Boolean Grammars Research Articles

A GLR-like Parsing Algorithm for Three-Valued Interpretations of Boolean Grammars with Strong Negation

A GLR-like Parsing Algorithm for Three-Valued Interpretations of Boolean Grammars with Strong Negation

Path Querying on Acyclic Graphs Using Boolean Grammars

Path Querying on Acyclic Graphs Using Boolean Grammars

Задача поиска путей в ациклических графах с ограничениями в терминах булевых грамматик

Graph data models are widely used in different areas of computer science such as bioinformatics, graph databases, social networks and static code analysis. One of the problems in graph data analysis is querying for specific paths. Such queries are usually performed by means of a formal grammar that describes the allowed edge-labeling of the paths. Path query is said to be calculated using relational query semantics if it is evaluated to triple , such that there is a path from to such that the labels on the edges of this path form a string derivable from the nonterminal . As the regular and context-free languages have limited expressive power, we focus on a more expressive languages, namely the Boolean languages that use Boolean grammars to describe the labeling of paths. Although path querying using relational query semantics and Boolean grammars is known to be undecidable, in this work we propose a path querying algorithm on acyclic graphs which uses relational query semantics and Boolean grammars and approximates the exact solution. To achieve better performance in compare with the naive algorithm, considered classes of graphs were limited to acyclic graphs.

Hardest languages for conjunctive and Boolean grammars

A famous theorem by Greibach (“The hardest context-free language”, SIAM J. Comp., 1973) states that there exists such a context-free language L0, that every context-free language over any alphabet is reducible to L0 by a homomorphic reduction—in other words, is representable as its inverse homomorphic image h−1(L0), for a suitable homomorphism h. This paper establishes similar characterizations for conjunctive grammars, that is, for grammars extended with a conjunction operator, as well as for Boolean grammars, which are further equipped with a negation operator. At the same time, it is shown that no such characterization is possible for several subclasses of linear grammars.

Open and Original Problems in Software Language Engineering 2015 Workshop Report

OOPSLE is a workshop co-located with a re-engineering conference and serving as a venue for software language engineers to meet outside the SLE conference to discuss either long-standing problems that remain unresolved for years or decades, or oftavoided problems that everyone is so used to work around that they stop noticing them at all. In 2015, it ran for the third time. The list of topics discussed at the workshop, included transformation in the presence of Boolean grammars, natural language software interfaces, formally supported model management, community-aware language design, domain-specific language design choices and uncertainty-aware development. A report such as this one was requested by the participants, but more condensed and general information can be found on our official webpage at http://oopsle.github.io

Parse views with Boolean grammars

We propose an enhancement to current parsing and transformation systems by leveraging the expressive power of Boolean grammars, a generalization of context-free grammars that adds conjunction and negation operators. In addition to naturally expressing a larger class of languages, Boolean grammars capture multiple parse trees of the same document simultaneously and the ability to switch between these parse “views”. In particular, source transformation and reengineering tasks can benefit from parse views by recasting the input text into whichever parse is most suitable for the task at hand.

Parsing by matrix multiplication generalized to Boolean grammars

The well-known parsing algorithm for context-free grammars due to Valiant (1975) [25] is analyzed and extended to handle the more general Boolean grammars, which are context-free grammars augmented with conjunction and negation operators in the rules. The algorithm reduces construction of a parsing table to computing multiple products of Boolean matrices of various sizes. Its time complexity on an input string of length n is O(BMM(n)logn), where BMM(n) is the number of operations needed to multiply two Boolean matrices of size n×n, which is O(nω) with ω<2.373 as per the current knowledge. A parse tree can be constructed in time MM(n)logO(1)n (where MM(n) is the complexity of multiplying two integer matrices), by applying a known efficient procedure for determining witnesses for Boolean matrix multiplication. The algorithm has a succinct proof of correctness and is ready to be implemented.

Parsing Boolean grammars over a one-letter alphabet using online convolution

Consider context-free grammars generating strings over a one-letter alphabet. For the membership problem for such grammars, stated as “Given a grammar G and a string an, determine whether an is generated by G”, only a naïve O(|G|⋅n2)-time algorithm is known. This paper develops a new algorithm solving this problem, which is based upon fast multiplication of integers, works in time |G|⋅nlog3n⋅2O(log∗n), and is applicable to context-free grammars augmented with Boolean operations, known as Boolean grammars. For unambiguous grammars, the running time of the algorithm is reduced to |G|⋅nlog2n⋅2O(log∗n). The algorithm is based upon (a simplification of) the online integer multiplication algorithm by Fischer and Stockmeyer [M.J. Fischer, L.J. Stockmeyer, Fast on-line integer multiplication, Journal of Computer and System Sciences 9 (3) (1974) 317–331].

Expressive power of LL( [formula omitted]) Boolean grammars

The paper studies the family of Boolean LL languages, generated by Boolean grammars and usable with the recursive descent parsing. It is demonstrated that over a one-letter alphabet, these languages are always regular, while Boolean LL subsets of Σ ∗ a ∗ obey a certain periodicity property, which, in particular, makes the language { a n b 2 n | n ⩾ 0 } non-representable. It is also shown that linear conjunctive LL grammars cannot generate any language of the form L ⋅ { a , b } , with L non-regular, and that no languages of the form L ⋅ c ∗ , with non-regular L , can be generated by any linear Boolean LL grammars. These results are used to establish a detailed hierarchy and closure properties of these and related families of formal languages.

A game-theoretic characterization of Boolean grammars

We obtain a simple, purely game-theoretic characterization of Boolean grammars [A. Okhotin, Boolean grammars, Information and Computation, 194(1) (2004) 19–48]. In particular, we propose a two-player infinite game of perfect information for Boolean grammars, which is equivalent to their well-founded semantics. The game is directly applicable to the simpler classes of conjunctive and context-free grammars, and offers a promising new connection between game theory and formal languages.

BOOLEAN GRAMMARS AND GSM MAPPINGS

It is proved that the language family generated by Boolean grammars is effectively closed under injective gsm mappings and inverse gsm mappings (where gsm stands for a generalized sequential machine). The same results hold for conjunctive grammars, unambiguous Boolean grammars and unambiguous conjunctive grammars.

A simple P-complete problem and its language-theoretic representations

A variant of the Circuit Value Problem is introduced, in which every gate implements the NOR function ¬ ( x ∨ y ) , and one of the inputs of every k th gate must be the ( k − 1 ) th gate. The problem, which remains P-complete, is encoded as a simple formal language over a two-letter alphabet, which can be succinctly represented by language equations of several types. Using this representation, a conjunctive grammar with 8 rules, a Boolean grammar with 5 rules and an LL(1) Boolean grammar with 8 rules for this language are constructed. Another encoding of the problem is represented by a trellis automaton with 11 states and a linear conjunctive grammar with 20 rules.

Representing a P-complete problem by small trellis automata

A restricted case of the Circuit Value Problem known as the Sequential NOR Circuit Value Problem was recently used to obtain very succinct examples of conjunctive grammars, Boolean grammars and language equations representing P-complete languages (Okhotin, http://dx.doi.org/10.1007/978-3-540-74593-8_23 "A simple P-complete problem and its representations by language equations", MCU 2007). In this paper, a new encoding of the same problem is proposed, and a trellis automaton (one-way real-time cellular automaton) with 11 states solving this problem is constructed.

Well-founded semantics for Boolean grammars

Boolean grammars [A. Okhotin, Boolean grammars, Information and Computation 194 (1) (2004) 19–48] are a promising extension of context-free grammars that supports conjunction and negation in rule bodies. In this paper, we give a novel semantics for Boolean grammars which applies to all such grammars, independently of their syntax. The key idea of our proposal comes from the area of negation in logic programming, and in particular from the so-called well-founded semantics which is widely accepted in this area to be the “correct” approach to negation. We show that for every Boolean grammar there exists a distinguished (three-valued) interpretation of the non-terminal symbols, which satisfies all the rules of the grammar and at the same time is the least fixed-point of an operator associated with the grammar. Then, we demonstrate that every Boolean grammar can be transformed into an equivalent (under the new semantics) grammar in normal form. Based on this normal form, we propose an O ( n 3 ) algorithm for parsing that applies to any such normalized Boolean grammar. In summary, the main contribution of this paper is to provide a semantics which applies to all Boolean grammars while at the same time retaining the complexity of parsing associated with this type of grammars.

Unambiguous Boolean grammars

Boolean grammars are an extension of context-free grammars, in which conjunction and negation may be explicitly used in the rules. In this paper, the notion of ambiguity in Boolean grammars is defined. It is shown that the known transformation of a Boolean grammar to the binary normal form preserves unambiguity, and that every unambiguous Boolean language can be parsed in time O( n 2). Linear conjunctive languages are shown to be unambiguous, while the existence of languages inherently ambiguous with respect to Boolean grammars is left open.

Locally stratified Boolean grammars

We introduce locally stratified Boolean grammars, a natural subclass of Boolean grammars with many desirable properties. Informally, if a grammar is locally stratified then the set of all pairs of the form ( nonterminal, string) of the grammar can be mapped to a (possibly infinite) set of strata so as that the following holds: if the membership of a string w in the language defined by nonterminal A depends on the membership of string w ′ in the language defined by nonterminal B, then ( B, w ′ ) cannot belong to a stratum higher than the stratum of ( A, w); furthermore, if the above dependency is obtained through negation, ( B , w ′ ) must belong to a stratum lower than the stratum of ( A, w). We prove that local stratifiability can be tested in linear time with respect to the size of the given grammar. We then develop the semantics of locally stratified grammars and prove that it is independent of the choice of the stratification mapping. We argue that the class of locally stratified Boolean grammars appears at present to be the broadest subclass of Boolean grammars that can be given a classical semantics (ie., without resorting to three-valued formal language theory).

Recursive descent parsing for Boolean grammars

The recursive descent parsing method for the context-free grammars is extended for their generalization, Boolean grammars, which include explicit set-theoretic operations in the formalism of rules and which are formally defined by language equations. The algorithm is applicable to a subset of Boolean grammars. The complexity of a direct implementation varies between linear and exponential, while memoization keeps it down to linear.

Scannerless Boolean Parsing

Scannerless generalized parsing techniques allow parsers to be derived directly from unified, declarative specifications. Unfortunately, in order to uniquely parse existing programming languages at the character level, disambiguation extensions beyond the usual context-free formalism are required.This paper explains how scannerless parsers for boolean grammars (context-free grammars extended with intersection and negation) can specify such languages unambiguously, and can also describe other interesting constructs such as indentation-based block structure.The sbp package implements this parsing technique and is publicly available as Java source code.

GENERALIZED LR PARSING ALGORITHM FOR BOOLEAN GRAMMARS

The generalized LR parsing algorithm for context-free grammars is extended for the case of Boolean grammars, which are a generalization of the context-free grammars with logical connectives added to the formalism of rules. In addition to the standard LR operations, Shift and Reduce, the new algorithm uses a third operation called Invalidate, which reverses a previously made reduction. This operation makes the mathematical justification of the algorithm significantly different from its prototype. On the other hand, the changes in the implementation are not very substantial, and the algorithm still works in time O(n4).

The dual of concatenation

A binary language-theoretic operation is proposed, which is dual to the concatenation of languages in the same sense as the universal quantifier in logic is dual to the existential quantifier; the dual of Kleene star is defined accordingly. These operations arise whenever concatenation or star appear in the scope of negation. The basic properties of the new operations are determined in the paper. Their use in regular expressions and in language equations is considered, and it is shown that they often eliminate the need of using negation, at the same time having an important technical advantage of being monotone. A generalization of context-free grammars featuring dual concatenation is introduced and proved to be equivalent to the recently studied Boolean grammars.