42 research outputs found
Solving Linear Constraints in Elementary Abelian p-Groups of Symmetries
Symmetries occur naturally in CSP or SAT problems and are not very difficult
to discover, but using them to prune the search space tends to be very
challenging. Indeed, this usually requires finding specific elements in a group
of symmetries that can be huge, and the problem of their very existence is
NP-hard. We formulate such an existence problem as a constraint problem on one
variable (the symmetry to be used) ranging over a group, and try to find
restrictions that may be solved in polynomial time. By considering a simple
form of constraints (restricted by a cardinality k) and the class of groups
that have the structure of Fp-vector spaces, we propose a partial algorithm
based on linear algebra. This polynomial algorithm always applies when k=p=2,
but may fail otherwise as we prove the problem to be NP-hard for all other
values of k and p. Experiments show that this approach though restricted should
allow for an efficient use of at least some groups of symmetries. We conclude
with a few directions to be explored to efficiently solve this problem on the
general case.Comment: 18 page
True Parallel Graph Transformations: an Algebraic Approach Based on Weak Spans
21 pages, 5 figuresWe address the problem of defining graph transformations by the simultaneous application of direct transformations even when these cannot be applied independently of each other. An algebraic approach is adopted, with production rules of the form , called weak spans. A parallel coherent transformation is introduced and shown to be a conservative extension of the interleaving semantics of parallel independent direct transformations. A categorical construction of finitely attributed structures is proposed, in which parallel coherent transformations can be built in a natural way. These notions are introduced and illustrated on detailed examples
Monographs, a Category of Graph Structures
International audienceDoes a graph necessarily have nodes? May an edge be adjacent to itself and be a self-loop? These questions arise in the study of graph structures, i.e., monadic many-sorted signatures and the corresponding algebras. A simple notion of monograph is proposed that generalizes the standard notion of directed graph and can be drawn consistently with them. It is shown that monadic many-sorted signatures can be represented by monographs, and that the corresponding algebras are isomorphic to the monographs typed by the corresponding signature monograph. Monographs therefore provide a simple unifying framework for working with monadic algebras. Their simplicity is illustrated by deducing some of their categorial properties from those of sets
Parallel Independence in Attributed Graph Rewriting
In order to define graph transformations by the simultaneous application of
concurrent rules, we have adopted in previous work a structure of attributed
graphs stable by unions. We analyze the consequences on parallel independence,
a property that characterizes the possibility to resort to sequential
rewriting. This property turns out to depend not only on the left-hand side of
rules, as in algebraic approaches to graph rewriting, but also on their
right-hand side. It is then shown that, of three possible definitions of
parallel rewriting, only one is convenient in the light of parallel
independence.Comment: In Proceedings TERMGRAPH 2020, arXiv:2102.0180
Subsumptions of Algebraic Rewrite Rules
What does it mean for an algebraic rewrite rule to subsume another rule (that may then be called a subrule)? We view subsumptions as rule morphisms such that the simultaneous application of a rule and a subrule (i.e. the application of a subsumption morphism) yields the same result as a single application of the subsuming rule. Simultaneous applications of categories of rules are obtained by Global Coherent Transformations and illustrated on graphs in the DPO approach. Other approaches are possible since these transformations are formulated in an abstract Rewriting Environment, and such environments exist for various approaches to Algebraic Rewriting, including DPO, SqPO and PBPO
Optimisation par renommage dans la méthode de résolution
La technique du renommage, appliquée exhaustivement, permet d'obtenir une forme clausale polynomiale. Nous choisissons de l'appliquer partiellement, de façon a minimiser certains criteres syntaxiques, principalement le nombre de clauses, tout en conservant une complexité polynomiale. Nous montrons qu'un algorithme efficace permet d'obtenir le nombre optimal de clauses sur les formules linéaires. Enfin, nous étudions l'influence de ces transformations sur les réfutations par la methode de resolution, autant théoriquement expérimentalemen
Algebraic Monograph Transformations
Equational logic and rewritingMonographs 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 8-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