Skip to main content

Fragments of Spider Diagrams of Order and Their Relative Expressiveness

  • Conference paper
Diagrammatic Representation and Inference (Diagrams 2010)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 6170))

Included in the following conference series:

  • 1434 Accesses


Investigating the expressiveness of a diagrammatic logic provides insight into how its syntactic elements interact at the semantic level. Moreover, it allows for comparisons with other notations. Various expressiveness results for diagrammatic logics are known, such as the theorem that Shin’s Venn-II system is equivalent to monadic first order logic. The techniques employed by Shin for Venn-II were adapted to allow the expressiveness of Euler diagrams to be investigated. We consider the expressiveness of spider diagrams of order (SDoO), which extend spider diagrams by including syntax that provides ordering information between elements. Fragments of SDoO are created by systematically removing each aspect of the syntax. We establish the relative expressiveness of the various fragments. In particular, one result establishes that spiders are syntactic sugar in any fragment that contains order, negation and shading. We also show that shading is syntactic sugar in any fragment containing negation and spiders. The existence of syntactic redundancy within the spider diagram of order logic is unsurprising however, we find it interesting that spiders or shading are redundant in fragments of the logic. Further expressiveness results are presented throughout the paper. The techniques we employ may well extend to related notations, such as the Euler/Venn logic of Swoboda et al. and Kent’s constraint diagrams.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others


  1. Kent, S.: Constraint diagrams: Visualizing invariants in object oriented modelling. In: Proceedings of OOPSLA 1997, pp. 327–341. ACM Press, New York (1997)

    Google Scholar 

  2. Dau, F.: The Logic System of Concept Graphs with Negations: And its Relationship to Prediacte Logic. Springer, Heidelberg (2003)

    Book  MATH  Google Scholar 

  3. Hammer, E.: Logic and Visual Information. CSLI Publications, Stanford (1995)

    MATH  Google Scholar 

  4. Swoboda, N., Allwein, G.: Using DAG transformations to verify Euler/Venn homogeneous and Euler/Venn FOL heterogeneous rules of inference. Journal on Software and System Modeling 3, 136–149 (2004)

    Article  Google Scholar 

  5. Howse, J., Stapleton, G., Taylor, J.: Spider diagrams. LMS Journal of Computation and Mathematics 8, 145–194 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Shin, S.J.: The Logical Status of Diagrams. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  7. Stapleton, G., Howse, J., Taylor, J.: A decidable constraint diagram reasoning system. Journal of Logic and Computation 15(6), 975–1008 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Delaney, A., Stapleton, G.: Spider diagrams of order. In: International Workshop on Visual Languages and Logic (September 2007)

    Google Scholar 

  9. Stapleton, G., Masthoff, J.: Incorporating negation into visual logics: A case study using Euler diagrams. In: Visual Languages and Computing 2007, pp. 187–194. Knowledge Systems Institute (2007)

    Google Scholar 

  10. Stapleton, G., Thompson, S., Howse, J., Taylor, J.: The expressiveness of spider diagrams. Journal of Logic and Computation 14(6), 857–880 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Delaney, A., Taylor, J., Thompson, S.: Spider diagrams of order and a hierarchy of star-free regular languages. In: Stapleton, G., Howse, J., Lee, J. (eds.) Diagrams 2008. LNCS (LNAI), vol. 5223, pp. 172–187. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  12. Thomas, W.: Classifying regular events in symbolic logic. Journal of Computer and System Sciences 25, 360–376 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ebbinghaus, H.D., Flum, J.: Finite Model Theory, 2nd edn. Springer, Heidelberg (1991)

    MATH  Google Scholar 

  14. Manzano, M.: Model Theory. Oxford University Press, Oxford (1999)

    MATH  Google Scholar 

  15. Delaney, A., Stapleton, G., Taylor, J., Thompson, S.: A diagrammatic characterisation of commutative star-free regular languages (in preparation)

    Google Scholar 

Download references

Author information

Authors and Affiliations


Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Delaney, A., Stapleton, G., Taylor, J., Thompson, S. (2010). Fragments of Spider Diagrams of Order and Their Relative Expressiveness. 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.

Download citation

  • DOI:

  • 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)

Publish with us

Policies and ethics