Context-free Research Articles

On restarting automata with auxiliary symbols and small window size

Here we show that for monotone RWW- (and RRWW-) automata, window size two is sufficient, both in the nondeterministic as well as in the deterministic case. For the former case, this is done by proving that each context-free language is already accepted by a monotone RWW-automaton of window size two. In the deterministic case, we first prove that each deterministic pushdown automaton can be simulated by a deterministic monotone RWW-automaton of window size three, and then we present a construction that transforms a deterministic monotone RWW-automaton of window size three into an equivalent automaton of the same type that has window size two. Furthermore, we study the expressive power of shrinking RWW- and RRWW-automata the window size of which is just one or two. We show that for shrinking RRWW-automata that are nondeterministic, window size one suffices, while for nondeterministic shrinking RWW-automata, we already need window size two to accept all growing context-sensitive languages. In the deterministic case, shrinking RWW- and RRWW-automata of window size one accept only regular languages, while those of window size two characterize the Church-Rosser languages.

House of Quality-Based Analysis of New Service Development Using Context Free Grammar Evaluation-Enhanced Fuzzy Hybrid Modelling

The purpose of this study is to analyze new service development competencies based on house of quality (HoQ) in the healthcare industry by using a hybrid interval-valued intuitionistic decision-making approach with fuzzy linguistic information modelled by hesitant fuzzy linguistic term sets (HFLTSs) and linguistic 2-tuples. The novelty of the manuscript is to identify a list of criteria, dimensions, and alternatives for HoQ-based new service development used for the investment decisions in healthcare industry. Thus, a hybrid model with HFLTSs has been proposed, considering 2-tuple linguistic information, Interval-Valued Intuitionistic Fuzzy Sets (IVIFSs) and a defuzzification process with an attitudinal expected score function. Hence, a DEMATEL extension dealing with HFLTSs, linguistic 2-tuple and IVIFS has been developed and used to weigh the criteria and dimensions of customer needs. Similarly, a TOPSIS extension dealing with the same type of information is introduced and applied in order to rank the technical requirements, the so- called “new service development competencies in healthcare industry”. Thus, this study contributes to the literature of health economics and management by proposing an original methodology. It is identified that financial expectations play the most significant role for demanders of healthcare services. Hence, competitive pricing strategies should be generated to obtain consumer loyalty in the healthcare industry. Moreover, technological infrastructure is found as the best ranked technical requirement of new service development competencies.

Weak Greibach Normal Form for Hyperedge Replacement Grammars

It is known that hyperedge replacement grammars are similar to string context-free grammars in the sense of definitions and properties. Therefore, we expect that there is a generalization of the well-known Greibach normal form from string grammars to hypergraph grammars. Such generalized normal forms are presented in several papers; however, they do not cover a large class of hypergraph languages (e.g. languages consisting of star graphs). In this paper, we introduce a weak Greibach normal form, whose definition corresponds to the lexicalized normal form for string grammars, and prove that every context-free hypergraph language (with nonsubstantial exceptions) can be generated by a grammar in this normal form. The proof presented in this paper generalizes a corresponding one for string grammars with a few more technicalities.

Boosting Reversible Pushdown and Queue Machines by Preprocessing

It is well known that reversible finite automata do not accept all regular languages, that reversible pushdown automata do not accept all deterministic context-free languages, and that reversible queue automata are less powerful than deterministic real-time queue automata. It is of significant interest from both a practical and theoretical point of view to close these gaps. We here extend these reversible models by a preprocessing unit which is basically a reversible injective and length-preserving finite state transducer. It turns out that preprocessing the input using such weak devices increases the computational power of reversible deterministic finite automata to the acceptance of all regular languages, whereas for reversible pushdown automata the accepted family of languages lies strictly in between the reversible deterministic context-free languages and the real-time deterministic context-free languages. For reversible queue automata the preprocessing of the input leads to machines that are stronger than real-time reversible queue automata, but less powerful than real-time deterministic (irreversible) queue automata. Moreover, it is shown that the computational power of all three types of machines is not changed by allowing the preprocessing finite state transducer to work irreversibly. Finally, we examine the closure properties of the family of languages accepted by such machines.

Logic for ω-pushdown automata

Context-free languages of infinite words have recently found increasing interest. Here, we will present a second-order logic with the same expressive power as Büchi or Muller pushdown automata for infinite words. This extends fundamental logical characterizations of Büchi, Elgot, Trakhtenbrot for regular languages of finite and infinite words and a more recent logical characterization of Lautemann, Schwentick and Thérien for context-free languages of finite words to ω-context-free languages. For our argument, we will investigate Greibach normal forms of ω-context-free grammars as well as a new type of Büchi pushdown automata which can alter their stack by at most one element and without ϵ-transitions. We show that they suffice to accept all ω-context-free languages. This enables us to use similar results recently developed for infinite nested words.

On the computing powers of [formula omitted]-reductions of insertion languages

We investigate the computing power of the following language operation %: Given two languages L1 over Σ and L2 over Γ with Γ⊂Σ, we consider the language operation L1%L2={u0u1⋯un|∃u=u0v1u1⋯vnun∈L1 and ∃vi∈L2(1≤∀i≤n)}. In this case we say that L(=L1%L2) is the L2-reduction of L1. This is extended to the language families as follows: L1%L2={L1%L2|L1∈L1,L2∈L2}. Among many works concerning Dyck-reductions, for the family of recursively enumerable languages RE, it was shown that LIN%{EQ}=RE (Jantzen & Petersen, 1994) with EQ={xnx‾n|n∈N} and that min-LIN%{D2}=RE (Hirose & Okawa, 1996, and Latteux & Turakainen, 1990), where LIN and min-LIN are the families of linear and minimal linear context-free languages, respectively.In this paper, we show that each recursively enumerable language L can be represented in the form L=K%D, for some K∈INS30 and a Dyck language D, where INS⁎0 (INS30) denotes the family of insertion languages (insertion languages where the maximum length of the string to be inserted is 3). We can refine it as INS⁎0%{D2}=RE, where D2 denotes the Dyck language over binary alphabet. For context-free languages, we show that INS30%F=CF, where F is the family of finite sets. This also derives that INS⁎0%{MIR}=CF with MIR={xx‾R|x∈{0,1}⁎}. Further, for regular languages, it is shown that each regular language R can be represented in the form R=K%F, for some K∈INS20 and a finite set F={abb‾a‾|a∈V}. We also present some results which characterize the computability and properties of L in the framework of L2-reduction of L1.It is intriguing to note that, from the DNA computing point of view, the notion of L-reduction is naturally motivated by a molecular biological functioning well-known as DNA(RNA) splicing occurring in most eukaryotic genes.

FlowCFL: generalized type-based reachability analysis: graph reduction and equivalence of CFL-based and type-based reachability

Reachability analysis is a fundamental program analysis with a wide variety of applications. We present FlowCFL, a type-based reachability analysis that accounts for mutable heap data. The underlying semantics of FlowCFL is Context-Free-Language (CFL)-reachability. We make three contributions. First, we define a dynamic semantics that captures the notion of flow commonly used in reachability analysis. Second, we establish correctness of CFL-reachability over graphs with inverse edges (inverse edges are necessary for the handling of mutable heap data). Our approach combines CFL-reachability with reference immutability to avoid the addition of certain inverse edges, which results in graph reduction and precision improvement. The key contribution of our work is the formal account of correctness, which extends to the case when inverse edges are removed. Third, we present a type-based reachability analysis and establish equivalence between a certain CFL-reachability analysis and the type-based analysis, thus proving correctness of the type-based analysis.

The computational origin of representation.

Each of our theories of mental representation provides some insight into how the mind works. However, these insights often seem incompatible, as the debates between symbolic, dynamical, emergentist, sub-symbolic, and grounded approaches to cognition attest. Mental representations-whatever they are-must share many features with each of our theories of representation, and yet there are few hypotheses about how a synthesis could be possible. Here, I develop a theory of the underpinnings of symbolic cognition that shows how sub-symbolic dynamics may give rise to higher-level cognitive representations of structures, systems of knowledge, and algorithmic processes. This theory implements a version of conceptual role semantics by positing an internal universal representation language in which learners may create mental models to capture dynamics they observe in the world. The theory formalizes one account of how truly novel conceptual content may arise, allowing us to explain how even elementary logical and computational operations may be learned from a more primitive basis. I provide an implementation that learns to represent a variety of structures, including logic, number, kinship trees, regular languages, context-free languages, domains of theories like magnetism, dominance hierarchies, list structures, quantification, and computational primitives like repetition, reversal, and recursion. This account is based on simple discrete dynamical processes that could be implemented in a variety of different physical or biological systems. In particular, I describe how the required dynamics can be directly implemented in a connectionist framework. The resulting theory provides an "assembly language" for cognition, where high-level theories of symbolic computation can be implemented in simple dynamics that themselves could be encoded in biologically plausible systems.

CYK Parsing over Distributed Representations

Parsing is a key task in computer science, with applications in compilers, natural language processing, syntactic pattern matching, and formal language theory. With the recent development of deep learning techniques, several artificial intelligence applications, especially in natural language processing, have combined traditional parsing methods with neural networks to drive the search in the parsing space, resulting in hybrid architectures using both symbolic and distributed representations. In this article, we show that existing symbolic parsing algorithms for context-free languages can cross the border and be entirely formulated over distributed representations. To this end, we introduce a version of the traditional Cocke–Younger–Kasami (CYK) algorithm, called distributed (D)-CYK, which is entirely defined over distributed representations. D-CYK uses matrix multiplication on real number matrices of a size independent of the length of the input string. These operations are compatible with recurrent neural networks. Preliminary experiments show that D-CYK approximates the original CYK algorithm. By showing that CYK can be entirely performed on distributed representations, we open the way to the definition of recurrent layer neural networks that can process general context-free languages.

The range of non-linear natural polynomials cannot be context-free

Suppose that some polynomial $f$ with rational coefficients takes only natural values at natural numbers, i.e., $L=\{f(n)\mid n\in \mathbb N\}\subset\mathbb N$. We show that the base-$q$ representation of $L$ is a context-free language if and only if $f$ is linear, answering a question of Shallit. The proof is based on a new criterion for context-freeness, which is a combination of the Interchange lemma and a generalization of the Pumping lemma.

Operator precedence temporal logic and model checking

In the last decades much research effort has been devoted to extending the success of model checking from the traditional field of finite state machines and various versions of temporal logics to suitable subclasses of context-free languages and appropriate extensions of temporal logics. To the best of our knowledge such attempts only covered structured languages, i.e. languages whose structure is immediately “visible” in their sentences, such as tree-languages or visibly pushdown ones. In this paper we present a new temporal logic suitable to express and automatically verify properties of operator precedence languages. This “historical” language family has been recently proved to enjoy fundamental algebraic and logic properties that make it suitable for model checking applications yet breaking the barrier of visible-structure languages (in fact the original motivation of its inventor Floyd was just to support efficient parsing, i.e. building the “hidden syntax tree” of language sentences). We prove that our logic is at least as expressive as analogous logics defined for visible pushdown languages yet covering a much more powerful family; we design a procedure that, given a formula in our logic builds an automaton recognizing the sentences satisfying the formula, whose size is at most exponential in the length of the formula. Our results cover both finite and infinite string languages.

Decidability of Right One-Way Jumping Finite Automata

We continue our investigation [S. Beier, M. Holzer: Properties of right one-way jumping finite automata. In Proc. 20th DCFS, LNCS, 2018] on (right) one-way jumping finite automata (ROWJFAs), a variant of jumping automata, which is an automaton model for discontinuous information processing. Here we focus on decision problems for ROWJFAs. It turns out that most problems such as, e.g., emptiness, finiteness, universality, the word problem and variants thereof, closure under permutation, etc., are decidable. Moreover, we show that the containment of a language within the strict hierarchy of ROWJFA permutation closed languages induced by the number of accepting states as well as whether permutation closed regular or jumping finite automata languages can be accepted by ROWJFAs is decidable, too. On the other hand, we prove that for (linear) context-free languages the corresponding ROWJFA acceptance problem becomes undecidable. Moreover, we discuss also some complexity results for the considered decision problems.

On the Complexity of the Universality and Inclusion Problems for Unambiguous Context-Free Grammars

We study the computational complexity of universality and inclusion problems for unambiguous finite automata and context-free grammars. We observe that several such problems can be reduced to the universality problem for unambiguous context-free grammars. The latter problem has long been known to be decidable and we propose a PSPACE algorithm that works by reduction to the zeroness problem of recurrence equations with convolution. We are not aware of any non-trivial complexity lower bounds. However, we show that computing the coin-flip measure of an unambiguous context-free language, a quantitative generalisation of universality, is hard for the long-standing open problem SQRTSUM.

The Coxeter relations and KP map for non-commuting symbols

We give an action of the symmetric group on non-commuting indeterminates in terms of series in the corresponding Mal’cev–Newmann division ring. The action is constructed from the non-Abelian Hirota–Miwa (discrete KP) system. The corresponding companion map, which gives generators of the action, is discussed in the generic case, and the corresponding explicit formulas have been found in the periodic reduction. We discuss also briefly connection of the companion to the KP map with context-free languages.

COMPLEXITY OF THE INFINITARY LAMBEK CALCULUS WITH KLEENE STAR

AbstractWe consider the Lambek calculus, or noncommutative multiplicative intuitionistic linear logic, extended with iteration, or Kleene star, axiomatised by means of an$\omega $-rule, and prove that the derivability problem in this calculus is$\Pi _1^0$-hard. This solves a problem left open by Buszkowski (2007), who obtained the same complexity bound for infinitary action logic, which additionally includes additive conjunction and disjunction. As a by-product, we prove that any context-free language without the empty word can be generated by a Lambek grammar with unique type assignment, without Lambek’s nonemptiness restriction imposed (cf. Safiullin, 2007).

DEADLOCK ALGORITHM FOR ADVANCED SYNTACTICAL ANALYSIS AND ITS APPLICATION TO PROGRAMMING LANGUAGES FOR QUANTUM COMPUTERS

When developing promising programming languages designed to support the work of supercomputers, including quantum ones, there is a need for research related to testing the developed language under conditions when parsers do not yet exist for it. In particular, in the process of developing a programming language for a quantum computer, it becomes necessary to parse a certain program written in a new programming language, which, like all programming languages, belongs to the class of context-free languages (cf-languages). The problem of syntactical analysis of the monomials of cf-languages was posed in the 50-60s of the last century, however, there are some discrepancies in its formulation, and therefore there is a need to clarify the formulation of this problem. In this regard, we will call the expanded problem of parsing the problem of developing a stupid (non-stop, irrevocable) algorithm that allows establishing whether a given monomial can be deduced using a system of products that form a cf-language grammar, and also find all the conclusions of this monomial at once if the latter exists. The description of the monomial inference is understood as follows: it is necessary to determine for which products from the grammar of the cf-language, how many times and in what order they are used to derive this monomial, which is equivalent to constructing all the output trees. The article has developed a deadlockless algorithm for solving the extended problem of parsing, based on the method of hierarchy of marked brackets. The marked brackets order shows what products they are assigned to, and allows you to trace the order of its use. The algorithm uses the method of successive approximations to solve the Chomsky-Schützenberger system of equations associated with the cf-language grammar. The developed algorithm has a simple software implementation; an assessment of the complexity of the algorithm is also given.

Formula omitted] Watson-Crick pushdown automata

Watson-Crick automata work on two-stranded tape, and, consequently, they have two reading heads. In the case of sensing 5′→3′ Watson-Crick automata, the two heads start from different ends of the input and they move in opposite directions until they meet. In this paper, an extension of these automata is presented by adding a pushdown stack. This model can also be seen as a 2-head extension of the pushdown automata. Acceptance by empty stack and acceptance by final states are proven to have the same recognition power in this new model. The language family recognized by this model contains only semi-linear context-sensitive languages and it includes the class of context-free languages. Some well-known non-context-free languages are shown as examples. Normal form like results, a pumping lemma and various closure properties are also proven.

On the Membership Problem of Permutation Grammars — A Direct Proof of NP-Completeness

One of the most essential classes of problems related to formal languages is the membership problem (also called word problem), i.e., to decide whether a given input word belongs to the language specified, e.g., by a generative grammar. For context-free languages the problem is solved efficiently by various well-known parsing algorithms. However, there are several important languages that are not context-free. The membership problem of the context-sensitive language class is PSPACE-complete, thus, it is believed that it is generally not solvable in an efficient way. There are various language classes between the above mentioned two classes having membership problems with various complexity. One of these classes, the class of permutation languages, is generated by permutation grammars, i.e., context-free grammars extended with permutation rules, where a permutation rule allows to interchange the position of two consecutive nonterminals in the sentential form. In this paper, the membership problem for permutation languages is studied. A proof is presented to show that this problem is NP-complete.

LTL model checking for communicating concurrent programs

We present in this paper a new approach to the static analysis of concurrent programs with procedures. To this end, we model multi-threaded programs featuring recursive procedure calls and synchronization by rendezvous between parallel threads with communicating pushdown systems (from now on CPDSs). The reachability problem for this particular class of automata is unfortunately undecidable. However, it has been shown that an efficient abstraction of the execution traces language can nonetheless be computed. To this end, an algebraic framework to over-approximate context-free languages has been introduced by Bouajjani et al. In this paper, we combine this framework with an automata-theoretic approach in order to approximate an answer to the model checking problem of the linear-time temporal logic (from now on LTL) on CPDSs. We then present an algorithm that, given a single-indexed or stutter-invariant LTL formula, allows us to prove that no run of a CPDS verifies this formula if the procedure ends.

Analog neuron hierarchy

In order to refine the analysis of the computational power of discrete-time recurrent neural networks (NNs) between the binary-state NNs which are equivalent to finite automata (level 3 in the Chomsky hierarchy), and the analog-state NNs with rational weights which are Turing-complete (Chomsky level 0), we study an intermediate model αANN of a binary-state NN that is extended with α≥0 extra analog-state neurons. For rational weights, we establish an analog neuron hierarchy 0ANNs ⊂ 1ANNs ⊂ 2ANNs ⊆ 3ANNs and separate its first two levels. In particular, 0ANNs coincide with the binary-state NNs (Chomsky level 3) being a proper subset of 1ANNs which accept at most context-sensitive languages (Chomsky level 1) including some non-context-free ones (above Chomsky level 2). We prove that the deterministic (context-free) language L#={0n1n∣n≥1} cannot be recognized by any 1ANN even with real weights. In contrast, we show that deterministic pushdown automata accepting deterministic languages can be simulated by 2ANNs with rational weights, which thus constitute a proper superset of 1ANNs. Finally, we prove that the analog neuron hierarchy collapses to 3ANNs by showing that any Turing machine can be simulated by a 3ANN having rational weights, with linear-time overhead.