There is extensive literature on automated test generation using reactive design models, where control is determined by events. In contrast, the (idealised) simulation paradigm defines control through cycles dictated by the passage of time. Within each cycle, inputs are read and processed, and outputs are provided, all instantaneously, and afterwards time progresses. To exercise a simulation using tests generated from a reactive design model requires changes to the tests to take into account this paradigm shift. This paper focuses on automation of the necessary changes and of the use of the resulting tests in a simulation campaign. Based on a notion of conformance that establishes whether a simulation is correct with respect to a reactive design, we (1) identify the reactive tests that are meaningful; (2) define a process to convert those tests; (3) provide an algorithm to execute those tests; and (4) prove soundness and completeness of our approach. Our work is described in the context of the RoboStar framework for model-based development of control software for robotics applications, and its process algebraic semantics. The testing approach we propose here represents a significant advancement in the current testing practices within the field of robotics, where simulations are widely used.

The Weisfeiler-Leman dimension of a graph \(G\) is the least number \(k\) such that the \(k\) -dimensional Weisfeiler-Leman algorithm distinguishes \(G\) from every other non-isomorphic graph, or equivalently, the least \(k\) such that \(G\) is definable in \((k+1)\) -variable logic with counting. The dimension is a standard measure of the descriptive or structural complexity of a graph and recently finds various applications in particular in the context of machine learning. This paper studies the complexity of computing the Weisfeiler-Leman dimension. We observe that deciding whether the Weisfeiler-Leman dimension of \(G\) is at most \(k\) is NP -hard, even if \(G\) is restricted to have \(4\) -bounded color classes. Therefore, we study parameterized versions of the problem. For each fixed \(k\geq 2\) , we give a polynomial-time algorithm that decides whether the Weisfeiler-Leman dimension of a given graph with \(5\) -bounded color classes is at most \(k\) . Moreover, we show that for these bounds on the color classes, this is optimal because the problem is P -hard under logspace-uniform AC 0 -reductions. Furthermore, for each larger bound \(c\) on the color classes and each fixed \(k\geq 2\) , we provide a polynomial-time decision algorithm for the abelian case, that is, for structures of which each color class has an abelian automorphism group. While the graph classes we consider may seem quite restrictive, graphs with \(4\) -bounded abelian colors include CFI-graphs and multipedes, which form the basis of almost all known hard instances and lower bounds related to the Weisfeiler-Leman algorithm.

We consider a class of formula equations in first-order logic, Horn formula equations, which are defined by a syntactic restriction on the occurrences of predicate variables. Horn formula equations play an important role in many applications in computer science. We state and prove a fixed-point theorem for Horn formula equations in first-order logic with a least fixed-point operator. Our fixed-point theorem is abstract in the sense that it applies to an abstract semantics which generalises standard semantics. We describe several corollaries of this fixed-point theorem in various areas of computational logic, ranging from the logical foundations of program verification to inductive theorem proving.

In this article, we consider the interplay of generalized quantifiers and built-in relations over finite structures, in particular, in the range of logics capturing the circuit complexity classes \(\mathrm{AC^{0}}\) and \(\mathrm{TC^{0}}\) . It is well known that for capturing \(\mathrm{AC^{0}}\) first-order logic has to be equipped with order and, e.g., predicates for addition and multiplication, whereas for \(\mathrm{TC^{0}}\) generalized quantifiers such as majority quantifiers are necessary. The sharp division between the classes \(\mathrm{AC^{0}}\) and \(\mathrm{TC^{0}}\) can be explained by the fact that \(\mathrm{AC^{0}}\) is not closed under restricting \(\mathrm{AC^{0}}\) -computable queries into simple subsequences of the input, whereas \(\mathrm{TC^{0}}\) is closed under such relativization as its queries can be expressed in terms of first-order formulas using universe-independent generalized quantifiers and order as the only built-in relation. In the terminology of abstract logics, the above means that logics capturing \(\mathrm{AC^{0}}\) do not have the relativization property, and hence, they are not regular logics unlike the logics capturing \(\mathrm{TC^{0}}\) . This weakness of \(\mathrm{AC^{0}}\) has been also elaborated in the line of research on the Crane Beach Conjecture. The conjecture (which was refuted by Barrington et al.) was that if a language \( L \) has a neutral letter, then \( L \) can be defined in \(\operatorname{FO}_{\mathcal{A}}\) , first-order logic with the collection of all numerical built-in relations \(\mathcal{A}\) , if and only if \( L \) can be already defined in \(\operatorname{FO}_{\leq}\) . Our approach is two-fold. First, we study universe-independent cardinality quantifiers \(\operatorname{\mathsf{Q}}\) defined by a parameter set \(S\subseteq\mathbb{N}\) and formulate a combinatorial criterion for \( S \) implying that all languages in \(\mathrm{DLOGTIME}\) -uniform \(\mathrm{TC^{0}}\) can be defined in \(\operatorname{FO}_{\leq}(\operatorname{\mathsf{Q}})\) . For instance, this criterion is satisfied if \( S \) is the range of some polynomial with positive integer coefficients of degree at least two. Second, by adapting the key properties of abstract logics to accommodate built-in relations, we define the regular interior \(\operatorname{\mathcal{R}-int}(\mathcal{L})\) (the largest regular \(\mathcal{L}^{*}\) such that \(\mathcal{L}^{*}\subseteq\mathcal{L}\) ) and regular closure \(\operatorname{\mathcal{R}-cl}(\mathcal{L})\) (the least regular \(\mathcal{L}^{*}\) such that \(\mathcal{L}\subseteq\mathcal{L}^{*}\) ), of a logic \(\mathcal{L}\) with built-in relations, and show that the Crane Beach Conjecture can be interpreted as a statement concerning the regular interior of \(\mathcal{L}\) . By extending the results of Barrington et al., we further show that if \(\mathcal{B}=\{+\}\) , or \(\mathcal{B}\) contains only unary relations besides \(\leq\) , then \(\operatorname{\mathcal{R}-int}(\operatorname{FO}_{\mathcal{B}})\equiv \operatorname{FO}_{\leq}\) . In contrast, our results from the first part of the article imply that if \(\mathcal{B}\) contains \(\leq\) and the range of a polynomial of degree at least two, then \(\operatorname{\mathcal{R}-cl}(\operatorname{FO}_{\mathcal{B}})\) includes all languages in \(\mathrm{DLOGTIME}\) -uniform \(\mathrm{TC^{0}}\) .

We introduce an operator on classes of regular languages, the star-free closure. Our motivation is to generalize standard results of automata theory within a unified framework. Given an arbitrary input class \(\mathscr{C}\) , the star-free closure operator outputs the least class closed under Boolean operations and language concatenation, and containing all languages of \(\mathscr{C}\) as well as all finite languages. We establish several equivalent characterizations of star-free closure: in terms of regular expressions, first-order logic, pure future and future-past temporal logic, and recognition by finite monoids. A key ingredient is that star-free closure coincides with another closure operator, defined in terms of regular operations where Kleene stars are allowed in restricted contexts. A consequence of this first result is that we can decide membership of a regular language in the star-free closure of a class whose separation problem is decidable. Moreover, we prove that separation itself is decidable for the star-free closure of any finite class, and of any class of group languages having itself decidable separation (plus mild additional properties). We actually show decidability of a stronger property, called covering.

The 1 -in- 3 and N ot -A ll -E qual satisfiability problems for Boolean CNF formulas are two well-known NP -hard problems. In contrast, the promise 1 -in- 3 vs . N ot -A ll -E qual problem can be solved in polynomial time. In the present work, we investigate this constraint satisfaction problem in a regime where the promise is weakened from either side by a rainbow-free structure and establish a complexity dichotomy for the resulting class of computational problems.

In this article, the QCSP (Quantified Constraint Satisfaction Problems) framework is shifted to the QCHR (Quantified Constraint Handling Rules) framework by enabling dynamic generation of quantifiers and access to user-defined constraints. QCSP offers a natural framework to express PSPACE problems as finite two-players games. But to define a QCSP model, the alternation of quantifiers must be formerly known and cannot be built dynamically even if the worst case will not occur. To overcome this issue, the new QCHR formalism, that allows to generate quantifiers dynamically during the solving, is defined. QCHR models exhibit state-of-the-art performances on a priori fixed alternation of quantifiers and outperforms previous QCSP approaches when the generation of quantifiers is dynamic.

We analyse abstract data types that model numerical structures with a concept of error. Specifically, we focus on arithmetic data types that contain an error value \(\bot\) whose main purpose is to always return a value for division. To rings and fields, we add a division operator \(x/y\) and study a class of algebras called common meadows wherein \(x/0=\bot\) . The set of equations true in all common meadows is named the equational theory of common meadows . We give a finite equational axiomatisation of the equational theory of common meadows and prove that it is complete and that the equational theory is decidable.

We propose a novel SAT-based approach to graph generation. Our approach utilizes the interaction between a CDCL SAT solver and a special symmetry propagator where the SAT solver runs on an encoding of the desired graph property. The symmetry propagator checks partially generated graphs for minimality with respect to a lexicographic ordering during the solving process. This approach has several advantages over a static symmetry breaking: (i) symmetries are detected early in the generation process, (ii) symmetry breaking is seamlessly integrated into the CDCL procedure, and (iii) the propagator performs a complete symmetry breaking without causing a prohibitively large initial encoding. We instantiate our approach by generating extremal graphs with certain restrictions in terms of forbidden subgraphs and diameter. In particular, we could confirm the Murty–Simon Conjecture (1979) on diameter-2-critical graphs for graphs up to 19 vertices and prove the exact number of Ramsey graphs \(\mathcal{R}(3,5,n)\) and \(\mathcal{R}(4,4,n)\) .

The k -spectrum is the set of all α > 0 such that G(n,n −α ) does not obey the 0-1 law for FO sentences with quantifier depth at most k . In this article, we prove that the minimum k such that the k -spectrum is infinite equals 5.