Recursively Enumerable Languages Research Articles

Classes of two-dimensional languages and recognizability conditions

The paper deals with some classes of two-dimensional recognizable languages of “high complexity”, in a sense specified in the paper and motivated by some necessary conditions holding for recognizable and unambiguous languages. For such classes we can solve some open questions related to unambiguity, finite ambiguity and complementation. Then we reformulate a necessary condition for recognizability stated by Matz, introducing a new complexity function. We solve an open question proposed by Matz, showing that all the known necessary conditions for recognizability of a language and its complement are not sufficient. The proof relies on a family of languages defined by functions.

Automata for Epistemic Temporal Logic with Synchronous Communication

We suggest that developing automata theoretic foundations is relevant for knowledge theory, so that we study not only what is known by agents, but also the mechanisms by which such knowledge is arrived at. We define a class of epistemic automata, in which agents' local states are annotated with abstract knowledge assertions about others. These are finite state agents who communicate synchronously with each other and information exchange is `perfect'. We show that the class of recognizable languages has good closure properties, leading to a Kleene-type theorem using what we call regular knowledge expressions. These automata model distributed causal knowledge in the following way: each agent in the system has a partial knowledge of the temporal evolution of the system, and every time agents synchronize, they update each other's knowledge, resulting in a more up-to-date view of the system state. Hence we show that these automata can be used to solve the satisfiability problem for a natural epistemic temporal logic for local properties. Finally, we characterize the class of languages recognized by epistemic automata as the regular consistent languages studied in concurrency theory.

Factorization forests for infinite words and applications to countable scattered linear orderings

The theorem of factorization forests of Imre Simon shows the existence of nested factorizations–à la Ramsey–for finite words. This theorem has important applications in semigroup theory, and beyond. We provide two improvements to the standard result. First we improve on all previously known bounds. Second, we extend it to ‘every linear ordering’. We use this last variant in a simplified proof of the translation of recognizable languages over countable scattered linear orderings to languages accepted by automata.

Trace monoids with idempotent generators and measure-only quantum automata

In this paper, we analyze a model of 1-way quantum automaton where only measurements are allowed ( MON -1qfa). The automaton works on a compatibility alphabet $$(\Sigma, E)$$ of observables and its probabilistic behavior is a formal series on the free partially commutative monoid $$\hbox{FI}(\Sigma, E)$$ with idempotent generators. We prove some properties of this class of formal series and we apply the results to analyze the class $${\bf LMO}(\Sigma, E)$$ of languages recognized by MON -1qfa's with isolated cut point. In particular, we prove that $${\bf LMO}(\Sigma, E)$$ is a boolean algebra of recognizable languages with finite variation, and that $${\bf LMO}(\Sigma, E)$$ is properly contained in the recognizable languages, with the exception of the trivial case of complete commutativity.

On the Tiling System Recognizability of Various Classes of Convex Polyominoes

We consider some problems concerning two relevant classes of two-dimensional languages, i.e., the tiling recognizable languages, and the local languages, recently introduced by Giammarresi and Restivo and already extensively studied. We show that various classes of convex and column-convex polyominoes can be naturally represented as two-dimensional words of tiling recognizable languages. Moreover, we investigate the nature of the generating function of a tiling recognizable language, providing evidence that such a generating function need not be D-finite.

Who Are You What Are You Why?

The two of us created “Who Are You What Are You Why?” together, as an experiment predicated on Wittgenstein’s statement: “Do not forget that a poem, even though it is composed in the language of information, is not used in the language-game of giving information.” We began this experiment by separately composing a series of non-contiguous questions and then “mashing” them together to form a virtual dialogue that provides only questions, never answers. What are the effects of composing using recognizable language in unrecognizable configurations? Where is the demarcation between sense and nonsense? How can we form meaning from seeming meaninglessness? These are a few of the questions we hope to raise with this experiment.

Highly Undecidable Problems about Recognizability by Tiling Systems

Altenbernd, Thomas and Wohrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with usual acceptance conditions, such as the Buchi andMuller ones, in [1]. It was proved in [9] that it is undecidable whether a Buchirecognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). We show here that these two decision problems are actually P$^{1}_{2}$-complete, hence located at the second level of the analytical hierarchy, and highly undecidable. We give the exact degree of numerous other undecidable problems for Buchi-recognizable languages of infinite pictures. In particular, the nonemptiness and the infiniteness problems are Σ$^{1}_{1}$-complete, and the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, are all P$^{1}_{2}$-complete. It is also P$^{1}_{2}$-complete to determine whether a given Buchi recognizable language of infinite pictures can be accepted row by row using an automaton model over ordinal words of length ω$^{2}$.

Deterministic and unambiguous two-dimensional languages over one-letter alphabet

The paper focuses on deterministic and unambiguous recognizable two-dimensional languages with particular attention to the case of a one-letter alphabet. The family DREC(1) of deterministic languages over a one-letter alphabet is characterized as both L (DOTA)(1), the class of languages accepted by deterministic on-line tessellation acceptors, and L (2AFA)(1), the class of languages recognized by 2-way alternating finite automata. We show that there are inherently ambiguous languages and unambiguously recognizable languages that cannot be deterministically recognized even in the case of a one-letter alphabet. In particular we show that on-line tessellation acceptors are more powerful than their deterministic counterpart, even in the case of a one-letter alphabet. Finally we show that DREC(1) is complex enough not to be characterized in terms of classical operations.

Descriptive Complexity Theories

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Coalgebraic Automata Theory: Basic Results

We generalize some of the central results in automata theory to the abstraction level of coalgebras and thus lay out the foundations of a universal theory of automata operating on infinite objects. Let F be any set functor that preserves weak pullbacks. We show that the class of recognizable languages of F-coalgebras is closed under taking unions, intersections, and projections. We also prove that if a nondeterministic F-automaton accepts some coalgebra it accepts a finite one of the size of the automaton. Our main technical result concerns an explicit construction which transforms a given alternating F-automaton into an equivalent nondeterministic one, whose size is exponentially bound by the size of the original automaton.

Rational subsets of partially reversible monoids

A class of monoids that can model partial reversibility allowing simultaneously instances of two-sided reversibility, one-sided reversibility and no reversibility is considered. Some of the basic decidability problems involving their rational subsets, syntactic congruences and characterization of recognizability, are solved using purely automata-theoretic techniques, giving further insight into the structure of recognizable languages.

Confusion of memory

It is a truism that for a machine to have a useful access to memory or workspace, it must “know” where its input ends and its working memory begins. Most machine models separate input from memory explicitly, in one way or another. We are interested here in computational models which do not separate input from working memory. We study the situation on deterministic single-queue machines working on a two symbol alphabet. We establish a negative result about such machines: they cannot compute the length of their input. This confirms the intuition that such machines are “unable to tell” where on the queue the input ends and the memory begins. On the positive side, we note that there are some interesting things that one can do with such queue machines: their halting problem is undecidable, there are self-replicating machines, and there are recognizable languages outside of the control hierarchy.

CLASSES OF TREE LANGUAGES DETERMINED BY CLASSES OF MONOIDS

In this paper finite state recognizers are considered as unary tree recognizers with unary operational symbols. We introduce translation recognizers of a tree recognizer, which are finite state recognizers whose operations are the elementary translations of the underlying algebra of the considered tree recognizer. In terms of translation recognizers we give general conditions under which a class of recognizable tree languages with a given property can be determined by a class of monoids determining the class of string languages having the same property.

Unambiguous recognizable two-dimensional languages

We consider the family UREC of unambiguous recognizable two-dimensional languages. We prove that there are recognizable languages that are inherently ambiguous, that is UREC family is a proper subclass of REC family. The result is obtained by showing a necessary condition for unambiguous recognizable languages. Further UREC family coincides with the class of picture languages defined by unambiguous 2OTA and it strictly contains its deterministic counterpart. Some closure and non-closure properties of UREC are presented. Finally we show that it is undecidable whether a given tiling system is unambiguous.

TRANSITIVITY IN TWO-DIMENSIONAL LOCAL LANGUAGES DEFINED BY DOT SYSTEMS

The paper investigates two-dimensional recognizable languages that are defined by the so-called "dot systems" that are special subgroups of (ℤ/2ℤ)ℤ2. The dot shapes that provide directional transitivity or mixing for the related language are investigated. It is shown that languages defined by parallelogram shapes fail to be transitive in the direction of a defining vector and hence fail to be mixing, while certain triangular shapes guarantee that the factor language of the associated dot system will be mixing. Dot systems belong to a class of two-dimensional shift spaces that have a factor language such that every admissable block can be extended to a configuration of the entire plane. For this class of shift spaces we introduce a finite graph (i.e., a finite state automaton) that recognizes two-dimensional local languages, then show that certain transitivity properties may be observed from the structure of the finite graph.

Recognizability of graph and pattern languages

Graph language recognizability is defined and investigated by virtue of the syntactic magmoid, analogously with the syntactic monoid of a word language. In this setup, the syntax complexity of a given recognizable graph language can be determined, giving rise to a syntactic classification inside the class of recognizable graph languages.

A Burnside Approach to the Finite Substitution Problem

We introduce a new model of weighted automata: the desert automata. We show that their limitedness problem is PSPACE-complete by solving the underlying Burnside problem. As an application of this result, we give a positive solution to the so-called finite substitution problem which was open for more than 10 years: given recognizable languages K and L, decide whether there exists a finite substitution σ such that σ(K) = L.

Minimizing of the only-insertion insdel systems

A more recent branch of natural computing is DNA computing. At the theoretical level, DNA computing is powerful. This is due to the fact that DNA structure and processing suggest a series of new data structures and operations, and to the fact of the massive parallelism. The insertion-deletion system (insdel system) is a DNA computing model based on two genetic operations: insertion and deletion which, working together, are very powerful, leading to characterizations of recursively enumerable languages. When designing an insdel computer, it is natural to try to keep the underlying model as simple as possible. One idea is to use either only insertion operations or only deletion operations. By helping with a weak coding and a morphism, the family INS 4 7 DEL 0 0 is equal to the family ofrecursively enumerable languages. It is an open problem proposed by Martin-Videet al. on whether or not the parameters 4 and 7 appearing here can be replaced by smaller numbers. In this paper, our positive answer to this question is that INS 2 4 DEL 0 0 can also play the same role as insertion and deletion. We suppose that the 1NS 2 4 DEL 0 0 may be the least only-insertion insdel system in this situation. We will give some reasons supporting this conjecture in our paper.

Traces of a Freed Language: Horace, Petronius, and the Rhetoric of Fable

Abstract This paper investigates the status that the genre of fable acquires when it is employed in literature. In particular, it surveys Horace's treatment of fables in the Satires andEpistles and the carefully controlled circumstances in which zoomorphic language is allowed to emerge during the banquet at Trimalchio's in Petronius' Satyrica. The analysis of the distribution of fables in Horace shows that for the Roman literary public the act of speaking through fables bore in itself a negative connotation, so much that the moral discourse of the satirist needed at first to provide additional justification in order to incorporate them: from vindication of ingenuitas and shifts in narrative voice, to use of rhetorical misnomers and eventually of philosophical frankness. Petronius' text, in turn, suggests that what is wrong with fable is precisely its being reminiscent of a servile past. During the dinner at Trimalchio's allusions to recognizable fable plots and zoomorphic language are allowed to surface only during a momentary absence of the host. This circumstance suggests that fable is not just another literary genre among the many genres abused in Trimalchio's house: for both the host and his freedmen guests fables are an uncomfortable reminder of an enduring past inscribed in their language.

Collage of two-dimensional words

We consider a new operation on one-dimensional (resp. two-dimensional) word languages, obtained by piling up, one on top of the other, words of a given recognizable language (resp. two-dimensional recognizable language) on a previously empty one-dimensional (resp. two-dimensional) array. The resulting language is the set of words “seen from above”: a position in the array is labeled by the topmost letter. We show that in the one-dimensional case, the language is always recognizable. This is no longer true in the two-dimensional case which is shown by a counter-example, and we investigate in which particular cases the result may still hold.