On polynomial grammars extended with substitution

A unifying approach to picture grammars

Several old and recent classes of picture grammars, that variously extend context-free string grammars in two dimensions, are based on rules that rewrite arrays of pixels. Such grammars can be unified and extended using a tiling based approach, whereby the right part of a rule is formalized by means of a finite set of permitted tiles. We focus on a simple type of tiling,named regional, and define the corresponding regional tile grammars. They include both Siromoney's (or Matz's) Kolam grammars and their generalization by Prusa, as well as Drewes's grid grammars. Regionally defined pictures can…

On the capabilities of grammars, automata, and transducers controlled by monoids

During the last decades, classical models in language theory have been extended by control mechanisms defined by monoids. We study which monoids cause the extensions of context-free grammars, finite automata, or finite state transducers to exceed the capacity of the original model. Furthermore, we investigate when, in the extended automata model, the nondeterministic variant differs from the deterministic one in capacity. We show that all these conditions are in fact equivalent and present an algebraic characterization. In particular, the open question of whether every language generated by a …

Synthesizing Context-free Grammars from Recurrent Neural Networks (Extended Version)

We present an algorithm for extracting a subclass of the context free grammars (CFGs) from a trained recurrent neural network (RNN). We develop a new framework, pattern rule sets (PRSs), which describe sequences of deterministic finite automata (DFAs) that approximate a non-regular language. We present an algorithm for recovering the PRS behind a sequence of such automata, and apply it to the sequences of automata extracted from trained RNNs using the L* algorithm. We then show how the PRS may converted into a CFG, enabling a familiar and useful presentation of the learned language. Extracting…

依存 Lambek 演算中解析器与形式文法理论的内在验证（扩展版）

Intrinsic Verification of Parsers and Formal Grammar Theory in Dependent Lambek Calculus (Extended Version)

提出 Dependent Lambek Calculus 依存类型理论，利用线性类型将形式文法表示为语法、解析器表示为线性项，实现解析器的内在验证，并在 Agda 中完成原型实现。

On LR(k)-parsers of polynomial size

Usually, a parser for an $LR(k)$-grammar $G$ is a deterministic pushdown transducer which produces backwards the unique rightmost derivation for a given input string $x \in L(G)$. The best known upper bound for the size of such a parser is $O(2^{|G||Σ|^k+k\log |Σ| + \log |G|})$ where $|G|$ and $|Σ|$ are the sizes of the grammar $G$ and the terminal alphabet $Σ$, respectively. If we add to a parser the possibility to manipulate a directed graph of size $O(|G|n)$ where $n$ is the length of the input then we obtain an extended parser. The graph is used for an efficient parallel simulation of all …

Parikh's theorem for infinite alphabets

We investigate commutative images of languages recognised by register automata and grammars. Semi-linear and rational sets can be naturally extended to this setting by allowing for orbit-finite unions instead of only finite ones. We prove that commutative images of languages of one-register automata are not always semi-linear, but they are always rational. We also lift the latter result to grammars: commutative images of one-register context-free languages are rational, and in consequence commutatively equivalent to register automata. We conjecture analogous results for automata and grammars w…