On the Density of Languages Accepted by Turing Machines and Other Machine Models

A language is dense if the set of all infixes (or subwords) of the language is the set of all words. Here, it is shown that it is decidable whether the language accepted by a nondeterministic Turing machine with a one-way read-only input and a reversal-bounded read/write worktape (the read/write head changes direction at most some fixed number of times) is dense. From this, it is implied that it is also decidable for one-way reversal-bounded queue automata, one-way reversal-bounded stack automata, and one-way reversal-bounded $k$-flip pushdown automata (machines that can "flip" their pushdowns up to $k$ times). However, it is undecidable for deterministic Turing machines with two 1-reversal-bounded worktapes (even when the two tapes are restricted to operate as 1-reversal-bounded pushdown stacks)

Variations of Checking Stack Automata: Obtaining Unexpected Decidability Properties

We introduce a model of one-way language acceptors (a variant of a checking stack automaton) and show the following decidability properties: (1) The deterministic version has a decidable membership problem but has an undecidable emptiness problem. (2) The nondeterministic version has an undecidable membership problem and emptiness problem. There are many models of accepting devices for which there is no difference with these problems between deterministic and nondeterministic versions, and the same holds for the emptiness problem. As far as we know, the model we introduce above is the first one-way model to exhibit properties (1) and (2). We define another family of one-way acceptors where the nondeterministic version has an undecidable emptiness problem, but the deterministic version has a decidable emptiness problem. We also know of no other model with this property in the literature. We also investigate decidability properties of other variations of checking stack automata (e.g., allowing multiple stacks, two-way input, etc.). Surprisingly, two-way deterministic machines with multiple checking stacks and multiple reversal-bounded counters are shown to have a decidable membership problem, a very general model with this property

On Store Languages of Language Acceptors

It is well known that the "store language" of every pushdown automaton -- the set of store configurations (state and stack contents) that can appear as an intermediate step in accepting computations -- is a regular language. Here many models of language acceptors with various data structures are examined, along with a study of their store languages. For each model, an attempt is made to find the simplest model that accepts their store languages. Some connections between store languages of one-way and two-way machines generally are demonstrated, as with connections between nondeterministic and deterministic machines. A nice application of these store language results is also presented, showing a general technique for proving families accepted by many deterministic models are closed under right quotient with regular languages, resolving some open questions (and significantly simplifying proofs for others that are known) in the literature. Lower bounds on the space complexity for recognizing store languages for the languages to be non-regular are obtained.Comment: 19 pages, preprint to be submitted to a journa

Deletion Operations on Deterministic Families of Automata

Many different deletion operations are investigated applied to languages accepted by one-way and two-way deterministic reversal-bounded multicounter machines, deterministic pushdown automata, and finite automata. Operations studied include the prefix, suffix, infix and outfix operations, as well as left and right quotient with languages from different families. It is often expected that language families defined from deterministic machines will not be closed under deletion operations. However, here, it is shown that one-way deterministic reversal-bounded multicounter languages are closed under right quotient with languages from many different language families; even those defined by nondeterministic machines such as the context-free languages. Also, it is shown that when starting with one-way deterministic machines with one counter that makes only one reversal, taking the left quotient with languages from many different language families -- again including those defined by nondeterministic machines such as the context-free languages -- yields only one-way deterministic reversal-bounded multicounter languages (by increasing the number of counters). However, if there are two more reversals on the counter, or a second 1-reversal-bounded counter, taking the left quotient (or even just the suffix operation) yields languages that can neither be accepted by deterministic reversal-bounded multicounter machines, nor by 2-way nondeterministic machines with one reversal-bounded counter.Comment: 20 pages, accepted version to Information and Computatio

On the Complexity and Decidability of Some Problems Involving Shuffle

The complexity and decidability of various decision problems involving the shuffle operation are studied. The following three problems are all shown to be $NP$-complete: given a nondeterministic finite automaton (NFA) $M$, and two words $u$ and $v$, is $L(M)$ not a subset of $u$ shuffled with $v$, is $u$ shuffled with $v$ not a subset of $L(M)$, and is $L(M)$ not equal to $u$ shuffled with $v$? It is also shown that there is a polynomial-time algorithm to determine, for $NFA$s $M_1, M_2$ and a deterministic pushdown automaton $M_3$, whether $L(M_1)$ shuffled with $L(M_2)$ is a subset of $L(M_3)$. The same is true when $M_1, M_2,M_3$ are one-way nondeterministic $l$-reversal-bounded $k$-counter machines, with $M_3$ being deterministic. Other decidability and complexity results are presented for testing whether given languages $L_1, L_2$ and $R$ from various languages families satisfy $L_1$ shuffled with $L_2$ is a subset of $R$, and $R$ is a subset of $L_1$ shuffled with $L_2$. Several closure results on shuffle are also shown.Comment: Preprint submitted to Information and Computatio

On counting functions and slenderness of languages

We study counting-regular languages -- these are languages $L$ for which there is a regular language $L'$ such that the number of strings of length $n$ in $L$ and $L'$ are the same for all $n$. We show that the languages accepted by unambiguous nondeterministic Turing machines with a one-way read-only input tape and a reversal-bounded worktape are counting-regular. Many one-way acceptors are a special case of this model, such as reversal-bounded deterministic pushdown automata, reversal-bounded deterministic queue automata, and many others, and therefore all languages accepted by these models are counting-regular. This result is the best possible in the sense that the claim does not hold for either $2$-ambiguous PDA's, unambiguous PDA's with no reversal-bound, and other models. We also study closure properties of counting-regular languages, and we study decidability problems in regards to counting-regularity. For example, it is shown that the counting-regularity of even some restricted subclasses of PDA's is undecidable. Lastly, $k$-slender languages -- where there are at most $k$ words of any length -- are also studied. Amongst other results, it is shown that it is decidable whether a language in any semilinear full trio is $k$-slender

Insertion Operations on Deterministic Reversal-Bounded Counter Machines

Several insertion operations are studied applied to languages accepted by one-way and two-way deterministic reversal-bounded multicounter machines. These operations are defined by the ideals obtained from relations such as the prefix, infix, suffix, and outfix relations, as well as operations defined from inverses of a type of deterministic transducer with reversal-bounded counters attached. The question of whether the resulting languages can always be accepted by deterministic machines with the same number (or larger number) of input-turns (resp., counters, counter-reversals, etc.) is investigated

On the Density of Context-Free and Counter Languages

A language $L$ is said to be dense if every word in the universe is an infix of some word in $L$. This notion has been generalized from the infix operation to arbitrary word operations $\varrho$ in place of the infix operation ($\varrho$-dense, with infix-dense being the standard notion of dense). It is shown here that it is decidable, for a language $L$ accepted by a one-way nondeterministic reversal-bounded pushdown automaton, whether $L$ is infix-dense. However, it becomes undecidable for both deterministic pushdown automata (with no reversal-bound), and for nondeterministic one-counter automata. When examining suffix-density, it is undecidable for more restricted families such as deterministic one-counter automata that make three reversals on the counter, but it is decidable with less reversals. Other decidability results are also presented on dense languages, and contrasted with a marked version called $\varrho$-marked-density. Also, new languages are demonstrated to be outside various deterministic language families after applying different deletion operations from smaller families. Lastly, bounded-dense languages are defined and examined

An low-cost spectrum analyzer for trouble shooting noise sources in scanning probe microscopy

Scanning probe microscopes are notoriously sensitive to many types of external and internal interference including electrical, mechanical and acoustic noise. Sometimes noise can even be misinterpreted as real features in the images. Therefore, quantification of the frequency and magnitude of any noise is key to discovering the source and eliminating it from the system. While commercial spectrum analyzers are perfect for this task, they are rather expensive and not always available. We present a simple, cost effective solution in the form of an audio output from the instrument coupled to a smart phone spectrum analyzer application. Specifically, the scanning probe signal, e.g. the tunneling current of a scanning tunneling microscope is fed to the spectrum analyzer which Fourier transforms the time domain acoustic signal into the frequency domain. When the scanning probe is in contact with the sample, but not scanning, the output is a spectrum containing both the amplitude and frequency of any periodic noise affecting the microscope itself, enabling troubleshooting to begin.Comment: 5 page

On Store Languages and Applications

The store language of a machine of some arbitrary type is the set of all store configurations (state plus store contents but not the input) that can appear in an accepting computation. New algorithms and characterizations of store languages are obtained, such as the result that any nondeterministic pushdown automaton augmented with reversal-bounded counters, where the pushdown can "flip" its contents up to a bounded number of times, can be accepted by a machine with only reversal-bounded counters. Then, connections are made between store languages and several model checking and reachability problems, such as accepting the set of all predecessor and successor configurations from a given set of configurations, and determining whether there are at least one, or infinitely many, common configurations between accepting computations of two machines. These are explored for a variety of different machine models often containing multiple parallel data stores. Many of the machine models studied can accept the set of predecessor configurations (of a regular set of configurations), the set of successor configurations, and the set of common configurations between two machines, with a machine model that is simpler than itself, with a decidable emptiness, infiniteness, and disjointness property. Store languages are key to showing these properties