Computational limitations of affine automata and generalized affine automata

We present new results on the computational limitations of affine automata (AfAs). First, we show that using the endmarker does not increase the computational power of AfAs. Second, we show that the computation of bounded-error rational-valued AfAs can be simulated in logarithmic space. Third, we identify some logspace unary languages that are not recognized by algebraic-valued AfAs. Fourth, we show that using arbitrary real-valued transition matrices and state vectors does not increase the computational power of AfAs in the unbounded-error model. When focusing only the rational values, we obtain the the same result also for bounded error. As a consequence, we show that the class of bounded-error affine languages remains the same when the AfAs are restricted to use rational numbers only.

State Complexity of Overlap Assembly

The state complexity of a regular language [Formula: see text] is the number [Formula: see text] of states in a minimal deterministic finite automaton (DFA) accepting [Formula: see text]. The state complexity of a regularity-preserving binary operation on regular languages is defined as the maximal state complexity of the result of the operation where the two operands range over all languages of state complexities [Formula: see text] and [Formula: see text], respectively. We determine, for [Formula: see text], [Formula: see text], the exact value of the state complexity of the binary operation overlap assembly on regular languages. This operation was introduced by Csuhaj-Varjú, Petre, and Vaszil to model the process of self-assembly of two linear DNA strands into a longer DNA strand, provided that their ends “overlap”. We prove that the state complexity of the overlap assembly of languages [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text], is at most [Formula: see text]. Moreover, for [Formula: see text] and [Formula: see text] there exist languages [Formula: see text] and [Formula: see text] over an alphabet of size [Formula: see text] whose overlap assembly meets the upper bound and this bound cannot be met with smaller alphabets. Finally, we prove that [Formula: see text] is the state complexity of the overlap assembly in the case of unary languages and that there are binary languages whose overlap assembly has exponential state complexity at least [Formula: see text].

Uncountable Realtime Probabilistic Classes

We investigate the minimal cases for realtime probabilistic machines that can define uncountably many languages with bounded error. We show that logarithmic space is enough for realtime PTMs on unary languages. On non-unary case, we obtain the same result for double logarithmic space, which is also tight. When replacing the work tape with a few counters, we can still achieve similar results for unary linear-space two-counter automata, unary sublinear-space three-counter automata, and non-unary sublinear-space two-counter automata. We also show how to slightly improve the sublinear-space constructions by using more counters.

Expressive capacity of subregular expressions

Different types of subregular expressions are studied. Each type is obtained by either omitting one of the regular operations or replacing it by complementation or intersection. For uniformity and in order to allow non-trivial languages to be expressed, the set of literals is a finite set of words instead of letters. The power and limitations as well as relations with each other are considered, which is often done in terms of unary languages. Characterizations of some of the language families are obtained. A finite hierarchy is shown that reveals that the operation complementation is generally stronger than intersection. Furthermore, we investigate the closures of language families described by regular expressions with omitted operation under that operation. While it is known that in case of union this closure captures all regular languages, for the cases of concatenation and star incomparability results are obtained with the corresponding language families where the operation is replaced by complementation.

Uncountable classical and quantum complexity classes

It is known that poly-time constant-space quantum Turing machines (QTMs) and logarithmic-space probabilistic Turing machines (PTMs) recognize uncountably many languages with bounded error (A.C. Cem Say and A. Yakaryılmaz, Magic coins are useful for small-space quantum machines. Quant. Inf. Comput. 17 (2017) 1027–1043). In this paper, we investigate more restricted cases for both models to recognize uncountably many languages with bounded error. We show that double logarithmic space is enough for PTMs on unary languages in sweeping reading mode or logarithmic space for one-way head. On unary languages, for quantum models, we obtain middle logarithmic space for counter machines. For binary languages, arbitrary small non-constant space is enough for PTMs even using only counter as memory. For counter machines, when restricted to polynomial time, we can obtain the same result for linear space. For constant-space QTMs, we obtain the result for a restricted sweeping head, known as restarting realtime.