Hierarchies over the context-free languages

Reinhardt, Klaus

doi:10.1007/3-540-53414-8_44

Cited by 6 publications

(9 citation statements)

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“…Reinhardt [25] related the aforementioned hierarchy of his to another hierarchy defined by alternating pushdown automata and he gave a characterization of the polynomial hierarchy in terms of this alternating hierarchy using logarithmic-space (or log-space) many-one reductions. Using an argument similar to his, we can establish in Section 5 an exact characterization of the eth level of the Boolean hierarchy over the kth level Σ P k of the polynomial hierarchy in terms of the corresponding eth level of the Boolean hierarchy over the k + 1st level of the CFL hierarchy.…”

Section: Backgrounds and Main Themesmentioning

confidence: 99%

“…The many-one CFL hierarchy consists of language families Σ CFL m,k and Π CFL m,k (k ∈ N + ) defined as follows: Σ CFL m,1 = CFL, Π CFL m,k = co-Σ CFL m,k , and Σ CFL m,k+1 = CFL m (Π CFL m,k ) for any level k ≥ 1, where the subscript "m" stands for "many-one" as before. Relative to oracle A, a relativized many-one NFA hierarchy, which was essentially formulated in [25], is defined as follows: Σ NFA,A Since the proof of Theorem 4.12 is involved, prior to the proof, we intend to demonstrate how to prove Proposition 4.11 using this theorem.…”

Section: Structural Properties Of the Cfl Hierarchymentioning

confidence: 99%

“…Symbolically, we write CFL A m to express the family of languages that are many-one CFL-reducible to a given oracle A. With a similar flavor, Reinhardt [25] considered many-one reductions, which are induced by nondeterministic finite automata (or nfa's) with no memory device. Notice that nondeterministic reducibility generally does not admit the transitivity property.…”

Section: Backgrounds and Main Themesmentioning

confidence: 99%

“…In the past literature, there were preceding ground works on many-one reducibilities within a framework of a theory of formal languages and automata. Based on deterministic/nondeterministic finite automata (or dfa's/nfa's), for instance, Reinhardt [25] discussed two many-one reducibilities between two languages. Tadaki, Yamakami, and Li [30] also studied the roles of various many-one reducibilities defined by one-tape linear-time Turing machines, which turn out to be closely related to finite automata.…”

Section: Many-one Reductions By Oracle Npda'smentioning

confidence: 99%

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Oracle Pushdown Automata, Nondeterministic Reducibilities, and the Hierarchy over the Family of Context-Free Languages

Yamakami

2014

SOFSEM 2014: Theory and Practice of Computer Science

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To expand a fundamental theory of context-free languages, we equip nondeterministic one-way pushdown automata with additional oracle mechanisms, which naturally induce various nondeterministic reducibilities among formal languages. As a natural restriction of NP-reducibility, we introduce a notion of many-one CFL reducibility and conduct a ground work to formulate a coherent framework for further expositions. Two more powerful reducibilities-bounded truth-table and Turing CFL-reducibilities-are also discussed in comparison. The Turing CFL-reducibility, in particular, helps us introduce an exquisite hierarchy, called the CFL hierarchy, built over the family CFL of context-free languages. For each level of this new hierarchy, its basic structural properties are proven and three alternative characterizations are presented. The second level is not included in NC 2 unless NP equals NC 2 . The first and second levels of the hierarchy are proven to be different. The rest of the hierarchy (more strongly, the Boolean hierarchy built over each level of the CFL hierarchy) is also infinite unless the polynomial (time) hierarchy over NP collapses. This follows from a characterization of the Boolean hierarchy over the kth level of the polynomial hierarchy in terms of the Boolean hierarchy over the k + 1st level of the CFL hierarchy using logarithmic-space many-one reductions. Similarly, the O(log n) query bounded complexity class Θ P k is related to the closure of the kth level of the CFL hierarchy under logarithmic truth-table reductions. We also argue that the CFL hierarchy coincides with a hierarchy over CFL built by application of many-one CFL-reductions. We show that BPCFL-a bounded-error probabilistic version of CFL-is not included in CFL even in the presence of advice. Employing a known circuit lower bound and a switching lemma, we exhibit a relativized world where BPCFL is not located within the second level of the CFL hierarchy.

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Section: Backgrounds and Main Themesmentioning

confidence: 99%

Section: Structural Properties Of the Cfl Hierarchymentioning

confidence: 99%

Section: Backgrounds and Main Themesmentioning

confidence: 99%

Section: Many-one Reductions By Oracle Npda'smentioning

confidence: 99%

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Oracle Pushdown Automata, Nondeterministic Reducibilities, and the Hierarchy over the Family of Context-Free Languages

Yamakami

2014

SOFSEM 2014: Theory and Practice of Computer Science

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“…Related line of study concerns the types of reductions used in finite or pushdown automata with oracle. For example, nondeterministic finite automata with oracle complying with many-one restriction have been applied to establishing oracle hierarchies over the context-free languages [7]. For the same purpose, oracle pushdown automata have been used for many-one, truth-table, and Turing reducibil-ities, respectively, inducing the underlying definitions also to oracle nondeterministic finite automata [13].…”

Section: Introductionmentioning

confidence: 99%

Simplest Non-Regular Deterministic Context-Free Language

Jancar,

Sima

2021

Preprint

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We introduce a new notion of C-simple problems for a class C of decision problems (i.e. languages), w.r.t. a particular reduction. A problem is C-simple if it can be reduced to each problem in C. This can be viewed as a conceptual counterpart to C-hard problems to which all problems in C reduce. Our concrete example is the class of non-regular deterministic context-free languages (DCFL ′ ), with a truth-table reduction by Mealy machines (which proves to be a preorder). The main technical result is a proof that the DCFL ′ language L # = {0 n 1 n | n ≥ 1} is DCFL ′ -simple, which can thus be viewed as the simplest problem in the class DCFL ′ . This result has already provided an application, to the computational model of neural networks 1ANN at the first level of analog neuron hierarchy. This model was proven not to recognize L # , by using a specialized technical argument that can hardly be generalized to other languages in DCFL ′ . By the result that L # is DCFL ′ -simple, w.r.t. the reduction that can be implemented by 1ANN, we immediately obtain that 1ANN cannot accept any language in DCFL ′ .It thus seems worthwhile to explore if looking for C-simple problems in other classes C under suitable reductions could provide effective tools for expanding the lower-bound results known for single problems to the whole classes of problems.

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