One-Tape Turing Machine Variants and Language Recognition

We present two restricted versions of one-tape Turing machines. Both characterize the class of context-free languages. In the first version, proposed by Hibbard in 1967 and called limited automata, each tape cell can be rewritten only in the first $d$ visits, for a fixed constant $d\geq 2$. Furthermore, for $d=2$ deterministic limited automata are equivalent to deterministic pushdown automata, namely they characterize deterministic context-free languages. Further restricting the possible operations, we consider strongly limited automata. These models still characterize context-free languages. However, the deterministic version is less powerful than the deterministic version of limited automata. In fact, there exist deterministic context-free languages that are not accepted by any deterministic strongly limited automaton.Comment: 20 pages. This article will appear in the Complexity Theory Column of the September 2015 issue of SIGACT New

Two-Way Automata Making Choices Only at the Endmarkers

The question of the state-size cost for simulation of two-way nondeterministic automata (2NFAs) by two-way deterministic automata (2DFAs) was raised in 1978 and, despite many attempts, it is still open. Subsequently, the problem was attacked by restricting the power of 2DFAs (e.g., using a restricted input head movement) to the degree for which it was already possible to derive some exponential gaps between the weaker model and the standard 2NFAs. Here we use an opposite approach, increasing the power of 2DFAs to the degree for which it is still possible to obtain a subexponential conversion from the stronger model to the standard 2DFAs. In particular, it turns out that subexponential conversion is possible for two-way automata that make nondeterministic choices only when the input head scans one of the input tape endmarkers. However, there is no restriction on the input head movement. This implies that an exponential gap between 2NFAs and 2DFAs can be obtained only for unrestricted 2NFAs using capabilities beyond the proposed new model. As an additional bonus, conversion into a machine for the complement of the original language is polynomial in this model. The same holds for making such machines self-verifying, halting, or unambiguous. Finally, any superpolynomial lower bound for the simulation of such machines by standard 2DFAs would imply LNL. In the same way, the alternating version of these machines is related to L =? NL =? P, the classical computational complexity problems.Comment: 23 page

Learning Residual Finite-State Automata Using Observation Tables

We define a two-step learner for RFSAs based on an observation table by using an algorithm for minimal DFAs to build a table for the reversal of the language in question and showing that we can derive the minimal RFSA from it after some simple modifications. We compare the algorithm to two other table-based ones of which one (by Bollig et al. 2009) infers a RFSA directly, and the other is another two-step learner proposed by the author. We focus on the criterion of query complexity.Comment: In Proceedings DCFS 2010, arXiv:1008.127

Nondeterministic one-tape off-line Turing machines and their time complexity

In this paper we consider the time and the crossing sequence complexities of one-tape off-line Turing machines. We show that the running time of each nondeterministic machine accepting a nonregular language must grow at least as n\log n, in the case all accepting computations are considered (accept measure). We also prove that the maximal length of the crossing sequences used in accepting computations must grow at least as \log n. On the other hand, it is known that if the time is measured considering, for each accepted string, only the faster accepting computation (weak measure), then there exist nonregular languages accepted in linear time. We prove that under this measure, each accepting computation should exhibit a crossing sequence of length at least \log\log n. We also present efficient implementations of algorithms accepting some unary nonregular languages.Comment: 18 pages. The paper will appear on the Journal of Automata, Languages and Combinatoric

Once-Marking and Always-Marking 1-Limited Automata

Single-tape nondeterministic Turing machines that are allowed to replace the symbol in each tape cell only when it is scanned for the first time are also known as 1-limited automata. These devices characterize, exactly as finite automata, the class of regular languages. However, they can be extremely more succinct. Indeed, in the worst case the size gap from 1-limited automata to one-way deterministic finite automata is double exponential. Here we introduce two restricted versions of 1-limited automata, once-marking 1-limited automata and always-marking 1-limited automata, and study their descriptional complexity. We prove that once-marking 1-limited automata still exhibit a double exponential size gap to one-way deterministic finite automata. However, their deterministic restriction is polynomially related in size to two-way deterministic finite automata, in contrast to deterministic 1-limited automata, whose equivalent two-way deterministic finite automata in the worst case are exponentially larger. For always-marking 1-limited automata, we prove that the size gap to one-way deterministic finite automata is only a single exponential. The gap remains exponential even in the case the given machine is deterministic. We obtain other size relationships between different variants of these machines and finite automata and we present some problems that deserve investigation.Comment: In Proceedings AFL 2023, arXiv:2309.0112

Descriptional Complexity of the Languages KaL: Automata, Monoids and Varieties

The first step when forming the polynomial hierarchies of languages is to consider languages of the form KaL where K and L are over a finite alphabet A and from a given variety V of languages, a being a letter from A. All such KaL's generate the variety of languages BPol1(V). We estimate the numerical parameters of the language KaL in terms of their values for K and L. These parameters include the state complexity of the minimal complete DFA and the size of the syntactic monoids. We also estimate the cardinality of the image of A* in the Schuetzenberger product of the syntactic monoids of K and L. In these three cases we obtain the optimal bounds. Finally, we also consider estimates for the cardinalities of free monoids in the variety of monoids corresponding to BPol1(V) in terms of sizes of the free monoids in the variety of monoids corresponding to V.Comment: In Proceedings DCFS 2010, arXiv:1008.127

Operational State Complexity of Deterministic Unranked Tree Automata

We consider the state complexity of basic operations on tree languages recognized by deterministic unranked tree automata. For the operations of union and intersection the upper and lower bounds of both weakly and strongly deterministic tree automata are obtained. For tree concatenation we establish a tight upper bound that is of a different order than the known state complexity of concatenation of regular string languages. We show that (n+1) ( (m+1)2^n-2^(n-1) )-1 vertical states are sufficient, and necessary in the worst case, to recognize the concatenation of tree languages recognized by (strongly or weakly) deterministic automata with, respectively, m and n vertical states.Comment: In Proceedings DCFS 2010, arXiv:1008.127