PUZZLE GRAMMARS AND CONTEXT-FREE ARRAY GRAMMARS

1991 ◽  
Vol 05 (05) ◽  
pp. 663-676 ◽  
Author(s):  
M. NIVAT ◽  
A. SAOUDI ◽  
K. G. SUBRAMANIAN ◽  
R. SIROMONEY ◽  
V. R. DARE
Keyword(s):  
Membership Problem ◽  
Generative Power ◽  
New Model ◽  
Regular Control ◽  
Emptiness Problem ◽  
Np Complete ◽  
Context Free ◽  
Rewriting Rules

We introduce a new model for generating finite, digitized, connected pictures called puzzle grammars and study its generative power by comparison with array grammars. We note how this model generalizes the classical Chomskian grammars and study the effect of direction-independent rewriting rules. We prove that regular control does not increase the power of basic puzzle grammars. We show that for basic and context-free puzzle grammars, the membership problem is NP-complete and the emptiness problem is undecidable.

On the Membership Problem of Permutation Grammars — A Direct Proof of NP-Completeness

2020 ◽  
Vol 31 (04) ◽  
pp. 515-525
Author(s):  
Benedek Nagy
Keyword(s):  
Direct Proof ◽  
Input Word ◽  
Membership Problem ◽  
Context Sensitive ◽  
Language Class ◽  
Language Classes ◽  
Sentential Form ◽  
Np Complete ◽  
Membership Problems ◽  
Context Free

One of the most essential classes of problems related to formal languages is the membership problem (also called word problem), i.e., to decide whether a given input word belongs to the language specified, e.g., by a generative grammar. For context-free languages the problem is solved efficiently by various well-known parsing algorithms. However, there are several important languages that are not context-free. The membership problem of the context-sensitive language class is PSPACE-complete, thus, it is believed that it is generally not solvable in an efficient way. There are various language classes between the above mentioned two classes having membership problems with various complexity. One of these classes, the class of permutation languages, is generated by permutation grammars, i.e., context-free grammars extended with permutation rules, where a permutation rule allows to interchange the position of two consecutive nonterminals in the sentential form. In this paper, the membership problem for permutation languages is studied. A proof is presented to show that this problem is NP-complete.

LANGUAGE FAMILIES DEFINED BY A CILIATE BIO-OPERATION: HIERARCHIES AND DECISION PROBLEMS

2005 ◽  
Vol 16 (04) ◽  
pp. 645-662 ◽  
Author(s):  
JÜRGEN DASSOW ◽  
MARKUS HOLZER
Keyword(s):  
Inverted Repeat ◽  
Decision Problems ◽  
Membership Problem ◽  
Chomsky Hierarchy ◽  
Context Sensitive ◽  
Recursively Enumerable ◽  
Enumerable Language ◽  
The Status ◽  
Emptiness Problem ◽  
Context Free

We formalize the hairpin inverted repeat excision, which is known in ciliate genetics as an operation on words and languages by defining [Formula: see text] as the set of all words xαyRαRz where w = xαyαRz and the pointer α is in P. We extend this concept to language families which results in families [Formula: see text]. For [Formula: see text] and [Formula: see text] be the families of finite, regular, context-free, context-sensitive or recursively enumerable language, respectively, we determine the hierarchy of the families [Formula: see text] and compare these families with those of the Chomsky hierarchy. Furthermore, we present the status of decidability of the membership problem, emptiness problem and finiteness problem for the families [Formula: see text].

WHEN CHURCH-ROSSER BECOMES CONTEXT FREE

2007 ◽  
Vol 18 (06) ◽  
pp. 1293-1302 ◽  
Author(s):  
MARTIN KUTRIB ◽  
ANDREAS MALCHER
Keyword(s):  
Linear Time ◽  
Membership Problem ◽  
Closure Properties ◽  
Context Free Language ◽  
Arithmetic Hierarchy ◽  
Context Free ◽  
Free Language

We investigate the intersection of Church-Rosser languages and (strongly) context-free languages. The intersection is still a proper superset of the deterministic context-free languages as well as of their reversals, while its membership problem is solvable in linear time. For the problem whether a given Church-Rosser or context-free language belongs to the intersection we show completeness for the second level of the arithmetic hierarchy. The equivalence of Church-Rosser and context-free languages is Π1-complete. It is proved that all considered intersections are pairwise incomparable. Finally, closure properties under several operations are investigated.

Membership Problem for Two-Dimensional General Row Jumping Finite Automata

2020 ◽  
Vol 31 (04) ◽  
pp. 527-538
Author(s):  
Grzegorz Madejski ◽  
Andrzej Szepietowski
Keyword(s):  
Computational Model ◽  
Turing Machine ◽  
Finite Automaton ◽  
Finite Automata ◽  
The Other ◽  
Membership Problem ◽  
Two Dimensional ◽  
Np Complete ◽  
Nondeterministic Turing Machine ◽  
Membership Problems

Two-dimensional general row jumping finite automata were recently introduced as an interesting computational model for accepting two-dimensional languages. These automata are nondeterministic. They guess an order in which rows of the input array are read and they jump to the next row only after reading all symbols in the previous row. In each row, they choose, also nondeterministically, an order in which segments of the row are read. In this paper, we study the membership problem for these automata. We show that each general row jumping finite automaton can be simulated by a nondeterministic Turing machine with space bounded by the logarithm. This means that the fixed membership problems for such automata are in NL, and so in P. On the other hand, we show that the uniform membership problem is NP-complete.

CONTEXT-SENSITIVITY OF TWO-DIMENSIONAL REGULAR ARRAY GRAMMARS

1989 ◽  
Vol 03 (03n04) ◽  
pp. 295-319 ◽  
Author(s):  
YASUNORI YAMAMOTO ◽  
KENICHI MORITA ◽  
KAZUHIRO SUGATA
Keyword(s):  
Context Sensitivity ◽  
The Other ◽  
Two Dimensional ◽  
Regular Array ◽  
Nonterminal Symbol ◽  
Left Hand ◽  
Other Hand ◽  
Context Free ◽  
Rewriting Rules

Regular array grammars (RAGs) are the lowest subclass in the Chomsky-like hierarchy of isometric array grammars. The left-hand side of each rewriting rule of RAGs has one nonterminal symbol and at most one "#" (a blank symbol). Therefore, the rewriting rules cannot sense contexts of non-# symbols. However, they can sense # as a kind of context. In this paper, we investigate this #-sensing ability. and study the language generating power of RAGs. Making good use of this ability, We show a method for RAGs to sense the contexts of local shapes of a host array in a derivation. Using this method, we give RAGs which generate the sets of all solid upright rectangles and all solid squares. On the other hand. it is proved that there is no context-free array grammar (and thus no RAG) which generates the set of all hollow upright rectangles.

scholarly journals Real-Valued GCS Classifier System

2007 ◽  
Vol 17 (4) ◽  
pp. 539-547 ◽  
Author(s):  
Łukasz Cielecki ◽  
Olgierd Unold
Keyword(s):  
Evolutionary Computation ◽  
Real World ◽  
Learning Classifier Systems ◽  
Classifier Systems ◽  
Context Free Grammar ◽  
New Model ◽  
Classifier System ◽  
Learning Classifier ◽  
Context Free ◽  
Real World Problems

Real-Valued GCS Classifier SystemLearning Classifier Systems (LCSs) have gained increasing interest in the genetic and evolutionary computation literature. Many real-world problems are not conveniently expressed using the ternary representation typically used by LCSs and for such problems an interval-based representation is preferable. A new model of LCSs is introduced to classify realvalued data. The approach applies the continous-valued context-free grammar-based system GCS. In order to handle data effectively, the terminal rules were replaced by the so-called environment probing rules. The rGCS model was tested on the checkerboard problem.

scholarly journals The subpower membership problem for semigroups

2016 ◽  
Vol 26 (07) ◽  
pp. 1435-1451 ◽  
Author(s):  
Andrei Bulatov ◽  
Marcin Kozik ◽  
Peter Mayr ◽  
Markus Steindl
Keyword(s):  
Finite Semigroup ◽  
Transformation Semigroup ◽  
Membership Problem ◽  
Clifford Semigroup ◽  
Direct Power ◽  
Nilpotent Semigroup ◽  
Full Transformation ◽  
Commutative Semigroups ◽  
Np Complete ◽  
A Finite Group

Fix a finite semigroup [Formula: see text] and let [Formula: see text] be tuples in a direct power [Formula: see text]. The subpower membership problem (SMP) asks whether [Formula: see text] can be generated by [Formula: see text]. If [Formula: see text] is a finite group, then there is a folklore algorithm that decides this problem in time polynomial in [Formula: see text]. For semigroups this problem always lies in PSPACE. We show that the [Formula: see text] for a full transformation semigroup on [Formula: see text] or more letters is actually PSPACE-complete, while on [Formula: see text] letters it is in P. For commutative semigroups, we provide a dichotomy result: if a commutative semigroup [Formula: see text] embeds into a direct product of a Clifford semigroup and a nilpotent semigroup, then [Formula: see text] is in P; otherwise it is NP-complete.

scholarly journals A Pumping Lemma for Permitting Semi-Conditional Languages

2019 ◽  
Vol 30 (01) ◽  
pp. 73-92
Author(s):  
Zsolt Gazdag ◽  
Krisztián Tichler ◽  
Erzsébet Csuhaj-Varjú
Keyword(s):  
Open Problem ◽  
Generative Power ◽  
Context Sensitive ◽  
Sentential Form ◽  
Pumping Lemma ◽  
Context Free ◽  
Context Free Grammars

Permitting semi-conditional grammars (pSCGs) are extensions of context-free grammars where each rule is associated with a word [Formula: see text] and such a rule can be applied to a sentential form [Formula: see text] only if [Formula: see text] is a subword of [Formula: see text]. We consider permitting generalized SCGs (pgSCGs) where each rule [Formula: see text] is associated with a set of words [Formula: see text] and [Formula: see text] is applicable only if every word in [Formula: see text] occurs in [Formula: see text]. We investigate the generative power of pgSCGs with no erasing rules and prove a pumping lemma for their languages. Using this lemma we show that pgSCGs are strictly weaker than context-sensitive grammars. This solves a long-lasting open problem concerning the generative power of pSCGs. Moreover, we give a comparison of the generating power of pgSCGs and that of forbidding random context grammars with no erasing rules.

The complexity of weakly recognizing morphisms

2018 ◽  
Vol 53 (1-2) ◽  
pp. 1-17
Author(s):  
Lukas Fleischer ◽  
Manfred Kufleitner
Keyword(s):  
Computational Complexity ◽  
Regular Languages ◽  
Decision Problems ◽  
Membership Problem ◽  
Surjective Morphism ◽  
Infinite Words ◽  
Descriptional Complexity ◽  
Emptiness Problem ◽  
Free Semigroups ◽  
Büchi Automaton

Weakly recognizing morphisms from free semigroups onto finite semigroups are a classical way for defining the class of ω-regular languages, i.e., a set of infinite words is weakly recognizable by such a morphism if and only if it is accepted by some Büchi automaton. We study the descriptional complexity of various constructions and the computational complexity of various decision problems for weakly recognizing morphisms. The constructions we consider are the conversion from and to Büchi automata, the conversion into strongly recognizing morphisms, as well as complementation. We also show that the fixed membership problem is NC1-complete, the general membership problem is in L and that the inclusion, equivalence and universality problems are NL-complete. The emptiness problem is shown to be NL-complete if the input is given as a non-surjective morphism.

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