Andrzej Szepietowski

Turing Machines with Sublogarithmic Space

Basic Notions.- Languages acceptable with logarithmic space.- Examples of languages acceptable with sublogarithmic space.- Lower bounds for accepting non-regular languages.- Space constructible functions.- Halting property and closure under complement.- Strong versus weak mode of space complexity.- Padding.- Deterministic versus nondeterministic Turing machines.- Space hierarchy.- Closure under concatenation.- Alternating hierarchy.- Independent complement.- Other models of Turing machines.

Algorithms counting monotone Boolean functions

We give a new algorithm couting monotone Boolean functions of n variables (or equivalently the elements of free distributive lattices of n generators). We computed the number of monotone functions of 8 variables which is 56 130 437 228 687 557 907 788.

If deterministic and nondeterministic space complexities are equal for log log n then they are also equal for log n

It is well known that for any „well behaved“ space function L(n) ≥ log n if DSPACE(L(n)) = NSPACE(L(n)) then also DSPACE(H(n)) = NSPACE(H(n)) for all „well behaved“ functions H(n) ≥ L(n). The aim of this paper is to show that also if DSPACE(log log n) = NSPACE(log log n) then L = NL (i.e. DSPACE(log n) = NSPACE(log n)).

On three-way two-dimensional Turing machines

Abstract This paper solves several open problems concerning closure properties of three-way tape-bounded Turing machines. It is shown that: (1) the class of sets of square tapes accepted by nondeterministic three-way L(m) tape-bounded Turing machines is closed under complementation if L(m)⩾m2 is constructible, (2) the class of sets of square tapes accepted by nondeterministic three-way L(m) tape-bounded Turing machines is closed neither under row nor column cyclic closure if L(m)

Fault tolerance of vertex pancyclicity in alternating group graphs

Abstract We study fault tolerance of vertex k pancyclicity of the alternating group graph AG n . A graph G is vertex k pancyclic, if for every vertex p ∈ G , there is a cycle going through p of every length from k to | G | . Xue and Liu [Z.-J. Xue, S.-Y. Liu, An optimal result on fault-tolerant cycle-embedding in alternating group graphs, Inform. Proc. Lett. 109 (2009) 1197–1201] put the conjecture that AG n is ( 2 n - 7 ) -fault-tolerant vertex pancyclic, which means that if the number of faults | F | ⩽ 2 n - 7 , then AG n - F is still vertex pancyclic. Chang and Yang [J.-M. Chang, J.-S. Yang, Fault-tolerant cycle-embedding in alternating group graphs, Appl. Math. Comput. 197 (2008) 760–767] showed that for the shortest cycles the fault-tolerance of AG n is much lower. They noted that with n - 2 faults one can cut all 3-cycles going through a given vertex p (it is easy to observe that the same set of faults cuts all 4- and 5-cycles going through p). On the other hand they show that AG n is n - 3 -fault tolerant vertex 3 pancyclic. In this paper we show that the cycles of length ⩾ 6 are much more fault-tolerant. More precisely, we show that AG n is ( 2 n - 6 ) -fault-tolerant vertex 6 pancyclic. This bound is optimal, because every vertex p has 2 n - 4 neighbors.

A note on the oriented chromatic number of grids

Oriented coloring is a coloring of the vertices of an oriented graph such that: (1) no two neighbors have the same color, (2) if there is an arc leading from the color β1 to β2, then no arc leads from β2 to β1. In this paper we discuss oriented chromatic number of 2-dimensional grids G(m,n). In [Inform. Process. Lett. 85 (2003) 261-266] Fertin et al. gave some results for small n and stated a conjecture that every grid is T7-colorable where T7 is an oriented graph with the set of vertices V = {0,1,...,6} and the set of arcs A = {(x,x+b mod 7) |x ∈ V, b = 1,2, or 4}. We give a negative answer to this conjecture. We describe an algorithm which we ran on a computer to find an example not colorable by T7, and another algorithm to check that all G(4,n) are T7-colorable.

Complexity of weak acceptance conditions in tree automata

Weak acceptance conditions for automata on infinite words or trees are defined in terms of the set of states that appear in the run. This is in contrast with, more usual, strong conditions that are defined in terms of states appearing infinitely often on the run. Weak conditions appear in the context of model-checking and translations of logical formalisms to automata. We study the complexity of the emptiness problem for tree automata with weak conditions. We also study the translations between automata with weak and strong conditions.

On the expressive power of the shuffle operator matched with intersection by regular sets

We investigate the complexity of languages described by some expressions containing shuffle operator and intersection. We show that deciding whether the shuffle of two words has a nonempty intersection with a regular set (or fulfills some regular pattern) is NL-complete. Furthermore we show that the class of languages of the form

The element distinctness problem on one-tape Turing machines

L\cap R

Some notes on strong and weak log log n space complexity

, with a shuffle language L and a regular language R , contains non-semilinear languages and does not form a family of mildly context- sensitive languages.