15 research outputs found
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 for deterministic automata over an
alphabet , in three different but equivalent ways: (i) by viewing
context-free grammars as -coalgebras; (ii) by defining a
format for behavioural differential equations (w.r.t. ) for
which the unique solutions are precisely the context-free languages; and
(iii) as the -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