Abstract
We present a system for deriving inclusions between graphs from a set of inclusions between graphs taken as hypotheses. The novel features are the extended notion of graph with an explicitly representation of complement, the more involved definition of the system, and its completeness proof due to the embedding of complements. This is an improvement on former work, where complement was introduced by definition. Our calculus provides a basis on which one can construct a wide range of graph calculi for several algebras of relations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Andréka, H., Bredikhin, D.A.: The equational theory of union-free algebras of relations. Algebra Universalis 33, 516–532 (1995)
Brown, C., Hutton, G.: Categories, allegories and circuit design. LICS 94, 372–381 (1994)
Brown, C., Jeffrey, A.: Allegories of circuits. In: Proc. Logical Foundations of Computer Science, pp. 56–68. St. Petersburg (1994)
Bredikhin, D.A.: On clones generated by primitive-positive operations of Tarskis relation algebras. Algebra Universalis 38, 165–174 (1997)
Cantone, D., Formisano, A., Omodeo, E.G., Zarba, C.G.: Compiling dyadic specifications into map algebra. Theoret. Comput. Sci. 293, 447–475 (2003)
Curtis, S., Lowe, G.: Proofs with graphs. Sci. Comput. Programming 26, 197–216 (1996)
Formisano, A., Omodeo, E.G., Simeoni, M.: A graphical approach to relational reasoning. ENTCS 44, 1–22 (2003)
Freyd, P.J., Scedrov, A.: Categories, Allegories. Elsevier, Amsterdam (1990)
de Freitas, R., Viana, P.: A note on proofs with graphs. Sci. Comput. Programming 73, 129–135 (2008)
de Freitas, R., Veloso, P.A.S., Veloso, S.R.M., Viana, P.: On positive relational calculi. Logic J. IGPL 15, 577–601 (2007)
de Freitas, R., Veloso, P.A.S., Veloso, S.R.M., Viana, P.: On a graph calculus for algebras of relations. In: Hodges, W., de Queiroz, R. (eds.) WoLLIC 2008. LNCS (LNAI), vol. 5110, pp. 298–312. Springer, Heidelberg (2008)
de Freitas, R., Veloso, P.A.S., Veloso, S.R.M., Viana, P.: Positive fork graph calculus. In: Artemov, S., Nerode, A. (eds.) LFCS 2009. LNCS, vol. 5407, pp. 152–163. Springer, Heidelberg (2008)
de Freitas, R., Veloso, P.A.S., Veloso, S.R.M., Viana, P.: On graph reasoning. Information and Computation 207, 1000–1014 (2009)
Grätzer, G.: Universal Algebra. Springer, New York (1979)
Hirsch, R., Hodkinson, I.: Relation Algebras by Games. Elsevier, Amsterdam (2002)
Hutton, G.: A relational derivation of a functional program. In: Lecture Notes of the STOP Summer School on Constructive Algorithms (1992)
Kahl, W.: Algebraic graph derivations for graphical calculi. In: D’Amore, F., Marchetti-Spaccamela, A., Franciosa, P.G. (eds.) WG 1996. LNCS, vol. 1197, pp. 224–238. Springer, Heidelberg (1997)
Kahl, W.: Relational treatment of term graphs with bound variables. Logic J. IGPL 6, 259–303 (1998)
Kahl, W.: Relational matching for graphical calculi of relations. Inform. Sciences 119, 253–273 (1999)
Maddux, R.: Relation Algebras. Elsevier, Amsterdam (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
de Freitas, R., Veloso, P.A.S., Veloso, S.R.M., Viana, P. (2010). A Calculus for Graphs with Complement. In: Goel, A.K., Jamnik, M., Narayanan, N.H. (eds) Diagrammatic Representation and Inference. Diagrams 2010. Lecture Notes in Computer Science(), vol 6170. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14600-8_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-14600-8_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14599-5
Online ISBN: 978-3-642-14600-8
eBook Packages: Computer ScienceComputer Science (R0)