Matrix Languages Research Articles

Finite-Turn Repetitive Checking Automata and Sequential/ Parallel Matrix Languages

A class of machines capable of recognizing two-dimensional sequential/parallel matrix languages is introduced. It is called finite-turn repetitive checking automata (FTRCA). This checking automaton is provided with three READ-WRITE heads, one READ only head, and a control register in its memory to keep track of the internal transition states and the stack symbols. The value of the register ranges from 0 to a finite positive integer k and can repetitively appear as many times as necessary. A one-to-one correspondence is established between the class of FIRCA and (R:R)ML–the smallest class of the sequential/parallel matrix languages. Several closure properties are also investigated.

Deux Familles de Langages Incomparables

We prove here that the context-free languages family (CFL) and the matrix languages of finite index family (MLIF) are incomparable, which answers Pun's question. The proof uses properties about the MLIF already known about the CLF. In particular, we prove analogs about the MLIF of Ogden's lemma and of the Transfer theorem for iterative pairs, due to Boasson. We also give another example of the previous Transfer theorem, by proving that the full-AFL MLIF is not principal.

On the family of finite index matrix languages

It is proved that: 1. (1) M ƒ λ is a substitution closed full AFL closed also under doubling ( D( L) = { xx | x ϵ L}). 2. (2) M ƒ = M ƒ λ = M ac ƒ = M ac ƒ λ = M left ƒ = M left ƒ λ = PM ƒ = PM ƒ λ! 3. (3) LM ƒ ⊂ M ƒ, strict inclusion. (The families M - M left λ are as in Salomaa's book [18], LM is the family or simple matrix languages of Ibarra [11] and PM, PM λ are the families of λ-free, respectively, arbitrary, parallel matrix languages [4]. For an arbitrary family of languages, L we denote by L ƒ the family of finite index languages in L , the index being considered with respect to the class of grammars generating languages in L .) 4. (4) The emptiness and the finiteness problems for M ƒ are algorithmically solvable, whereas many other decision problems about M ƒ are unsolvable. 5. (5) The composition does not increase the generative capacity of (context-free), arbitrary or of finite index, matrix grammars. A conjecture with many significant implications is formulated in the last part of the paper.

Some further remarks on the family of finite index matrix languages

Some further remarks on the family of finite index matrix languages

A homomorphic representation of simple matrix languages

In this paper we given a homomorphic representation of simple matrix languages similar in form to those of context-free languages, namely: Any simple matrix language L can be written as the homomorphic image of the intersection of the Dyck context-free language and an equal matrix language. We do not obtain a characterization, because the converse is not true.

On the index of grammars and languages

We prove the following results concerning the index of grammars and languages. (1) Any context-free grammar with the terminal vocabulary containing only one letter has a finite index. (2) Any type-O grammar of finite index such that any rule is of the form αAβ → αγβ , generates a context-free language of finite index. (3) The family of matrix languages of finite index is a full AFL.

The complexity of the membership problem for some extensions of context-free languagest†

The time and tape complexity of some families of languages defined in the literature by altering methods of generation by context-free grammars is considered. Specifically; it is shown that the following families of languages can be recognized by deterministic multitape Turing machines either in polynomial time or within (log n)2 tape: 1) the context independent developmental (EOL) languages; 2) the simple matrix languages; 3) the languages generated by derivation restricted state grammars.: 4) the languages generated by linear context-free grammars with certain non-regular control sets; 5) the languages generated by certain classes of vector grammars. In fact, these languages are of the same tape complexity as context-free languages. Other results indicate the complexity of EDOL languages and the effects on complexity of applying the homomorphic replication operator to regular and context-free languages.

Image theorems for simple matrix languages andn-parallel languages

Three image theorems are proved for three families of languages in terms of prototype languages and (nondeterministic) generalized sequential machine maps. Further, for one family, then-right linear simple matrix languages of Ibarra, a new characterization theorem is proved.

On the complexity of regulated context-free rewriting

Some complexity measures which are well-known for context-free languages are generalized in order to classify matrix languages and programmed languages. It is shown that the complexity of some context-free languages decreases if they are generated by matrix grammars or programmed grammars. An arithmetic characterization is given for infinite languages generated by two matrices. The number of matrices (as a complexity measure) is shown to be independent from any other complexity measure regarded in this paper.

Characterizations of regular and context-free matrices

The u-v theorem for context-free languages is extended to prove an intercalation theorem for the family of context-free matrix languages. A row-wise iteration factor theorem is proved for the families of regular and context-free matrix languages. Characterizations of regular and context-free matrix languages are given in terms of vertical regular sequences and simple operations on vertical regular sequences. Closure of regular and context-free matrix languages under array nondeterministic finite state transducer mappings is established and an image theorem proved. This is used to give another characterization of regular matrix languages. Further it is shown that the family of regular matrix languages is a principal abstract family of matrices (AFM). The effect of string control and array control on these families are examined.

Simple matrix languages with a leftmost restriction

It is shown that every type 1 language can be generated by an ε -free simple matrix grammar (of order 2), if a certain leftmost restriction is imposed on the application of matrices. Dropping the ε -free restriction, every type 0 language can be obtained. This strengthens a result previously established for matrix grammars.

Controlled pushdown automata

A device called controlled pushdown automaton (CPA) is introduced and special classes of it are shown to characterize the families of simple matrix languages (SML's) and right-linear simple matrix languages (RLSML's). Other special classes are shown to define, respectively, the smallest full AFL containing the SML's and the smallest full AFL containing the RLSML's. Hierarchy theorems and other related results are also obtained. In particular, it is shown that the smallest full AFL containing the homomorphic replications of the context-free languages is not principal.

Unambiguous equal matrix languages

A necessary and sufficient algebraic condition is given for a subclass of equal matrix languages to be unambiguous. It is shown that all languages from this class are unambiguous. Recursive unsolvability of ambiguity in equal matrix grammar and decidability of ambiguity in bounded equal matrix grammar are proved.

Finite-turn checking automata

A one-to-one correspondence is established between the class of all equal matrix languages and the class of finite-turn checking automata. This checking automaton is provided with a counter in its memory to keep track of the number of turns of the stack-head. Several closure properties of this family are established.

Equal matrix languages and Kleene closure

It is shown that the family of equal matrix languages (EML) which intersects the family of context-sensitive and context-free languages is not closed under Kleene closure, intersection and complementation. The family of bounded EML whose generative power lies between that of bounded context-sensitive languages and bounded context-free languages is shown to be a class of unambiguous languages. Operations that preserve unambiguity and certain decidability questions are discussed.

Simple matrix languages

Simple matrix languages and right-linear simple matrix languages are defined as subfamilies of matrix languages by putting restrictions on the form and length (degree) of the rewriting rules associated with matrix grammars. For each n ⩾ 1, let L ( n ) [ ℛ ( n ) ] be the class of simple matrix languages [right-linear simple matrix languages] of degree n , and let L = ⋃ n ⩾ 1 L ( n ) [ ℛ = ⋃ n ⩾ 1 ℛ ( n ) ] . It is shown that L ( 1 ) [ ℛ ( 1 ) ] coincides with the class of context-free languages [regular sets] and that L is a proper subset of the family of languages accepted by deterministic linear bounded automata. It is proved that L ( n ) [ ℛ ( n ) ] forms a hierarchy of classes of languages in L [ ℛ ] . The closure properties and decision problems associated with L ( n ) , L , ℛ ( n ) , and ℛ are thoroughly investigated. Let L B [ ℛ B ] be the bounded languages in L [ ℛ ] . It is shown that L B = ℛ B and that most of the positive closure and decision results which are true for bounded context-free languages are carried over in L B . A characterization of L B as the smallest family of languages which contains the bounded context-free languages and which is closed under the operations of union and intersection is proved.

Channel capacity of equal matrix languages

The channel capacity of the class of equal matrix languages which includes several well-known context-sensitive and context-free languages is found out.