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Graph grammars form an interesting area of research because of their versatility in modelling diverse situations with graphs as the structures which are to be manipulated. A new class of graph grammars, nc-eNCE Graph Grammars has been introduced recently with an aim of restricting the order of application of graph production rules, thereby generating different graph classes using the same set of rules. On the other hand 2D game design using an algorithmic approach known as procedural content generation has been of interest recently. In this paper we modify the structure of nc-eNCE graph grammars with the aim of generating directed graphs. We show that employing these graph grammars simplifies the design of 2D games. We have also developed an algorithm which makes use of these graph grammars for generating random game level layouts ensuring that the players will get a different gaming experience each time they play., Comment: 14 pages, 10 figures

This ongoing project aims to define and investigate, from the standpoint of category theory, order theory and universal algebra, the notions of higher-order many-sorted rewriting system and of higher-order many-sorted categorial algebra and their relationships, via the higher-order Curry-Howard isomorphisms. The ultimate goal, to be developed in future versions of this work, is to define and investigate the category of towers, whose objects will consist of families, indexed by $\mathbb{N}$, of higher-order many-sorted rewriting systems and of higher-order many-sorted categorial algebras, including higher-order Curry-Howard type results for the latter, together with an additional structure that intertwines such $\mathbb{N}$-families; and whose morphism from a tower to another will be families, indexed by $\mathbb{N}$, of morphisms between its higher-order many-sorted rewriting systems and of higher-order many-sorted categorial algebras compatible with their structures. All feedback is appreciated., Comment: 814 pages, three parts and an appendix

Context-free languages can be characterized in several ways. This article studies projective linearisations of languages of simple dependency trees, i.e., dependency trees in which a node can govern at most one node with a given syntactic function. We prove that the projective linearisations of local languages of simple dependency trees coincide with the context-free languages. Simple dependency trees suggest alternative dual notions of locality and projectivity, which permits defining a dual language for each context-free language. We call this new class of languages anti-context-free. These languages are related to some linguistic constructions exhibiting the so-called cross-serial dependencies that were historically important for the development of computational linguistics. We propose that this duality could be a relevant linguistic phenomenon., Comment: 26 pages, 17 figures, to appear at Journal of Automata, languages and combinatorics

A new family of categorial grammars is proposed, defined by enriching basic categorial grammars with a conjunction operation. It is proved that the formalism obtained in this way has the same expressive power as conjunctive grammars, that is, context-free grammars enhanced with conjunction. It is also shown that categorial grammars with conjunction can be naturally embedded into the Lambek calculus with conjunction and disjunction operations. This further implies that a certain NP-complete set can be defined in the Lambek calculus with conjunction. We also show how to handle some subtle issues connected with the empty string. Finally, we prove that a language generated by a conjunctive grammar can be described by a Lambek grammar with disjunction (but without conjunction)., Comment: This article is an extended version of the conference presentation "Conjunctive categorial grammars" at the Mathematics of Language 2017 meeting (London, UK, July 13-14, 2017; proceedings published in ACL Anthology, W17-3414)

Combinatorial generation of expander families and Lindenmayer-style development models are both parallel in nature. Both can be handled within proposed parallel graph grammar formalism. Their first-order properties can then be checked by encompassing the generated graph language into an appropriate automatic structure.

There are many types of automata and grammar models that have been studied in the literature, and for these models, it is common to determine whether certain problems are decidable. One problem that has been difficult to answer throughout the history of automata and formal language theory is to decide whether a given system $M$ accepts a bounded language (whether there exist words $w_1, \ldots,w_k$ such that $L(M) \subseteq w_1 \cdots w_k$?). Decidability of this problem has gone unanswered for the majority of automata/grammar models in the literature. Boundedness was only known to be decidable for regular and context-free languages until recently when it was shown to also be decidable for finite-automata and pushdown automata augmented with reversal-bounded counters, and for vector addition systems with states. In this paper, we develop new techniques to show that the boundedness problem is decidable for larger classes of one-way nondeterministic automata and grammar models, by reducing the problem to the decidability of boundedness for simpler classes of automata. One technique involves characterizing the models in terms of multi-tape automata. We give new characterizations of finite-turn Turing machines, finite-turn Turing machines augmented with various storage structures (like a pushdown, multiple reversal-bounded counters, partially-blind counters, etc.), and simple matrix grammars. The characterizations are then used to show that the boundedness problem for these models is decidable. Another technique uses the concept of the store language of an automaton. This is used to show that the boundedness problem is decidable for pushdown automata that can "flip" their pushdown a bounded number of times, and boundedness remains decidable even if we augment this device with additional stores., Comment: 23 pages,2 figures

Rijkswaterstaat, the National Dutch body responsible for infrastructure, recognised the importance of formal modelling and set up a program to model the control of road tunnels. This is done to improve the standardisation of tunnel control and make communication with suppliers smoother. A subset of SysML is used to formulate the models, which are substantial. In an earlier paper we have shown that these models can be used to prove behavioural properties by manually translating the models to mCRL2. In this paper we report on an automatic translation to mCRL2. As the results of the translation became unwieldy, we also investigated modelling tunnel control in the specification language Dezyne which has built-in verification capabilities and compared the results., Comment: In Proceedings MARS 2024, arXiv:2403.17862

Engelfriet and Vereijken have shown that linear graph grammars based on hyperedge replacement generate graph languages that can be considered as interpretations of regular string languages over typed symbols. In this paper we show that finite automata can be lifted from strings to graphs within the same framework. For the efficient recognition of graphs with these automata, we make them deterministic by a modified powerset construction, and state sufficient conditions under which deterministic finite graph automata recognize graphs without the need to use backtracking., Comment: 23 pages, accepted for publication in the Postproceedings of the 14th International Workshop on Graph Computational Models (GCM 2023)

Monographs are graph-like structures with directed edges of unlimited length that are freely adjacent to each other. The standard nodes are represented as edges of length zero. They can be drawn in a way consistent with standard graphs and many others, like E-graphs or $\infty$-graphs. The category of monographs share many properties with the categories of graph structures (algebras of monadic many-sorted signatures), except that there is no terminal monograph. It is universal in the sense that its slice categories (or categories of typed monographs) are equivalent to the categories of graph structures. Type monographs thus emerge as a natural way of specifying graph structures. A detailed analysis of single and double pushout transformations of monographs is provided, and a notion of attributed typed monographs generalizing typed attributed E-graphs is analyzed w.r.t. attribute-preserving transformations.

In the research on computational effects, defined algebraically, effect symbols are often expected to obey certain equations. If we orient these equations, we get a rewrite system, which may be an effective way of transforming or optimizing the effects in a program. In order to do so, we need to establish strong normalization, or termination, of the rewrite system. Here we define a framework for carrying out such proofs, and extend the well-known Recursive Path Ordering of Dershowitz to show termination of some effect systems., Comment: 15 pages, 2 figures. Submitted to FSCD 2023

Higher-dimensional rewriting is founded on a duality of rewrite systems and cell complexes, connecting computational mathematics to higher categories and homotopy theory: the two sides of a rewrite rule are two halves of the boundary of an (n+1)-cell, which are diagrams of n-cells. We study higher-dimensional diagram rewriting as a mechanism of computation, focussing on the matching problem for rewritable subdiagrams within the combinatorial framework of diagrammatic sets. We provide an algorithm for subdiagram matching in arbitrary dimensions, based on new results on layerings of diagrams, and derive upper bounds on its time complexity. We show that these superpolynomial bounds can be improved to polynomial bounds under certain acyclicity conditions, and that these conditions hold in general for diagrams up to dimension 3. We discuss the challenges that arise in dimension 4., Comment: Accepted for LICS 2023. 13 pages + appendix 9 pages

Polygraphs are a higher-dimensional generalization of the notion of directed graph. Based on those as unifying concept, this monograph on polygraphs revisits the theory of rewriting in the context of strict higher categories, adopting the abstract point of view offered by homotopical algebra. The first half explores the theory of polygraphs in low dimensions and its applications to the computation of the coherence of algebraic structures. It is meant to be progressive, with little requirements on the background of the reader, apart from basic category theory, and is illustrated with algorithmic computations on algebraic structures. The second half introduces and studies the general notion of n-polygraph, dealing with the homotopy theory of those. It constructs the folk model structure on the category of strict higher categories and exhibits polygraphs as cofibrant objects. This allows extending to higher dimensional structures the coherence results developed in the first half.

It is well established that formulating an effective constraint model of a problem of interest is crucial to the efficiency with which it can subsequently be solved. Following from the observation that it is difficult, if not impossible, to know a priori which of a set of candidate models will perform best in practice, we envisage a system that explores the space of models through a process of reformulation from an initial model, guided by performance on a set of training instances from the problem class under consideration. We plan to situate this system in a refinement-based approach, where a user writes a constraint specification describing a problem above the level of abstraction at which many modelling decisions are made. In this position paper we set out our plan for an exploratory reformulation system, and discuss progress made so far., Comment: 13 pages, 6 figures

We investigate the correspondence between the time and space recognition complexity of languages; for this purpose, we will code the long-continued computations of deterministic two-tape Turing machines by the relatively short-length quantified Boolean formulae. The modified Stockmeyer and Meyer method will appreciably be used for this simulation. It will be proved using this modeling that the complexity classes $\mathbf{EXP}$ and $\mathbf{PSPACE}$ coincide; and more generally, the class $(k\!+\!1)$-fold Deterministic Exponential Time equals to the class $k$-fold Deterministic Exponential Space for each $k\geqslant1$; the space complexity of the languages of the class $\mathbf{P}$ will also be studied. Furthermore, this allows us to slightly improve the early founded lower complexity bound of decidable theories that are nontrivial relative to some equivalence relation (this relation may be equality) -- each of these theories is consistent with the formula, which asserts that there are two non-equivalent elements. Keywords: computational complexity, the coding of computations through formulae, exponential time, polynomial space, lower complexity bound of the language recognition, Comment: 44 pages, 26 references bibliography; text overlap with arXiv:1907.04521 because the paper is created in the same method

We continue our study of ordered context-free grammars, a grammar formalism that places an order on the parse trees produced by the corresponding context-free grammar. In particular, we simplify our previous definition of a derivation of a string for a given ordered context-free grammar, and present a parsing algorithm, using shared packed parse forests, with time complexity O(n^4), where n is the length of the input string being parsed., Comment: In Proceedings NCMA 2023, arXiv:2309.07333

A classical theorem states that the set of languages given by a pushdown automaton coincides with the set of languages given by a context-free grammar. In previous work, we proved the pendant of this theorem in a setting with interaction: the set of processes given by a pushdown automaton coincides with the set of processes given by a finite guarded recursive specification over a process algebra with actions, choice, sequencing and guarded recursion, if and only if we add sequential value passing. In this paper, we look what happens if we consider parallel pushdown automata instead of pushdown automata, and a process algebra with parallelism instead of sequencing., Comment: In Proceedings EXPRESS/SOS2023, arXiv:2309.05788. arXiv admin note: text overlap with arXiv:2203.01713

Pumping lemmata are the main tool to prove that a certain language does not belong to a class of languages like the recognizable languages or the context-free languages. Essentially two pumping lemmata exist for the recognizable weighted languages: the classical one for the Boolean semiring (i.e., the unweighted case), which can be generalized to zero-sum free semirings, and the one for fields. A joint generalization of these two pumping lemmata is provided that applies to all Artinian semirings, over which all finitely generated semimodules have a finite bound on the length of chains of strictly increasing subsemimodules. Since Artinian rings are exactly those that satisfy the Descending Chain Condition, the Artinian semirings include all fields and naturally also all finite semirings (like the Boolean semiring). The new pumping lemma thus covers most previously known pumping lemmata for recognizable weighted languages., Comment: In Proceedings AFL 2023, arXiv:2309.01126

Graph-controlled insertion-deletion (GCID) systems are regulated extensions of insertion-deletion systems. Such a system has several components and each component contains some insertion-deletion rules. The components are the vertices of a directed control graph. A rule is applied to a string in a component and the resultant string is moved to the target component specified in the rule. The language of the system is the set of all terminal strings collected in the final component. We impose the restriction in the structure of the underlying graph to be a star structure where there is a central, control component which acts like a master and transmits a string (after applying one of its rules) to one of the components specified in the (applied) rule. A component which receives the string can process the obtained string with any applicable rule available in it and sends back the resultant string only to the center component. With this restriction, we obtain computational completeness for some descriptional complexity measures, Comment: In Proceedings AFL 2023, arXiv:2309.01126

Single-tape nondeterministic Turing machines that are allowed to replace the symbol in each tape cell only when it is scanned for the first time are also known as 1-limited automata. These devices characterize, exactly as finite automata, the class of regular languages. However, they can be extremely more succinct. Indeed, in the worst case the size gap from 1-limited automata to one-way deterministic finite automata is double exponential. Here we introduce two restricted versions of 1-limited automata, once-marking 1-limited automata and always-marking 1-limited automata, and study their descriptional complexity. We prove that once-marking 1-limited automata still exhibit a double exponential size gap to one-way deterministic finite automata. However, their deterministic restriction is polynomially related in size to two-way deterministic finite automata, in contrast to deterministic 1-limited automata, whose equivalent two-way deterministic finite automata in the worst case are exponentially larger. For always-marking 1-limited automata, we prove that the size gap to one-way deterministic finite automata is only a single exponential. The gap remains exponential even in the case the given machine is deterministic. We obtain other size relationships between different variants of these machines and finite automata and we present some problems that deserve investigation., Comment: In Proceedings AFL 2023, arXiv:2309.01126

In the study of automata and grammars, closure properties of the associated languages have been studied extensively. In particular, closure properties of various types of graph grammars have been examined in (Rozenberg and Welzl, Inf. and Control,1986) and (Rozenberg and Welzl, Acta Informatica,1986). In this paper we examine some critical closure properties of the nc-eNCE graph grammars discussed in (Jayakrishna and Mathew, Symmetry 2023) and (Jayakrishna and Mathew, ICMICDS 2022)., Comment: 14 pages,9 figures, to be submitted to Theory of Computing

We formalise the basics of the double-pushout approach to graph transformation in the proof assistant Isabelle/HOL and provide associated machine-checked proofs. Specifically, we formalise graphs, graph morphisms and rules, and a definition of direct derivations based on deletion and gluing. We then formalise graph pushouts and prove with Isabelle's help that both deletions and gluings are pushouts. We also prove that pushouts are unique up to isomorphism. The formalisation comprises around 2000 lines of source text. Our motivation is to pave the way for rigorous, machine-checked proofs in the theory of the double-pushout approach, and to lay the foundations for verifying graph transformation systems and rule-based graph programs by interactive theorem proving., Comment: In Proceedings GCM 2022, arXiv:2212.10975

State grammars are context-free grammars where the productions have states associated with them, and a production can only be applied to a nonterminal if the current state matches the state in the production. Once states are added to grammars, it is natural to add various stores, similar to machine models. With such extensions, productions can only be applied if both the state and the value read from each store matches between the current sentential form and the production. Here, generative capacity results are presented for different derivation modes, with and without additional stores. In particular, with the standard derivation relation, it is shown that adding reversal-bounded counters does not increase the capacity, and states are enough. Also, state grammars with reversal-bounded counters that operate using leftmost derivations are shown to coincide with languages accepted by one-way machines with a pushdown and reversal-bounded counters, and these are surprisingly shown to be strictly weaker than state grammars with the standard derivation relation (and no counters). The complexity of the emptiness problem involving state grammars with reversal-bounded counters is also studied., Comment: 21 pages

It is shown that for every language family that is a trio containing only semilinear languages, all bounded languages in it can be accepted by one-way deterministic reversal-bounded multicounter machines (DCM). This implies that for every semilinear trio (where these properties are effective), it is possible to decide containment, equivalence, and disjointness concerning its bounded languages. A condition is also provided for when the bounded languages in a semilinear trio coincide exactly with those accepted by DCM machines, and it is used to show that many grammar systems of finite index -- such as finite-index matrix grammars and finite-index ETOL -- have identical bounded languages as DCM. Then connections between ambiguity, counting regularity, and commutative regularity are made, as many machines and grammars that are unambiguous can only generate/accept counting regular or commutatively regular languages. Thus, such a system that can generate/accept a non-counting regular or non-commutatively regular language implies the existence of inherently ambiguous languages over that system. In addition, it is shown that every language generated by an unambiguous finite-index matrix grammar has a rational characteristic series in commutative variables, and is counting regular. This result plus the connections are used to demonstrate that finite-index matrix grammars and finite-index ETOL can generate inherently ambiguous languages (over their grammars), as do several machine models. It is also shown that all bounded languages generated by these two grammar systems (those in any semilinear trio) can be generated unambiguously within the systems. Finally, conditions on languages generated by finite-index matrix grammars and finite-index ETOL implying commutative regularity are obtained. In particular, it is shown that every finite-index EDOL language is commutatively regular., Comment: 28 pages, 4 images

In this electronic appendix to our paper "Input Invariants," accepted at ESEC/FSE'22, we provide additional examples, formal definitions, theorems, and proof sketches to complement our paper. Furthermore, we show the invariants that ISLearn mined in our evaluation., Comment: 9 pages. The main paper "Input Invariants" appeared at ESEC/FSE 2022

Take any word over some alphabet. If it is non-empty, go to any position and print out the letter being scanned. Now repeat the following any number of times (possibly zero): either stay at the current letter, or move one letter leftwards (if possible) or move one letter rightwards (if possible); then print out the letter being scanned. In effect, we are going for a walk on the input word. Let u be the infix of the input word comprising the visited positions, and w the word printed out (empty if the input word is). Since any unvisited prefix or suffix of the input word cannot influence w, we may as well discard them, and say that u generates w. We ask: given a word w, what words u generate it? The answer is surprising. Call u a primitive generator of w if u generates w and is not generated by any word shorter than u. We show that, excepting some degenerate cases, every word has precisely two primitive generators., Comment: Considerably extended from v1. (Secs. 4, 5 and 6 added.)

In this submission, we explore the use of equality saturation to optimize concurrent computations. A concurrent environment gives rise to new optimization opportunities, like extracting a common concurrent subcomputation. To our knowledge, no existing equality saturation framework allows such an optimization. The challenge with concurrent environments is that they require non-local reasoning since parallel computations are inherently unrelated and disjoint. This submission presents a new approach to optimizing equivalent concurrent computations: extending e-graphs to capture equal concurrent computations in order to replace them with a single computation.

Tree-adjoining grammar (TAG) and combinatory categorial grammar (CCG) are two well-established mildly context-sensitive grammar formalisms that are known to have the same expressive power on strings (i.e., generate the same class of string languages). It is demonstrated that their expressive power on trees also essentially coincides. In fact, CCG without lexicon entries for the empty string and only first-order rules of degree at most 2 are sufficient for its full expressive power., Comment: 30 pages, 6 figures, revised and extended version of paper appearing in Trans. ACL (2021) 9: 707-720

This paper presents an extension of the GLL parsing algorithm for context-free grammars which also supports parsing expression grammars with ordered choice and lookahead. The new PEGLL algorithm retains support for unordered choice, and thus parses a common superset of context-free grammars and parsing expression grammars. As part of this work, the authors have modified an existing GLL parser-generator to support parsing expression grammars, adding operators for common parsing expressions and modifying the lexer algorithm to better support ordered choice., Comment: 10 pages, 11 figures

Compilers are a prime target for formal verification, since compiler bugs invalidate higher-level correctness guarantees, but compiler changes may become more labor-intensive to implement, if they must come with proof patches. One appealing approach is to present compilers as sets of algebraic rewrite rules, which a generic engine can apply efficiently. Now each rewrite rule can be proved separately, with no need to revisit past proofs for other parts of the compiler. We present the first realization of this idea, in the form of a framework for the Coq proof assistant. Our new Coq command takes normal proved theorems and combines them automatically into fast compilers with proofs. We applied our framework to improve the Fiat Cryptography toolchain for generating cryptographic arithmetic, producing an extracted command-line compiler that is about 1000$\times$ faster while actually featuring simpler compiler-specific proofs., Comment: 13th International Conference on Interactive Theorem Proving (ITP 2022)

We develop a general model theoretic semantics to rewriting beyond the usual confluence and termination assumptions. This is based on preordered algebra which is a model theory that extends many sorted algebra. In this framework we characterise rewriting in arbitrary algebras rather than term algebras (called algebraic rewriting) as a persistent adjunction and use this result, on the one hand for proving the soundness and the completeness of an abstract computational model of rewriting that underlies the non-deterministic programming with Maude and CafeOBJ, and on the other hand for developing a compositionality result for algebraic rewriting in the context of the pushout-based modularisation technique.

A foundational theory of compositional categorical rewriting theory is presented, based on a collection of fibration-like properties that collectively induce and intrinsically structure the large collection of lemmata used in the proofs of theorems such as concurrency and associativity. The resulting highly generic proofs of these theorems are given. It is noteworthy that the proof of the concurrency theorem takes only a few lines and, while that of associativity remains somewhat longer, it would be unreadably long if written directly in terms of the basic lemmata. In essence, our framework improves the readability and ease of comprehension of these proofs by exposing latent modularity. A curated list of known instances of our framework is used to conclude the paper with a detailed discussion of the conditions under which the Double Pushout and Sesqui-Pushout semantics of graph transformation are compositional., Comment: 85 pages; invited and substantially extended journal version of arXiv:2105.02842, accepted for publication in JLAMP

In structural proof theory, designing and working on large calculi make it difficult to get intuitions about each rule individually and as part of a whole system. We introduce two novel tools to help working on calculi using the approach of graph theory and automata theory. The first tool is a Proof Tree Automaton (PTA): a tree automaton which language is the derivation language of a calculus. The second tool is a graphical representation of a calculus called Proof Tree Graph (PTG). In this directed hypergraph, vertices are sets of terms (e.g. sequents) and hyperarcs are rules. We explore properties of PTA and PTGs and how they relate to each other. We show that we can decompose a PTA as a partial map from a calculus to a traditional tree automaton. We formulate that statement in the theory of refinement systems. Finally, we compare our framework to proof nets and string diagrams., Comment: Extended abstract accepted at SLSS 2022

We describe a formalization of higher-order rewriting theory and formally prove that an AFS is strongly normalizing if it can be interpreted in a well-founded domain. To do so, we use Coq, which is a proof assistant based on dependent type theory. Using this formalization, one can implement several termination techniques, like the interpretation method or dependency pairs, and prove their correctness. Those implementations can then be extracted to OCaml, which results in a verified termination checker.

We develop a multiset query and update language executable in a term rewriting system. Its most remarkable feature, besides non-standard approach to quantification and introduction of fresh values, is non-determinism - a query result is not uniquely determined by the database. We argue that this feature is very useful, e.g., in modelling user choices during simulation or reachability analysis of a data-centric business process - the intended application of our work. Query evaluation is implemented by converting the query into a terminating term rewriting system and normalizing the initial term which encapsulates the current database. A normal form encapsulates a query result. We prove that our language can express any relational algebra query. Finally, we present a simple business process specification framework (and an example specification). Both syntax and semantics of our query language is implemented in Maude., Comment: 40 pages, version edited by Fundamenta Informaticae

We report our experience in the formal verification of the reference implementation of the Beacon Chain. The Beacon Chain is the backbone component of the new Proof-of-Stake Ethereum 2.0 network: it is in charge of tracking information about the validators, their stakes, their attestations (votes) and if some validators are found to be dishonest, to slash them (they lose some of their stakes). The Beacon Chain is mission-critical and any bug in it could compromise the whole network. The Beacon Chain reference implementation developed by the Ethereum Foundation is written in Python, and provides a detailed operational description of the state machine each Beacon Chain's network participant (node) must implement. We have formally specified and verified the absence of runtime errors in (a large and critical part of) the Beacon Chain reference implementation using the verification-friendly language Dafny. During the course of this work, we have uncovered several issues, proposed verified fixes. We have also synthesised functional correctness specifications that enable us to provide guarantees beyond runtime errors. Our software artefact is available at https://github.com/ConsenSys/eth2.0-dafny.

Designers increasingly rely on procedural generation for automatic generation of content in various industries. These techniques require extensive knowledge of the desired content, and about how to actually implement such procedural methods. Algorithms for learning interpretable generative models from example content could alleviate both difficulties. We propose SIGI, a novel method for inferring shapes and inducing a shape grammar from grid-based 3D building examples. This interpretable grammar is well-suited for co-creative design. Applied to Minecraft buildings, we show how the shape grammar can be used to automatically generate new buildings in a similar style.

Following a recently considered generalisation of linear equations to unordered-data vectors and to ordered-data vectors, we perform a further generalisation to data vectors that are functions from k-element subsets of the unordered-data set to vectors of integer numbers. These generalised equations naturally appear in the analysis of vector addition systems (or Petri nets) extended so that each token carries a set of unordered data. We show that nonnegative-integer solvability of linear equations is in nondeterministic exponential time while integer solvability is in polynomial time.

Reinforcement learning has recently shown promise as a technique for training an artificial neural network to parse sentences in some unknown format, through a body of work known as RL-GRIT. A key aspect of the RL-GRIT approach is that rather than explicitly inferring a grammar that describes the format, the neural network learns to perform various parsing actions (such as merging two tokens) over a corpus of sentences, with the goal of maximizing the estimated frequency of the resulting parse structures. This can allow the learning process to more easily explore different action choices, since a given choice may change the optimality of the parse (as expressed by the total reward), but will not result in the failure to parse a sentence. However, this also presents a limitation: because the trained neural network can successfully parse any sentence, it cannot be directly used to identify sentences that deviate from the format of the training sentences, i.e., that are anomalous. In this paper, we address this limitation by presenting procedures for extracting production rules from the neural network, and for using these rules to determine whether a given sentence is nominal or anomalous. When a sentence is anomalous, an attempt is made to identify the location of the anomaly. We empirically demonstrate that our approach is capable of grammatical inference and anomaly detection for both non-regular formats and those containing regions of high randomness/entropy. While a format with high randomness typically requires large sets of production rules, we propose a two pass grammatical inference method to generate parsimonious rule sets for such formats. By further improving parser learning, and leveraging the presented rule extraction and anomaly detection algorithms, one might begin to understand common errors, either benign or malicious, in practical formats., Comment: 10 pages, 3 figures

The Kappa biochemistry and the M{\O}D organic chemistry frameworks are amongst the most intensely developed applications of rewriting-based methods in the life sciences to date. A typical feature of these types of rewriting theories is the necessity to implement certain structural constraints on the objects to be rewritten (a protein is empirically found to have a certain signature of sites, a carbon atom can form at most four bonds, ...). In this paper, we contribute a number of original developments that permit to implement a universal theory of continuous-time Markov chains (CTMCs) for stochastic rewriting systems. Our core mathematical concepts are a novel rule algebra construction for the relevant setting of rewriting rules with conditions, both in Double- and in Sesqui-Pushout semantics, augmented by a suitable stochastic mechanics formalism extension that permits to derive dynamical evolution equations for pattern-counting statistics. A second main contribution of our paper is a novel framework of restricted rewriting theories, which comprises a rule-algebra calculus under the restriction to so-called constraint-preserving completions of application conditions (for rules considered to act only upon objects of the underlying category satisfying a globally fixed set of structural constraints). This novel framework in turn renders a faithful encoding of bio- and organo-chemical rewriting in the sense of Kappa and M{\O}D possible, which allows us to derive a rewriting-based formulation of reaction systems including a full-fledged CTMC semantics as instances of our universal CTMC framework. While offering an interesting new perspective and conceptual simplification of this semantics in the setting of Kappa, both the formal encoding and the CTMC semantics of organo-chemical reaction systems as motivated by the M{\O}D framework are the first such results of their kind., Comment: 62 pages; long version of arXiv:2003.09395

Sesqui-pushout (SqPO) rewriting along non-linear rules and for monic matches is well-known to permit the modeling of fusing and cloning of vertices and edges, yet to date, no construction of a suitable concurrency theorem was available. The lack of such a theorem, in turn, rendered compositional reasoning for such rewriting systems largely infeasible. We develop in this paper a suitable concurrency theorem for non-linear SqPO-rewriting in categories that are quasi-topoi (subsuming the example of adhesive categories) and with matches required to be regular monomorphisms of the given category. Our construction reveals an interesting "backpropagation effect" in computing rule compositions. We derive in addition a concurrency theorem for non-linear double pushout (DPO) rewriting in rm-adhesive categories. Our results open non-linear SqPO and DPO semantics to the rich static analysis techniques available from concurrency, rule algebra and tracelet theory., Comment: 19+14 pages, LNCS style; ICGT 2021 conference paper extended version

"Spider" is a nickname of special Frobenius algebras, a fundamental structure from mathematics, physics, and computer science. Pregroups are a fundamental structure from linguistics. Pregroups and spiders have been used together in natural language processing: one for syntax, the other for semantics. It turns out that pregroups themselves can be characterized as pointed spiders in the category of preordered relations, where they naturally arise from grammars. The other way around, preordered spider algebras in general can be characterized as unions of pregroups. This extends the characterization of relational spider algebras as disjoint unions of groups. The compositional framework that emerged with the results suggests new ways to understand and apply the basis structures in machine learning and data analysis., Comment: 21 pages, 16 diagrams. Final journal version: DOI kindly inserted by Fosco Loregian

There are many open questions surrounding the characterisation of groups with context-sensitive word problem. Only in 2018 was it shown that all finitely generated virtually Abelian groups have multiple context-free word problems, and it is a long-standing open question as to where to place the word problems of hyperbolic groups in the formal language hierarchy. In this paper, we introduce a new language class called the parallel hyperedge replacement string languages, show that it contains all multiple context-free and ET0L languages, and lay down the foundations for future work that may be able to place the word problems of many hyperbolic groups in this class., Comment: In Proceedings TERMGRAPH 2020, arXiv:2102.01804. arXiv admin note: substantial text overlap with arXiv:2101.02310

In 2018, it was shown that all finitely generated virtually Abelian groups have multiple context-free word problems, and it is still an open problem as to where to precisely place the word problems of hyperbolic groups in the formal language hierarchy. Motivated by this, we introduce a new language class, the parallel hyperedge replacement string languages, containing all multiple context-free and ET0L languages. We show that parallel hyperedge replacement grammars can be "synchronised", which allows us to establish many useful formal language closure results relating to both the hypergraph and string languages generated by various families of parallel hyperedge replacement grammars, laying the foundations for future work in this area., Comment: 40 pages, 2021

In weighted automata theory, many classical results on formal languages have been extended into a quantitative setting. Here, we investigate weighted context-free languages of infinite words, a generalization of $\omega$-context-free languages (Cohen, Gold 1977) and an extension of weighted context-free languages of finite words (Chomsky, Sch\"utzenberger 1963). As in the theory of formal grammars, these weighted context-free languages, or $\omega$-algebraic series, can be represented as solutions of mixed $\omega$-algebraic systems of equations and by weighted $\omega$-pushdown automata. In our first main result, we show that (mixed) $\omega$-algebraic systems can be transformed into Greibach normal form. We use the Greibach normal form in our second main result to prove that simple $\omega$-reset pushdown automata recognize all $\omega$-algebraic series. Simple $\omega$-reset automata do not use $\epsilon$-transitions and can change the stack only by at most one symbol. These results generalize fundamental properties of context-free languages to weighted context-free languages.

A recursive descent parser is built from a set of mutually-recursive functions, where each function directly implements one of the nonterminals of a grammar. A packrat parser uses memoization to reduce the time complexity for recursive descent parsing from exponential to linear in the length of the input. Recursive descent parsers are extremely simple to write, but suffer from two significant problems: (i) left-recursive grammars cause the parser to get stuck in infinite recursion, and (ii) it can be difficult or impossible to optimally recover the parse state and continue parsing after a syntax error. Both problems are solved by the pika parser, a novel reformulation of packrat parsing as a dynamic programming algorithm, which requires parsing the input in reverse: bottom-up and right to left, rather than top-down and left to right. This reversed parsing order enables pika parsers to handle grammars that use either direct or indirect left recursion to achieve left associativity, simplifying grammar writing, and also enables optimal recovery from syntax errors, which is a crucial property for IDEs and compilers. Pika parsing maintains the linear-time performance characteristics of packrat parsing as a function of input length. The pika parser was benchmarked against the widely-used Parboiled2 and ANTLR4 parsing libraries. The pika parser performed significantly better than the other parsers for an expression grammar, although for a complex grammar implementing the Java language specification, a large constant performance impact was incurred per input character. Therefore, if performance is important, pika parsing is best applied to simple to moderate-sized grammars, or to very large inputs, if other parsing alternatives do not scale linearly in the length of the input. Several new insights into precedence, associativity, and left recursion are presented., Comment: Submitted to ACM

The Kappa biochemistry and the M{\O}D organo-chemistry frameworks are amongst the most intensely developed applications of rewriting theoretical methods in the life sciences to date. A typical feature of these types of rewriting theories is the necessity to implement certain structural constraints on the objects to be rewritten (a protein is empirically found to have a certain signature of sites, a carbon atom can form at most four bonds, ...). In this paper, we contribute to the theoretical foundations of these types of rewriting theory a number of conceptual and technical developments that permit to implement a universal theory of continuous-time Markov chains (CTMCs) for stochastic rewriting systems. Our core mathematical concepts are a novel rule algebra construction for the relevant setting of rewriting rules with conditions, both in Double- and in Sesqui-Pushout semantics, augmented by a suitable stochastic mechanics formalism extension that permits to derive dynamical evolution equations for pattern-counting statistics., Comment: 18+6 pages, LNCS style; ICGT 2020 conference paper extended version

Feature Structures (FSs) are a widespread tool used for decompositional frameworks of Attribute-Value associations. Even though they thrive in simple systems, they lack a way of representing higher-order entities and relations. This is however needed in Frame Semantics, where semantic dependencies should be able to connect groups of individuals and their properties, especially to model quantification. To answer this issue, this master report introduces wrappings as a way to envelop a sub-FS and treat it as a node. Following the work of [Kallmeyer, Osswald 2013], we extend its syntax, semantics and some properties (translation to FOL, subsumption, unification). We can then expand the proposed pipeline. Lexical minimal model sets are generated from formulas. They unify by FS value equations obtained by LTAG parsing to an underspecified sentence representation. The syntactic approach of quantifiers allows us to use existing methods to produce any possible reading. Finally, we give a transcription to type-logical formulas to interact with the context in the view of dynamic semantics. Supported by ideas of Frame Types, this system provides a workable and tractable tool for higher-order relations with FS., Comment: Master report

Lambek Grammars (LG) are a computational modelling of natural language, based on non-commutative compositional types. It has been widely studied, especially for languages where the syntax plays a major role (like English). The goal of this internship report is to demonstrate that every Lambek Grammar can be, not entirely but efficiently, expressed in Abstract Categorial Grammars (ACG). The latter is a novel modelling based on higher-order signature homomorphisms (using $\lambda$-calculus), aiming at uniting the currently used models. The main idea is to transform the type rewriting system of LGs into that of Context-Free Grammars (CFG) by erasing introduction and elimination rules and generating enough axioms so that the cut rule suffices. This iterative approach preserves the derivations and enables us to stop the possible infinite generative process at any step. Although the underlying algorithm was not fully implemented, this proof provides another argument in favour of the relevance of ACGs in Natural Language Processing., Comment: Bachelor internship report, in French

Binary "YES-NO" notions of process compliance are not very helpful to managers for assessing the operational performance of their company because a large number of cases fall in the grey area of partial compliance. Hence, it is necessary to have ways to quantify partial compliance in terms of metrics and be able to classify actual cases by assigning a numeric value of compliance to them. In this paper, we formulate an evaluation framework to quantify the level of compliance of business processes across different levels of abstraction (such as task,trace and process level) and across multiple dimensions of each task (such as temporal, monetary, role-, data-, and quality-related) to provide managers more useful information about their operations and to help them improve their decision making processes. Our approach can also add social value by making social services provided by local, state and federal governments more flexible and improving the lives of citizens., Comment: 15 page, 4 figures, 2 tables; Under consideration at AICOL 2020, co-located with Jurix

We consider the recommendations of the World Wide Web Consortium (W3C) about RDF framework and its associated query language SPARQL. We propose a new formal framework based on category theory which provides clear and concise formal definitions of the main basic features of RDF and SPARQL. We define RDF graphs as well as SPARQL basic graph patterns as objects of some nested categories. This allows one to clarify, in particular, the role of blank nodes. Furthermore, we consider basic SPARQL CONSTRUCT and SELECT queries and formalize their operational semantics following a novel algebraic graph transformation approach called POIM., Comment: In Proceedings GCM 2020, arXiv:2012.01181. arXiv admin note: substantial text overlap with arXiv:1910.07519