Unsafe Order-2 Tree Languages Are Context-Sensitive

Kobayashi, Naoki; Inaba, Kazuhiro; Tsukada, Takeshi

doi:10.1007/978-3-642-54830-7_10

Cited by 6 publications

(4 citation statements)

References 17 publications

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“…However, the languages of the oi-hierarchy are generated by so-called "safe" high-level grammars, and it is not known whether the same results hold for unsafe high-level grammars. It is proved in [54] that the languages generated by unsafe level-2 grammars, the unsafe version of OI (2), are in NSPACE(n).…”

Section: Resultsmentioning

confidence: 99%

Linear-bounded composition of tree-walking tree transducers: linear size increase and complexity

Engelfriet

Inaba²,

Maneth

2019

Acta Informatica

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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. All the above results also hold for compositions of macro tree transducers. 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.

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Section: Resultsmentioning

confidence: 99%

Linear-bounded composition of tree-walking tree transducers: linear size increase and complexity

Engelfriet

Inaba²,

Maneth

2019

Acta Informatica

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“…A problem still open is whether languages defined by such grammars are context sensitive, or in other words if they belong to the complexity class NLINSPACE (non-deterministic linear space). Recent progresses on this problem have been achieved by Kobayashi et al [17], who showed that this is at least the case up to order 2 for tree languages and order 3 for word languages. Beside the fact that this line of research does not target a capturing result, the most significant difference with our work is that we consider a polyadic μ-calculus, or in different words, an automaton model that uses multiple tapes, whereas collapsible pushdown automata (the automaton model for higher-order grammars) only work with one tape.…”

Section: Contributionsmentioning

confidence: 99%

Capturing Bisimulation-Invariant Complexity Classes with Higher-Order Modal Fixpoint Logic

Lange

Lozes

2014

Advanced Information Systems Engineering

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Polyadic Higher-Order Fixpoint Logic (PHFL) is a modal fixpoint logic obtained as the merger of Higher-Order Fixpoint Logic (HFL) and the Polyadic μ-Calculus. Polyadicity enables formulas to make assertions about tuples of states rather than states only. Like HFL, PHFL has the ability to formalise properties using higher-order functions. We consider PHFL in the setting of descriptive complexity theory: its fragment using no functions of higher-order is exactly the Polyadic μ-Calculus, and it is known from Otto's Theorem that it captures the bisimulation-invariant fragment of PTIME. We extend this and give capturing results for the bisimulation-invariant fragments of EXP-TIME, PSPACE, and NLOGSPACE.

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“…Intersection type systems were intensively used in the context of recursion schemes, for several purposes like model-checking [18,21,5,29], pumping [19,2], transformations of HORSes [20,1,8], etc. Interestingly, constructions very similar to intersection types were used also on the side of collapsible pushdown systems; they were alternating stack automata [4], and types of stacks [23,16].…”

Section: Introductionmentioning

confidence: 99%

Intersection Types for Unboundedness Problems

Parys

2019

Electron. Proc. Theor. Comput. Sci.

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Intersection types have been originally developed as an extension of simple types, but they can also be used for refining simple types. In this survey we concentrate on the latter option; more precisely, on the use of intersection types for describing quantitative properties of simply typed lambda-terms. We present two type systems. The first allows to estimate (by appropriately defined value of a derivation) the number of appearances of a fixed constant a in the beta-normal form of a considered lambdaterm. The second type system is more complicated, and allows to estimate the maximal number of appearances of the constant a on a single branch. * Work supported by the National Science Centre, Poland (grant no. 2016/22/E/ST6/00041). 1 Following the convention in this area, we use the word "sort" for simple types, and the word "type" for intersection types refining them.

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Unsafe Order-2 Tree Languages Are Context-Sensitive

Cited by 6 publications

References 17 publications

Linear-bounded composition of tree-walking tree transducers: linear size increase and complexity

Linear-bounded composition of tree-walking tree transducers: linear size increase and complexity

Capturing Bisimulation-Invariant Complexity Classes with Higher-Order Modal Fixpoint Logic

Intersection Types for Unboundedness Problems

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