Unambiguous Languages Research Articles

Non-closure under complementation for unambiguous linear grammars

The paper demonstrates the non-closure of the family of unambiguous linear languages (that is, those defined by unambiguous linear context-free grammars) under complementation. To be precise, a particular unambiguous linear grammar is presented, and it is proved that the complement of this language is not defined by any context-free grammar. This also constitutes an alternative proof for the result of Hibbard and Ullian (“The independence of inherent ambiguity from complementedness among context-free languages”, JACM, 1966) on the non-closure of the unambiguous languages under complementation.

A Unifying Approach to Algebraic Systems Over Semirings

A fairly general definition of canonical solutions to algebraic systems over semirings is proposed. This is based on the notion of summation semirings, traditionally known as ${\Sigma }$ -semirings, and on assigning unambiguous context-free languages to variables of each system. The presented definition applies to all algebraic systems over continuous or complete semirings and to all proper algebraic systems over power series semirings, for which it coincides with the usual definitions of their canonical solutions. As such, it unifies the approaches to algebraic systems over semirings studied in literature. An equally general approach is adopted to study pushdown automata, for which equivalence with algebraic systems is proved. Finally, the Chomsky-Schutzenberger theorem is generalised to the setting of summation semirings.

The Algebraic Theory of Parikh Automata

The Parikh automaton model equips a finite automaton with integer registers and imposes a semilinear constraint on the set of their final settings. Here the theories of typed monoids and of rational series are used to characterize the language classes that arise algebraically. Complexity bounds are derived, such as containment of the unambiguous Parikh automata languages in NC1. Affine Parikh automata, where each transition applies an affine transformation on the registers, are also considered. Relying on these characterizations, the landscape of relationships and closure properties of the classes at hand is completed, in particular over unary languages.

Sofic-Dyck shifts

We define the class of sofic-Dyck shifts which extends the class of Markov–Dyck shifts introduced by Inoue, Krieger and Matsumoto. Sofic-Dyck shifts are shifts of sequences whose finite factors form unambiguous context-free languages. We show that they correspond exactly to the class of shifts of sequences whose sets of factors are visibly pushdown languages. We give an expression of the zeta function of a sofic-Dyck shift.

HOMOMORPHISMS PRESERVING DETERMINISTIC CONTEXT-FREE LANGUAGES

The paper characterizes the family of homomorphisms, under which the deterministic context-free languages, the LL context-free languages and the unambiguous context-free languages are closed. The family of deterministic context-free languages is closed under a homomorphism h if and only if h is either a code of bounded deciphering delay, or the images of all symbols under h are powers of the same string. The same characterization holds for LL context-free languages. The unambiguous context-free languages are closed under h if and only if either h is a code, or the images of all symbols under h are powers of the same string.

Unambiguous Tree Languages Are Topologically Harder Than Deterministic Ones

The paper gives an example of a tree language G that is recognised by an unambiguous parity automaton and is analytic-complete as a set in Cantor space. This already shows that the unambiguous languages are topologically more complex than the deterministic ones, that are all coanalytic. Using set G as a building block we construct an unambiguous language that is topologically harder than any countable boolean combination of analytic and coanalytic sets. In particular the language is harder than any set in difference hierarchy of analytic sets considered by O.Finkel and P.Simonnet in the context of nondeterministic automata.

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.

Biologie des systèmes appliqués aux cancers

Cancer Systems Biology is now accepted and recognized as a promising field both in biological and clinical research. It relies on a rigorous formalization of regulation networks into precise and unambiguous languages. It provides both detailed and modular views of the complex biological system of interest (which in cancer research is typically an interaction network governing essential cellular events such as proliferation, differentiation, cell death...) in order to facilitate the interpretation of molecular profiles of tumors. The translation of these networks into mathematical models allows prediction of the evolution of the system in time and under certain perturbations. As a result, it can not only propose specific target points for pharmaceutical purposes, but also anticipate the evolution of tumors as well as their classifications. These characteristics emphasize the important role of Systems Biology of Cancer in the future of biomedical research.

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.

THE BENFORD-NEWCOMB DISTRIBUTION AND UNAMBIGUOUS CONTEXT-FREE LANGUAGES

For a string w ∈ {0,1, 2,…, d-1}*, let vald(w) denote the integer whose base d representation is the string w and let MSDd(x) denote the most significant or the leading digit of a positive integer x when x is written as a base d integer. Hirvensalo and Karhumäki [9] studied the problem of computing the leading digit in the ternary representation of 2x ans showed that this problem has a polynomial time algorithm. In [16], some applications are presented for the problem of computing the leading digit of AB for given inputs A and B. In this paper, we study this problem from a formal language point of view. Formally, we consider the language Lb,d,j = {w|w ∈ {0,1, 2,…, d-1}*, 1 ≤ j ≤ 9, MSDb(dvalb(w))) = j} (and some related classes of languages) and address the question of whether this and some related languages are context-free. Standard pumping lemma proofs seem to be very difficult for these languages. We present a unified and simple combinatorial technique that shows that these languages are not unambiguous context-free languages. The Benford-Newcomb distribution plays a central role in our proofs.

Lower bound technique for length-reducing automata

The class of growing context-sensitive languages (GCSL) was proposed as a naturally defined subclass of context-sensitive languages whose membership problem is solvable in polynomial time. Growing context-sensitive languages and their deterministic counterpart called Church–Rosser languages (CRL) complement the Chomsky hierarchy in a natural way, as the classes filling the gap between context-free languages and context-sensitive languages. They possess characterizations by a natural machine model, length-reducing two-pushdown automata (lrTPDA). We introduce a lower bound technique for lrTPDAs. Using this technique, we prove the conjecture of McNaughton, Narendran and Otto that the set of palindromes is not in CRL. As a consequence we obtain that CFL∩coCFL as well as UCFL∩coUCFL are not included in CRL, where UCFL denotes the class of unambiguous context-free languages; this solves an open problem posed by Beaudry, Holzer, Niemann and Otto. Another corollary is that CRL is a strict subset of GCSL∩coGCSL.

Growth and ergodicity of context-free languages II: The linear case

A language L over a finite alphabet E is called growth-sensitive if forbidding any non-empty set F of subwords yields a sub-language L F whose exponential growth rate is smaller than that of L. Say that a context-free grammar (and associated language) is ergodic if its dependency di-graph is strongly connected. It is known that regular and unambiguous non-linear context-free languages which are ergodic are growth-sensitive. In this note it is shown that ergodic unambiguous linear languags are growth-sensitive, closing the gap that remained open.

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.

On the growth of linear languages

It is well known that the growth of a context-free language is either polynomial or exponential. However no algorithm for such an alternative is known. In this article we determine such an algorithm for the subclass of unambiguous linear languages.

Unambiguous Auxiliary Pushdown Automata and Semi-unbounded Fan-in Circuits

Notions of unambiguity for uniform circuits and AuxPDA′s are studied and related to each other. In particular, a coincidence for counting and unambiguous versions of AuxPDA′s and semi-unbounded fan-in circuits is shown. Moreover, an improved simulation of LOGUCFL (the class of languages logspace many-one reducible to unambiguous context-free languages) by unambiguous circuits and AuxPDA′s is developed. Next, an inductive counting technique on semi-unbounded fan-in circuits is presented and employed for several applications, especially an alternative proof for the closure under complementation of LOGCFL. A cost-free simulation of polynomially ambiguity bounded AuxPDA′s by unambiguous ones is given. A first nontrivial upper bound for a circuit class defined by Lange and its closure under complementation are indicated. Finally, a normal form for AuxPDA′s is investigated. Inter alia, it is shown that for unambiguous AuxPDA′s operating in polynomial time and logarithmic space a pushdown height of O(log2n) suffices, thus paralleling results for deterministic and nondeterministic AuxPDA′s. It is pointed out that without loss of generality the underlying machines of the most important AuxPDA classes work obliviously.

ON THE STAR HEIGHT OF RATIONAL LANGUAGES

Two problems concerning the star height of a rational language are investigated: the star height one problem and the relationships between the unambiguity of an expression and its star height. For this purpose we consider the class of factorial, transitive and rational (FTR) languages. From the algebraic point of view a FTR language is the set of factors of a rational submonoid M. Two subclasses of FTR languages are introduced: renewal languages, corresponding to the case of M finitely generated, and unambiguous renewal languages, corresponding to the case of M finitely generated and free. We prove that a FTR language has star height one if and only if it is renewal. This gives a simple decision procedure for the star height one problem for this class of languages. As concerns the relationships with the ambiguity, we introduce the notion of unambiguous star height of a rational language and prove that it can be strictly greater than the star height. This is a consequence of the following results: i) a FTR language of unambiguous star height one is unambiguous renewal; ii) there exist renewal languages which are not unambiguous renewal. This last result is obtained by using arguments involving the entropy of a language.

On the Existence of Minimum Asynchronous Automata and on the Equivalence Problem for Unambiguous Regular Trace Languages

Given a recognizable trace language T there always exists a minimum (up to isomorphism) monoid automation which recognizes T . In this paper we consider finite state asynchronous automata, introduced by W. Zielonka ("Notes on Finite Asynchronous Automata," Technical Report, Institute of Mathematics, Warsaw Technical University, 1984) and we show that this result cannot be extended to this model. More precisely, we exhibit a recognizable trace language which does not admit a unique minimum asynchronous automaton. This result solves an open problem posed by Zielonka. We further prove that every recognizable trace language defined on a concurrent alphabet with a transitive dependency relation admits a unique minimum asynchronous automation. In the second part of this paper we investigate the equivalence problem for regular trace languages. This problem has been investigated by many authors: in particular, Aalbersberg and Hoogeboom (1987) have shown that the equivalence problem for regular trace languages defined over a concurrent alphabet 〈 Σ , C 〉 is decidable if and only if the concurrency relation C is transitive. We show that this result does not hold when we restrict our attention to unambiguous regular trace languages we exhibit a concurrent alphabet 〈 Σ ′, C ′〉 with a non transitive concurrency relation such that the equivalence problem for unambiguous regular trace languages on 〈 Σ ′, C ′〉 can be decided in polynomial time. Further we show that the problem of distinguishing whether or not a regular expression on a concurrent alphabet 〈 Σ , C 〉 defines an unambiguous regular trace language is undecidable .

Observations on log ( n) time parallel recognition of unambiguous cfl's

We present two simpler versions of the log( n) time parallel recognition of unambiguous context-free languages on CREW PRAM's. The first algorithm combines the ideas of the algorithms given in [3] and [7]. The result related to that algorithm is an unambiguous decomposition of trees into smaller ones. The analysis of this algorithm is quite simple. The second algorithm is based on unique paths decomposing trees into smaller subtrees. It has a shorter description and it is simpler to explain how it works. However, its correctness is nontrivial. Both algorithms reduce the numbers of processors by a linear factor in the case of general unambiguous cfl's. For linear unambiguous cfl's the number of processors is reduced to O( n 2) by a very simple algorithm. We show how to remove write conflicts in parallel computations on trees by using two kinds of unique decompositions of trees. Finally we show that our results can be generalized for finitely ambiguous cfl's.

The complexity of computing the number of strings of given length in context-free languages

Computing the number of strings of given length contained in a language is related to classical problems of combinatorics, formal languages and computational complexity. Here we study the complexity of this problem in the case of context-free languages. It is shown that, for unambiguous context-free languages such a computation is “easy” and can be carried out by efficient parallel algorithms. On the contrary, for some context-free languages of ambiguity degree two, the problem becomes intractable. These results are related to other classical subjects concerning counting problems, exponential time recognizable languages and sparse sets.

Compression and Ranking

A complexity-theoretic approach to the classical data compression problem is presented. A notion of language compressibility is defined, and it is shown that essentially all strings in a sufficiently sparse “easy” (e.g., polynomial-time) language can be compressed efficiently. A notion of ranking as a form of optimal compression is also defined, and it is shown that some “very easy” languages (e.g., unambiguous context-free languages) can be ranked efficiently. Languages that cannot be compressed or ranked efficiently under various complexity-theoretic assumptions are exhibited.The notion of compressibility is closely related to Kolmogorov complexity and randomness. This relationship and the complexity-theoretic implications of our results are discussed.