Membership Problem for Two-Dimensional General Row Jumping Finite Automata

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.

Testing a new idea to solve the P = NP problem with mathematical induction

Background. P and NP are two classes (sets) of languages in Computer Science. An open problem is whether P = NP. This paper tests a new idea to compare the two language sets and attempts to prove that these two language sets consist of same languages by elementary mathematical methods and basic knowledge of Turing machine. Methods. By introducing a filter function C(M,w) that is the number of configurations which have more than one children (nondeterministic moves) in the shortest accept computation path of a nondeterministic Turing machine M for input w, for any language L(M) ∈ NP, we can define a series of its subsets, Li(M) = {w | w ∈ L(M) ∧ C(M,w) ≤ i}, and a series of the subsets of NP as Li = {Li(M) | ∀M ∙ L(M) ∈ NP}. The nondeterministic multi-tape Turing machine is used to bridge two language sets Li and Li+1, by simulating the (i+1)-th nondeterministic move deterministically in multiple work tapes, to reduce one (the last) nondeterministic move. Results. The main result is that, with the above methods, the language set Li+1, which seems more powerful, can be proved to be a subset of Li. This result collapses Li ⊆ P for all i ∈ N. With NP = ⋃i∈NLi, it is clear that NP ⊆ P. Because by definition P ⊆ NP, we have P = NP. Discussion. There can be other ways to define the subsets Li and prove the same result. The result can be extended to cover any sets of time functions C, if ∀f ∙ f ∈ C ⇒ f2 ∈ C, then DTIME(C) = NTIME(C). This paper does not show any ways to find a solution in P for the problem known in NP.

Testing a new idea to solve the P = NP problem with mathematical induction

Background. P and NP are two classes (sets) of languages in Computer Science. An open problem is whether P = NP. This paper tests a new idea to compare the two language sets and attempts to prove that these two language sets consist of same languages by elementary mathematical methods and basic knowledge of Turing machine. Methods. By introducing a filter function C(M,w) that is the number of configurations which have more than one children (nondeterministic moves) in the shortest accept computation path of a nondeterministic Turing machine M for input w, for any language L(M) ∈ NP, we can define a series of its subsets, Li(M) = {w | w ∈ L(M) ∧ C(M,w) ≤ i}, and a series of the subsets of NP as Li = {Li(M) | ∀M ∙ L(M) ∈ NP}. The nondeterministic multi-tape Turing machine is used to bridge two language sets Li and Li+1, by simulating the (i+1)-th nondeterministic move deterministically in multiple work tapes, to reduce one (the last) nondeterministic move. Results. The main result is that, with the above methods, the language set Li+1, which seems more powerful, can be proved to be a subset of Li. This result collapses Li ⊆ P for all i ∈ N. With NP = ⋃i∈NLi, it is clear that NP ⊆ P. Because by definition P ⊆ NP, we have P = NP. Discussion. There can be other ways to define the subsets Li and prove the same result. The result can be extended to cover any sets of time functions C, if ∀f ∙ f ∈ C ⇒ f2 ∈ C, then DTIME(C) = NTIME(C). This paper does not show any ways to find a solution in P for the problem known in NP.

ON THE DESCRIPTIONAL COMPLEXITY OF ACCEPTING NETWORKS OF EVOLUTIONARY PROCESSORS WITH FILTERED CONNECTIONS

In this paper we consider, from the descriptional complexity point of view, a model of computation introduced in [1], namely accepting network of evolutionary processors with filtered connections (ANEPFCs). First we show that for each morphism h : V → W*, with V ∩ W = ∅, one can effectively construct an ANEPFC, of size 6 + |W|, which accepts every input word w and, at the end of the computation on this word, obtains h(w) in its output node. This result can be applied in constructing two different ANEPFCs, with 27 and, respectively, 26 processors, recognizing a given recursively enumerable language. The first architecture, based on the construction of a universal ANEPFC, has the property that only 7 of its 27 processors depend on the accepted language. On the other hand, all the 26 processors of the second architecture depend on the accepted language, but, differently from the first one, this network simulates efficiently (from both time and space perspectives) a nondeterministic Turing machine accepting the given language.

THE SIMULATION OF TWO-DIMENSIONAL ONE-MARKER AUTOMATA BY THREE-WAY TURING MACHINES

We denote a two-dimensional deterministic (nondeterministic) one-marker automaton by 2-DM1 (2-NM1), and a three-way two-dimensional deterministic (nondeterministic) Turing machine by TR2-DTM (TR2-NTM). In this paper, we show that the necessary and sufficient space for TR2-NTMs to simulate 2-DM1s (2-NM1s) is n log n (n2), and the necessary and sufficient space for TR2-DTMs to simulate 2-DM1s (2-NM1s) is 2O(n log n) (2 O(n2)), where n is the number of columns of rectangular input tapes.