph, unfolds it, and contracts its silent transitions, which leads to an infinite tree, which is finally node labelled thanks to a map from the set of control states of the CPDA to a ranked alphabet. Our contribution is to prove that these two models, higher-order recursion schemes and collapsible pushdown automata, are equi-expressive for generating infinite ranked trees. This is achieved by giving effective transformations in both directions.

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Collapsible Pushdown Automata and Recursion Schemes

Review: Roland Fraisse, Sur Quelques Classifications des Systemes de Relations

Let R be a class of generators of node-labelled infinite trees, and Lbe a logical language for describing correctness properties of the setrees. Given r in R and phi in L, we say that r_phi is aphi-reflection of r just if (i) r and r_phi generate the same underlying tree, and (ii) suppose a node u of the tree t(r) generated by r has label f, then the label of the node u of t(r_phi) is f* if uin t(r) satisfies phi; it is f otherwise. Thus if t(r) is the computation tree of a program r, we may regard r_phi as a transform of R that can internally observe its behaviour against a specification phi. We say that R is (constructively) reflective w.r.t. L just if there is an algorithm that transforms a given pair (r,phi) to r_phi. In this paper, we prove that higher-order recursion schemes are reflective w.r.t. both modal mu-calculus and monadic second order(MSO) logic. To obtain this result, we give the first characterisation of the winning regions of parity games over the transition graphs of collapsible pushdown automata (CPDA): they are regular sets defined by a new class of automata. (Order-n recursion schemes are equi-expressive with order-n CPDA for generating trees.) As a corollary, we show that these schemes are closed under the operation of MSO-interpretation followed by tree unfolding a la Caucal.

/pdf/recursion-schemes-and-logical-reflection-3qn0z5cqx6.pdf

Recursion Schemes and Logical Reflection

University of Edinburgh, PhD in Informatics, September 2006 – August 2010 Thesis: Model Checking Infinite-state Systems: Generic and Specific Approaches Supervisors: Prof. Leonid Libkin and Dr. Richard Mayr Examiners: Prof. Chih-Hao Luke Ong and Prof. Colin Stirling University of Melbourne, BSc (Honours) in Computer Science & Mathematics, February 2001 – December 2004 Thesis: On Unary-conjunctive-view Queries Adviser: Prof. James Bailey

Model Checking Infinite-State Systems: Generic and Specific Approaches

We introduce a natural extension of collapsible pushdown systems called annotated pushdown systems that replaces collapse links with stack annotations. We believe this new model has many advantages. We present a saturation method for global backwards reachability analysis of these models that can also be used to analyse collapsible pushdown systems. Beginning with an automaton representing a set of configurations, we build an automaton accepting all configurations that can reach this set. We also improve upon previous saturation techniques for higher-order pushdown systems by significantly reducing the size of the automaton constructed and simplifying the algorithm and proofs.

/pdf/a-saturation-method-for-collapsible-pushdown-systems-55464zayso.pdf

A saturation method for collapsible pushdown systems

We present a pumping lemma for each level of the collapsible pushdown graph hierarchy in analogy to the second author's 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.

/pdf/strictness-of-the-collapsible-pushdown-hierarchy-2cr8dm63rh.pdf

Strictness of the collapsible pushdown hierarchy

We show that graphs generated by collapsible pushdown systems of level $2$ are tree-automatic. Even when we allow $\varepsilon$-contractions and add a reachability predicate (with regular constraints) for pairs of configurations, the structures remain tree-automatic. Hence, their \FO theories are decidable, even when expanded by a reachability predicate. As a corollary, we obtain the tree-automaticity of the second level of the Caucal-hierarchy.

/pdf/collapsible-pushdown-graphs-of-level-2-are-tree-automatic-qzt01jm58s.pdf

Collapsible Pushdown Graphs of Level 2 are Tree-Automatic

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.

/pdf/satisfiability-of-ctl-with-constraints-3trpuigfjy.pdf

Satisfiability of CTL* with constraints

Recently, we proved that satisfiability for $\mathsf {ECTL}^*$ with constraints over $\mathbb {Z}$ is decidable using a new technique based on weak monadic second-order logic with the bounding quantifier ($\mathsf {WMSO\!+\!B}$). Here we apply this approach to concrete domains that are tree-like. We show that satisfiability of $\mathsf {ECTL}^*$ with constraints is decidable over (i) semi-linear orders, (ii) ordinal trees (semi-linear orders where the branches form ordinals), and (iii) infinitely branching order trees of height h for each fixed $h\in \mathbb {N}$. In contrast, we introduce Ehrenfeucht-Fraisse-games for $\mathsf {WMSO\!+\!B}$ (weak $\mathsf {MSO}$ with the bounding quantifier) and use them to show that our approach cannot deal with the class of order trees. Missing proofs and details can be found in the long version [6].

/pdf/satisfiability-of-ectl-with-tree-constraints-hk7q8r45fj.pdf

Satisfiability of ECTL* with Tree Constraints

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 Delhomme's 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 $\Delta^0_{\omega^\omega}$ of the hyperarithmetical hierarchy (under Turing-reductions). Full proofs can be found in the arXiv-version [11] of this paper.

/pdf/tree-automatic-well-founded-trees-19f4o38ksf.pdf

Alexander Kartzow

Papers

Strictness of the collapsible pushdown hierarchy

Collapsible Pushdown Graphs of Level 2 are Tree-Automatic

Satisfiability of CTL* with constraints

Satisfiability of ECTL* with Tree Constraints

Tree-Automatic well-founded trees