Collapsible Pushdown Graphs of Level 2 are Tree-Automatic

We show that graphs generated by collapsible pushdown systems of level 2 are tree-automatic. Even when we allow $\epsilon$-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.Comment: 12 pages Accepted for STACS 201

An Experiment in Ping-Pong Protocol Verification by Nondeterministic Pushdown Automata

An experiment is described that confirms the security of a well-studied class of cryptographic protocols (Dolev-Yao intruder model) can be verified by two-way nondeterministic pushdown automata (2NPDA). A nondeterministic pushdown program checks whether the intersection of a regular language (the protocol to verify) and a given Dyck language containing all canceling words is empty. If it is not, an intruder can reveal secret messages sent between trusted users. The verification is guaranteed to terminate in cubic time at most on a 2NPDA-simulator. The interpretive approach used in this experiment simplifies the verification, by separating the nondeterministic pushdown logic and program control, and makes it more predictable. We describe the interpretive approach and the known transformational solutions, and show they share interesting features. Also noteworthy is how abstract results from automata theory can solve practical problems by programming language means.Comment: In Proceedings MARS/VPT 2018, arXiv:1803.0866

Marking Shortest Paths On Pushdown Graphs Does Not Preserve MSO Decidability

In this paper we consider pushdown graphs, i.e. infinite graphs that can be described as transition graphs of deterministic real-time pushdown automata. We consider the case where some vertices are designated as being final and we built, in a breadth-first manner, a marking of edges that lead to such vertices (i.e., for every vertex that can reach a final one, we mark all out-going edges laying on some shortest path to a final vertex). Our main result is that the edge-marked version of a pushdown graph may itself no longer be a pushdown graph, as we prove that this enrich graph may have an undecidable MSO theory. In this paper we consider pushdown graphs, i.e. infinite graphs that can be described as transition graphs of deterministic real-time pushdown automata. We consider the case where some vertices are designated as being final and we build, in a breadth-first manner, a marking of edges that lead to such vertices (i.e., for every vertex that can reach a final one, we mark all out-going edges laying on some shortest path to a final vertex). Our main result is that the edge-marked version of a pushdown graph may itself no longer be a pushdown graph, as we prove that the MSO theory of this enriched graph may be undecidable.Comment: 11 pages, 2 figure

Separation of Test-Free Propositional Dynamic Logics over Context-Free Languages

For a class L of languages let PDL[L] be an extension of Propositional Dynamic Logic which allows programs to be in a language of L rather than just to be regular. If L contains a non-regular language, PDL[L] can express non-regular properties, in contrast to pure PDL. For regular, visibly pushdown and deterministic context-free languages, the separation of the respective PDLs can be proven by automata-theoretic techniques. However, these techniques introduce non-determinism on the automata side. As non-determinism is also the difference between DCFL and CFL, these techniques seem to be inappropriate to separate PDL[DCFL] from PDL[CFL]. Nevertheless, this separation is shown but for programs without test operators.Comment: In Proceedings GandALF 2011, arXiv:1106.081

Memoization for Unary Logic Programming: Characterizing PTIME

We give a characterization of deterministic polynomial time computation based on an algebraic structure called the resolution semiring, whose elements can be understood as logic programs or sets of rewriting rules over first-order terms. More precisely, we study the restriction of this framework to terms (and logic programs, rewriting rules) using only unary symbols. We prove it is complete for polynomial time computation, using an encoding of pushdown automata. We then introduce an algebraic counterpart of the memoization technique in order to show its PTIME soundness. We finally relate our approach and complexity results to complexity of logic programming. As an application of our techniques, we show a PTIME-completeness result for a class of logic programming queries which use only unary function symbols.Comment: Soumis {\`a} LICS 201

On external presentations of infinite graphs

The vertices of a finite state system are usually a subset of the natural numbers. Most algorithms relative to these systems only use this fact to select vertices. For infinite state systems, however, the situation is different: in particular, for such systems having a finite description, each state of the system is a configuration of some machine. Then most algorithmic approaches rely on the structure of these configurations. Such characterisations are said internal. In order to apply algorithms detecting a structural property (like identifying connected components) one may have first to transform the system in order to fit the description needed for the algorithm. The problem of internal characterisation is that it hides structural properties, and each solution becomes ad hoc relatively to the form of the configurations. On the contrary, external characterisations avoid explicit naming of the vertices. Such characterisation are mostly defined via graph transformations. In this paper we present two kind of external characterisations: deterministic graph rewriting, which in turn characterise regular graphs, deterministic context-free languages, and rational graphs. Inverse substitution from a generator (like the complete binary tree) provides characterisation for prefix-recognizable graphs, the Caucal Hierarchy and rational graphs. We illustrate how these characterisation provide an efficient tool for the representation of infinite state systems

Two-Way Visibly Pushdown Automata and Transducers

Automata-logic connections are pillars of the theory of regular languages. Such connections are harder to obtain for transducers, but important results have been obtained recently for word-to-word transformations, showing that the three following models are equivalent: deterministic two-way transducers, monadic second-order (MSO) transducers, and deterministic one-way automata equipped with a finite number of registers. Nested words are words with a nesting structure, allowing to model unranked trees as their depth-first-search linearisations. In this paper, we consider transformations from nested words to words, allowing in particular to produce unranked trees if output words have a nesting structure. The model of visibly pushdown transducers allows to describe such transformations, and we propose a simple deterministic extension of this model with two-way moves that has the following properties: i) it is a simple computational model, that naturally has a good evaluation complexity; ii) it is expressive: it subsumes nested word-to-word MSO transducers, and the exact expressiveness of MSO transducers is recovered using a simple syntactic restriction; iii) it has good algorithmic/closure properties: the model is closed under composition with a unambiguous one-way letter-to-letter transducer which gives closure under regular look-around, and has a decidable equivalence problem

A B\"uchi-Elgot-Trakhtenbrot theorem for automata with MSO graph storage

We introduce MSO graph storage types, and call a storage type MSO-expressible if it is isomorphic to some MSO graph storage type. An MSO graph storage type has MSO-definable sets of graphs as storage configurations and as storage transformations. We consider sequential automata with MSO graph storage and associate with each such automaton a string language (in the usual way) and a graph language; a graph is accepted by the automaton if it represents a correct sequence of storage configurations for a given input string. For each MSO graph storage type, we define an MSO logic which is a subset of the usual MSO logic on graphs. We prove a B\"uchi-Elgot-Trakhtenbrot theorem, both for the string case and the graph case. Moreover, we prove that (i) each MSO graph transduction can be used as storage transformation in an MSO graph storage type, (ii) every automatic storage type is MSO-expressible, and (iii) the pushdown operator on storage types preserves the property of MSO-expressibility. Thus, the iterated pushdown storage types are MSO-expressible

Highly Undecidable Problems For Infinite Computations

We show that many classical decision problems about 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are $\Pi_2^1$-complete, hence located at the second level of the analytical hierarchy, and "highly undecidable". In particular, the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and the unambiguity problem are all $\Pi_2^1$-complete for context-free omega-languages or for infinitary rational relations. Topological and arithmetical properties of 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are also highly undecidable. These very surprising results provide the first examples of highly undecidable problems about the behaviour of very simple finite machines like 1-counter automata or 2-tape automata.Comment: to appear in RAIRO-Theoretical Informatics and Application