Precedence Automata and Languages

Operator precedence grammars define a classical Boolean and deterministic context-free family (called Floyd languages or FLs). FLs have been shown to strictly include the well-known visibly pushdown languages, and enjoy the same nice closure properties. We introduce here Floyd automata, an equivalent operational formalism for defining FLs. This also permits to extend the class to deal with infinite strings to perform for instance model checking.Comment: Extended version of the paper which appeared in Proceedings of CSR 2011, Lecture Notes in Computer Science, vol. 6651, pp. 291-304, 2011. Theorem 1 has been corrected and a complete proof is given in Appendi

Generalizing input-driven languages: theoretical and practical benefits

Regular languages (RL) are the simplest family in Chomsky's hierarchy. Thanks to their simplicity they enjoy various nice algebraic and logic properties that have been successfully exploited in many application fields. Practically all of their related problems are decidable, so that they support automatic verification algorithms. Also, they can be recognized in real-time. Context-free languages (CFL) are another major family well-suited to formalize programming, natural, and many other classes of languages; their increased generative power w.r.t. RL, however, causes the loss of several closure properties and of the decidability of important problems; furthermore they need complex parsing algorithms. Thus, various subclasses thereof have been defined with different goals, spanning from efficient, deterministic parsing to closure properties, logic characterization and automatic verification techniques. Among CFL subclasses, so-called structured ones, i.e., those where the typical tree-structure is visible in the sentences, exhibit many of the algebraic and logic properties of RL, whereas deterministic CFL have been thoroughly exploited in compiler construction and other application fields. After surveying and comparing the main properties of those various language families, we go back to operator precedence languages (OPL), an old family through which R. Floyd pioneered deterministic parsing, and we show that they offer unexpected properties in two fields so far investigated in totally independent ways: they enable parsing parallelization in a more effective way than traditional sequential parsers, and exhibit the same algebraic and logic properties so far obtained only for less expressive language families

Toward a theory of input-driven locally parsable languages

If a context-free language enjoys the local parsability property then, no matter how the source string is segmented, each segment can be parsed independently, and an efficient parallel parsing algorithm becomes possible. The new class of locally chain parsable languages (LCPLs), included in the deterministic context-free language family, is here defined by means of the chain-driven automaton and characterized by decidable properties of grammar derivations. Such automaton decides whether to reduce or not a substring in a way purely driven by the terminal characters, thus extending the well-known concept of input-driven (ID) alias visibly pushdown machines. The LCPL family extends and improves the practically relevant Floyd's operator-precedence (OP) languages which are known to strictly include the ID languages, and for which a parallel-parser generator exists

Left Recursion in Parsing Expression Grammars

Parsing Expression Grammars (PEGs) are a formalism that can describe all deterministic context-free languages through a set of rules that specify a top-down parser for some language. PEGs are easy to use, and there are efficient implementations of PEG libraries in several programming languages. A frequently missed feature of PEGs is left recursion, which is commonly used in Context-Free Grammars (CFGs) to encode left-associative operations. We present a simple conservative extension to the semantics of PEGs that gives useful meaning to direct and indirect left-recursive rules, and show that our extensions make it easy to express left-recursive idioms from CFGs in PEGs, with similar results. We prove the conservativeness of these extensions, and also prove that they work with any left-recursive PEG. PEGs can also be compiled to programs in a low-level parsing machine. We present an extension to the semantics of the operations of this parsing machine that let it interpret left-recursive PEGs, and prove that this extension is correct with regards to our semantics for left-recursive PEGs.Comment: Extended version of the paper "Left Recursion in Parsing Expression Grammars", that was published on 2012 Brazilian Symposium on Programming Language

Parallel parsing made practical

The property of local parsability allows to parse inputs through inspecting only a bounded-length string around the current token. This in turn enables the construction of a scalable, data-parallel parsing algorithm, which is presented in this work. Such an algorithm is easily amenable to be automatically generated via a parser generator tool, which was realized, and is also presented in the following. Furthermore, to complete the framework of a parallel input analysis, a parallel scanner can also combined with the parser. To prove the practicality of a parallel lexing and parsing approach, we report the results of the adaptation of JSON and Lua to a form fit for parallel parsing (i.e. an operator-precedence grammar) through simple grammar changes and scanning transformations. The approach is validated with performance figures from both high performance and embedded multicore platforms, obtained analyzing real-world inputs as a test-bench. The results show that our approach matches or dominates the performances of production-grade LR parsers in sequential execution, and achieves significant speedups and good scaling on multi-core machines. The work is concluded by a broad and critical survey of the past work on parallel parsing and future directions on the integration with semantic analysis and incremental parsing

Beyond operator-precedence grammars and languages

Operator Precedence Languages (OPL) are deterministic context-free and have desirable properties. OPL are parallely parsable, and, when structurally compatible, are closed under Boolean operations, concatenation and star; they include the Input Driven languages. OPL use three relations between two terminal symbols, to assign syntax structure to words. We extend such relations to k-tuples of consecutive symbols, in agreement with strictly locally testable regular languages. For each k, the new corresponding class of Higher-order Operator Precedence languages properly includes the OPL and enjoy many of their properties. OPL are a strict hierarchy based on k, which contains maximal languages

Commutative Languages and their Composition by Consensual Methods

Commutative languages with the semilinear property (SLIP) can be naturally recognized by real-time NLOG-SPACE multi-counter machines. We show that unions and concatenations of such languages can be similarly recognized, relying on -- and further developing, our recent results on the family of consensually regular (CREG) languages. A CREG language is defined by a regular language on the alphabet that includes the terminal alphabet and its marked copy. New conditions, for ensuring that the union or concatenation of CREG languages is closed, are presented and applied to the commutative SLIP languages. The paper contributes to the knowledge of the CREG family, and introduces novel techniques for language composition, based on arithmetic congruences that act as language signatures. Open problems are listed.Comment: In Proceedings AFL 2014, arXiv:1405.527

Weighted Operator Precedence Languages

In the last years renewed investigation of operator precedence languages (OPL) led to discover important properties thereof: OPL are closed with respect to all major operations, are characterized, besides the original grammar family, in terms of an automata family (OPA) and an MSO logic; furthermore they significantly generalize the well-known visibly pushdown languages (VPL). In another area of research, quantitative models of systems are also greatly in demand. In this paper, we lay the foundation to marry these two research fields. We introduce weighted operator precedence automata and show how they are both strict extensions of OPA and weighted visibly pushdown automata. We prove a Nivat-like result which shows that quantitative OPL can be described by unweighted OPA and very particular weighted OPA. In a Büchi-like theorem, we show that weighted OPA are expressively equivalent to a weighted MSO-logic for OPL

Complexity of Two-Dimensional Patterns

In dynamical systems such as cellular automata and iterated maps, it is often useful to look at a language or set of symbol sequences produced by the system. There are well-established classification schemes, such as the Chomsky hierarchy, with which we can measure the complexity of these sets of sequences, and thus the complexity of the systems which produce them. In this paper, we look at the first few levels of a hierarchy of complexity for two-or-more-dimensional patterns. We show that several definitions of ``regular language'' or ``local rule'' that are equivalent in d=1 lead to distinct classes in d >= 2. We explore the closure properties and computational complexity of these classes, including undecidability and L-, NL- and NP-completeness results. We apply these classes to cellular automata, in particular to their sets of fixed and periodic points, finite-time images, and limit sets. We show that it is undecidable whether a CA in d >= 2 has a periodic point of a given period, and that certain ``local lattice languages'' are not finite-time images or limit sets of any CA. We also show that the entropy of a d-dimensional CA's finite-time image cannot decrease faster than t^{-d} unless it maps every initial condition to a single homogeneous state.Comment: To appear in J. Stat. Phy