Spatial Existential Positive Logics for Hyperedge Replacement Grammars

We study a (first-order) spatial logic based on graphs of conjunctive queries for expressing (hyper-)graph languages. In this logic, each primitive positive (resp. existential positive) formula plays a role of an expression of a graph (resp. a finite language of graphs) modulo graph isomorphism. First, this paper presents a sound- and complete axiomatization for the equational theory of primitive/existential positive formulas under this spatial semantics. Second, we show Kleene theorems between this logic and hyperedge replacement grammars (HRGs), namely that over graphs, the class of existential positive first-order (resp. least fixpoint, transitive closure) formulas has the same expressive power as that of non-recursive (resp. all, linear) HRGs

Context-free languages, coalgebraically

We give a coalgebraic account of context-free languages using the functor ${\cal D}(X) = 2 \times X^A$ for deterministic automata over an alphabet $A$, in three different but equivalent ways: (i) by viewing context-free grammars as ${\cal D}$-coalgebras; (ii) by defining a format for behavioural differential equations (w.r.t. ${\cal D}$) for which the unique solutions are precisely the context-free languages; and (iii) as the ${\cal D}$-coalgebra of generalized regular expressions in which the Kleene star is replaced by a unique fixed point operator. In all cases, semantics is defined by the unique homomorphism into the final coalgebra of all languages, thus paving the way for coinductive proofs of context-free language equivalence. Furthermore, the three characterizations are elementary to the extent that they can serve as the basis for the definition of a general coalgebraic notion of context-freeness, which we see as the ultimate long-term goal of the present study

LL(1) Parsing with Derivatives and Zippers

In this paper, we present an efficient, functional, and formally verified parsing algorithm for LL(1) context-free expressions based on the concept of derivatives of formal languages. Parsing with derivatives is an elegant parsing technique, which, in the general case, suffers from cubic worst-case time complexity and slow performance in practice. We specialise the parsing with derivatives algorithm to LL(1) context-free expressions, where alternatives can be chosen given a single token of lookahead. We formalise the notion of LL(1) expressions and show how to efficiently check the LL(1) property. Next, we present a novel linear-time parsing with derivatives algorithm for LL(1) expressions operating on a zipper-inspired data structure. We prove the algorithm correct in Coq and present an implementation as a parser combinators framework in Scala, with enumeration and pretty printing capabilities.Comment: Appeared at PLDI'20 under the title "Zippy LL(1) Parsing with Derivatives

Sequencing and Intermediate Acceptance: Axiomatisation and Decidability of Bisimilarity

The Theory of Sequential Processes includes deadlock, successful termination, action prefixing, alternative and sequential composition. Intermediate acceptance, which is important for the integration of classical automata theory, can be expressed through a combination of alternative composition and successful termination. Recently, it was argued that complications arising from the interplay between intermediate acceptance and sequential composition can be eliminated by replacing sequential composition by sequencing. In this paper we study the equational theory of the recursion-free fragment of the resulting process theory modulo bisimilarity, proving that it is not finitely based, but does afford a ground-complete axiomatisation if a unary auxiliary operator is added. Furthermore, we prove that bisimilarity is decidable for processes definable by means of a finite guarded recursive specification over the process theory

Sequencing and intermediate acceptance: Axiomatisation and decidability of bisimilarity

The Theory of Sequential Processes includes deadlock, successful termination, action prefixing, alternative and sequential composition. Intermediate acceptance, which is important for the integration of classical automata theory, can be expressed through a combination of alternative composition and successful termination. Recently, it was argued that complications arising from the interplay between intermediate acceptance and sequential composition can be eliminated by replacing sequential composition by sequencing. In this paper we study the equational theory of the recursion-free fragment of the resulting process theory modulo bisimilarity, proving that it is not finitely based, but does afford a ground-complete axiomatisation if a unary auxiliary operator is added. Furthermore, we prove that bisimilarity is decidable for processes definable by means of a finite guarded recursive specification over the process the

Greibach Normal Form in Algebraically Complete Semirings

We give inequational and equational axioms for semirings with a fixed-point operator and formally develop a fragment of the theory of context-free languages. In particular, we show that Greibach's normal form theorem depends only on a few equational properties of least pre-fixed-points in semirings, and elimination of chain- and deletion rules depend on their inequational properties (and the idempotency of addition). It follows that these normal form theorems also hold in non-continuous semirings having enough fixed-points

A proof theory of right-linear (omega-)grammars via cyclic proofs

Right-linear (or left-linear) grammars are a well-known class of context-free grammars computing just the regular languages. They may naturally be written as expressions with (least) fixed points but with products restricted to letters as left arguments, giving an alternative to the syntax of regular expressions. In this work, we investigate the resulting logical theory of this syntax. Namely, we propose a theory of right-linear algebras (RLA) over of this syntax and a cyclic proof system CRLA for reasoning about them. We show that CRLA is sound and complete for the intended model of regular languages. From here we recover the same completeness result for RLA by extracting inductive invariants from cyclic proofs, rendering the model of regular languages the free right-linear algebra. Finally, we extend system CRLA by greatest fixed points, nuCRLA, naturally modelled by languages of omega-words thanks to right-linearity. We show a similar soundness and completeness result of (the guarded fragment of) nuCRLA for the model of omega-regular languages, employing game theoretic techniques.Comment: 34 pages, 3 figure

Coalgebraic characterizations of context-free languages

Article / Letter to editorLeiden Inst Advanced Computer Science