We study classes of graphs with bounded clique-width that are well-quasi-ordered by the induced subgraph relation, in the presence of labels on the vertices. We prove that, given a finite presentation of a class of graphs, one can decide whether the class is labelled-well-quasi-ordered. This answers positively to two conjectures of Pouzet in the restricted case of bounded clique-width classes. Namely, we prove that being labelled-well-quasi-ordered by a set of size 2 or by a well-quasi-ordered infinite set are equivalent conditions, and that in such cases, one can freely assume that the graphs are equipped with a total ordering on their vertices. Finally, we provide a structural characterization of those classes as those that are of bounded clique-width and do not existentially transduce the class of all finite paths.

Abstract We introduce a high-level language with Python-like syntax for string-to-string, polyregular, first-order definable transductions. This language features function calls, boolean variables, and nested for-loops. We devise and implement a complete decision procedure for the verification of such programs against a first-order specification. The decision procedure reduces the verification problem to the decidable first-order theory of finite words (extensively studied in automata theory), which we discharge using either complete tools specific to this theory (MONA), or to general-purpose SMT solvers (Z3, CVC5).

We construct an algorithm that inputs an MSO-interpretation from finite words to graphs, and decides if there exists a k ∈ ℕ such that the class of graphs induced by the interpretation is not well-quasi-ordered by the induced subgraph relation when vertices are freely labelled using {1, …, k}. In case no such k exists, we also prove that the class of graphs is not well-quasi-ordered by the induced subgraph relation when vertices are freely labelled using any well-quasi-ordered set of labels. As a byproduct of our analysis, we prove that for classes of bounded linear clique-width, a weak version of a conjecture by Pouzet holds.

This paper studies which functions computed by ℤ-weighted automata can be realised by ℕ-weighted automata, under two extra assumptions: commutativity (the order of letters in the input does not matter) and polynomial growth (the output of the function is bounded by a polynomial in the size of the input). We leverage this effective characterization to decide whether a function computed by a commutative ℕ-weighted automaton of polynomial growth is star-free, a notion borrowed from the theory of regular languages that has been the subject of many investigations in the context of string-to-string functions during the last decade.

Abstract Noetherian spaces are a generalisation of well-quasi-orderings to topologies, that can be used to prove termination of programs. They find applications in the verification of transition systems, some of which are better described using topology. The goal of this paper is to allow the systematic description of computations using inductively defined datatypes via Noetherian spaces. This is achieved through a fixed point theorem based on a topological minimal bad sequence argument.

This paper studies a robust class of functions from finite words to integers that we call ℤ-polyregular functions. We show that it admits natural characterizations in terms of logics, ℤ-rational expressions, ℤ-rational series and transducers.We then study two subclass membership problems. First, we show that the asymptotic growth rate of a function is computable, and corresponds to the minimal number of variables required to represent it using logical formulas. Second, we show that first-order definability of ℤ-polyregular functions is decidable. To show the latter, we introduce an original notion of residual transducer, and provide a semantic characterization based on aperiodicity.

This paper investigates the expressiveness of a fragment of first-order sentences in Gaifman normal form, namely the positive Boolean combinations of basic local sentences. We show that they match exactly the first-order sentences preserved under local elementary embeddings, thus providing a new general preservation theorem and extending the Łós-Tarski Theorem.

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The "generic operational metatheory" of Johann, Simpson and Voigtländer (LiCS 2010) defines contextual equivalence, in the presence of algebraic effects, in terms of a basic operational preorder on ground-type effect trees. We propose three general approaches to specifying such preorders: (i) operational (ii) denotational, and (iii) axiomatic; coinciding with the three major styles of program semantics. We illustrate these via a nontrivial case study: the combination of probabilistic choice with nondeterminism, for which we show that natural instantiations of the three specification methods (operational in terms of Markov decision processes, denotational using a powerdomain, and axiomatic) all determine the same canonical preorder. We do this in the case of both angelic and demonic nondeterminism.

We introduce a general diagrammatic theory of digital circuits, based on connections between monoidal categories and graph rewriting. The main achievement of the paper is conceptual, filling a foundational gap in reasoning syntactically and symbolically about a large class of digital circuits (discrete values, discrete delays, feedback). This complements the dominant approach to circuit modelling, which relies on simulation. The main advantage of our symbolic approach is the enabling of automated reasoning about parametrised circuits, with a potentially interesting new application to partial evaluation of digital circuits. Relative to the recent interest and activity in categorical and diagrammatic methods, our work makes several new contributions. The most important is establishing that categories of digital circuits are Cartesian and admit, in the presence of feedback expressive iteration axioms. The second is producing a general yet simple graph-rewrite framework for reasoning about such categories in which the rewrite rules are computationally efficient, opening the way for practical applications.