Linear Bounded Composition of Tree-Walking Tree Transducers: Linear Size Increase and Complexity

Compositions of tree-walking tree transducers form a hierarchy with respect to the number of transducers in the composition. As main technical result it is proved that any such composition can be realized as a linear bounded composition, which means that the sizes of the intermediate results can be chosen to be at most linear in the size of the output tree. This has consequences for the expressiveness and complexity of the translations in the hierarchy. First, if the computed translation is a function of linear size increase, i.e., the size of the output tree is at most linear in the size of the input tree, then it can be realized by just one, deterministic, tree-walking tree transducer. For compositions of deterministic transducers it is decidable whether or not the translation is of linear size increase. Second, every composition of deterministic transducers can be computed in deterministic linear time on a RAM and in deterministic linear space on a Turing machine, measured in the sum of the sizes of the input and output tree. Similarly, every composition of nondeterministic transducers can be computed in simultaneous polynomial time and linear space on a nondeterministic Turing machine. Their output tree languages are deterministic context-sensitive, i.e., can be recognized in deterministic linear space on a Turing machine. The membership problem for compositions of nondeterministic translations is nondeterministic polynomial time and deterministic linear space. The membership problem for the composition of a nondeterministic and a deterministic tree-walking tree translation (for a nondeterministic IO macro tree translation) is log-space reducible to a context-free language, whereas the membership problem for the composition of a deterministic and a nondeterministic tree-walking tree translation (for a nondeterministic OI macro tree translation) is possibly NP-complete

Streaming Property Testing of Visibly Pushdown Languages

In the context of language recognition, we demonstrate the superiority of streaming property testers against streaming algorithms and property testers, when they are not combined. Initiated by Feigenbaum et al., a streaming property tester is a streaming algorithm recognizing a language under the property testing approximation: it must distinguish inputs of the language from those that are $\varepsilon$-far from it, while using the smallest possible memory (rather than limiting its number of input queries). Our main result is a streaming $\varepsilon$-property tester for visibly pushdown languages (VPL) with one-sided error using memory space $\mathrm{poly}((\log n) / \varepsilon)$. This constructions relies on a (non-streaming) property tester for weighted regular languages based on a previous tester by Alon et al. We provide a simple application of this tester for streaming testing special cases of instances of VPL that are already hard for both streaming algorithms and property testers. Our main algorithm is a combination of an original simulation of visibly pushdown automata using a stack with small height but possible items of linear size. In a second step, those items are replaced by small sketches. Those sketches relies on a notion of suffix-sampling we introduce. This sampling is the key idea connecting our streaming tester algorithm to property testers.Comment: 23 pages. Major modifications in the presentatio

A Characterization for Decidable Separability by Piecewise Testable Languages

The separability problem for word languages of a class $\mathcal{C}$ by languages of a class $\mathcal{S}$ asks, for two given languages $I$ and $E$ from $\mathcal{C}$, whether there exists a language $S$ from $\mathcal{S}$ that includes $I$ and excludes $E$, that is, $I \subseteq S$ and $S\cap E = \emptyset$. In this work, we assume some mild closure properties for $\mathcal{C}$ and study for which such classes separability by a piecewise testable language (PTL) is decidable. We characterize these classes in terms of decidability of (two variants of) an unboundedness problem. From this, we deduce that separability by PTL is decidable for a number of language classes, such as the context-free languages and languages of labeled vector addition systems. Furthermore, it follows that separability by PTL is decidable if and only if one can compute for any language of the class its downward closure wrt. the scattered substring ordering (i.e., if the set of scattered substrings of any language of the class is effectively regular). The obtained decidability results contrast some undecidability results. In fact, for all (non-regular) language classes that we present as examples with decidable separability, it is undecidable whether a given language is a PTL itself. Our characterization involves a result of independent interest, which states that for any kind of languages $I$ and $E$, non-separability by PTL is equivalent to the existence of common patterns in $I$ and $E$

Practical Run-time Checking via Unobtrusive Property Caching

The use of annotations, referred to as assertions or contracts, to describe program properties for which run-time tests are to be generated, has become frequent in dynamic programing languages. However, the frameworks proposed to support such run-time testing generally incur high time and/or space overheads over standard program execution. We present an approach for reducing this overhead that is based on the use of memoization to cache intermediate results of check evaluation, avoiding repeated checking of previously verified properties. Compared to approaches that reduce checking frequency, our proposal has the advantage of being exhaustive (i.e., all tests are checked at all points) while still being much more efficient than standard run-time checking. Compared to the limited previous work on memoization, it performs the task without requiring modifications to data structure representation or checking code. While the approach is general and system-independent, we present it for concreteness in the context of the Ciao run-time checking framework, which allows us to provide an operational semantics with checks and caching. We also report on a prototype implementation and provide some experimental results that support that using a relatively small cache leads to significant decreases in run-time checking overhead.Comment: 30 pages, 1 table, 170 figures; added appendix with plots; To appear in Theory and Practice of Logic Programming (TPLP), Proceedings of ICLP 201

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

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

Quantum Property Testing

A language L has a property tester if there exists a probabilistic algorithm that given an input x only asks a small number of bits of x and distinguishes the cases as to whether x is in L and x has large Hamming distance from all y in L. We define a similar notion of quantum property testing and show that there exist languages with quantum property testers but no good classical testers. We also show there exist languages which require a large number of queries even for quantumly testing

Finite Countermodel Based Verification for Program Transformation (A Case Study)

Both automatic program verification and program transformation are based on program analysis. In the past decade a number of approaches using various automatic general-purpose program transformation techniques (partial deduction, specialization, supercompilation) for verification of unreachability properties of computing systems were introduced and demonstrated. On the other hand, the semantics based unfold-fold program transformation methods pose themselves diverse kinds of reachability tasks and try to solve them, aiming at improving the semantics tree of the program being transformed. That means some general-purpose verification methods may be used for strengthening program transformation techniques. This paper considers the question how finite countermodels for safety verification method might be used in Turchin's supercompilation method. We extract a number of supercompilation sub-algorithms trying to solve reachability problems and demonstrate use of an external countermodel finder for solving some of the problems.Comment: In Proceedings VPT 2015, arXiv:1512.0221