Abstract
We show how edge-labelled graphs can be used to represent first-order logic formulae. This gives rise to recursively nested structures, in which each level of nesting corresponds to the negation of a set of existentials. The model is a direct generalisation of the negative application conditions used in graph rewriting, which count a single level of nesting and are thereby shown to correspond to the fragment ∃ ¬ ∃ of first-order logic. Vice versa, this generalisation may be used to strengthen the notion of application conditions. We then proceed to show how these nested models may be flattened to (sets of) plain graphs, by allowing some structure on the labels. The resulting formulae-as-graphs may form the basis of a unification of the theories of graph transformation and predicate transformation.
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
Bauderon, M., Courcelle, B.: Graph expressions and graph rewritings. Mathematical Systems Theory 20(2-3), 83–127 (1987)
Chein, M., Mugnier, M.-L., Simonet, G.: Nested graphs: A graph-based knowledge representation model with FOL semantics. In: Cohn, Schubert, Shapiro (eds.) Principles of Knowledge Representation and Reasoning, pp. 524–535. Morgan Kaufmann, San Francisco (1998)
Corradini, A., Montanari, U., Rossi, F., Ehrig, H., Heckel, R., Löwe, M.: Algebraic approaches to graph transformation, part I: Basic concepts and double pushout approach. In: Rozenberg [16], ch. 3, pp. 163–246
Courcelle, B.: The monadic second-order logic of graphs XI: Hierarchical decompositions of connected graphs. Theoretical Comput. Sci. 224(1-2), 35–58 (1999)
Dau, F.: The Logic System of Concept Graphs with Negation. LNCS, vol. 2892. Springer, Heidelberg (2003)
Ehrig, H., Habel, A.: Graph grammars with application conditions. In: Rozenberg, Salomaa (eds.) The Book of L, pp. 87–100. Springer, Heidelberg (1986)
Ehrig, H., Ehrig, K., Habel, A., Pennemann, K.-H.: Constraints and application conditions: From graphs to high-level structures. In: Ehrig, H., Engels, G., Parisi-Presicce, F., Rozenberg, G. (eds.) ICGT 2004. LNCS, vol. 3256, pp. 287–303. Springer, Heidelberg (2004)
Ehrig, H., Heckel, R., Korff, M., Löwe, M., Ribeiro, L., Wagner, A., Corradini, A.: Algebraic approaches to graph transformation, part II: Single pushout approach and comparison with double pushout approach. In: Rozenberg, G. [16], pp. 247–312
Ermel, C., Rudolf, M., Taentzer, G.: The AGG approach: language and environment. In: Ehrig, et al. (eds.) Handbook of Graph Grammars and Computing by Graph Transformation. Applications, Languages and Tools, vol. II, World Scientific, Singapore (1999)
Habel, A., Heckel, R., Taentzer, G.: Graph grammars with negative application conditions. Fundamenta Informaticae 26(3/4), 287–313 (1996)
Heckel, R., Wagner, A.: Ensuring consistency of conditional graph grammars — a constructive approach. ENTCS, vol. 2 (1995)
Koch, M., Mancini, L.V., Parisi-Presicce, F.: Conflict detection and resolution in access control policy specifications. In: Nielsen, M., Engberg, U. (eds.) FOSSACS 2002. LNCS, vol. 2303, pp. 223–238. Springer, Heidelberg (2002)
Pierik, C., de Boer, F.S.: A syntax-directed hoare logic for object-oriented programming concepts. In: Najm, E., Nestmann, U., Stevens, P. (eds.) FMOODS 2003. LNCS, vol. 2884, pp. 64–78. Springer, Heidelberg (2003)
Roberts, D.D.: The Existential Graphs of Charles S. Peirce. Mouton and Co., Netherlands (1973)
Roberts, D.D.: The existential graphs. Computers and Mathematics with Applications 6, 639–663 (1992)
Rozenberg, G. (ed.): Handbook of Graph Grammars and Computing by Graph Transformation. Foundations, vol. I. World Scientific, Singapore (1997)
Sowa, J.F.: Conceptual Structures: Information Processing in Mind and Machine. Addison-Wesley, Reading (1984)
Wermelinger, M.: Conceptual graphs and first-order logic. In: Ellis, G., Rich, W., Levinson, R., Sowa, J.F. (eds.) ICCS 1995. LNCS, vol. 954, pp. 323–337. Springer, Heidelberg (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Rensink, A. (2004). Representing First-Order Logic Using Graphs. In: Ehrig, H., Engels, G., Parisi-Presicce, F., Rozenberg, G. (eds) Graph Transformations. ICGT 2004. Lecture Notes in Computer Science, vol 3256. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30203-2_23
Download citation
DOI: https://doi.org/10.1007/978-3-540-30203-2_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-23207-0
Online ISBN: 978-3-540-30203-2
eBook Packages: Springer Book Archive