# Separating Graph Logic from MSO

• Timos Antonopoulos
• Anuj Dawar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5504)

## Abstract

Graph logic (GL) is a spatial logic for querying graphs introduced by Cardelli et al. It has been observed that in terms of expressive power, this logic is a fragment of Monadic Second Order Logic (MSO), with quantification over sets of edges. We show that the containment is proper by exhibiting a property that is not GL definable but is definable in MSO, even in the absence of quantification over labels. Moreover, this holds when the graphs are restricted to be forests and thus strengthens in several ways a result of Marcinkowski. As a consequence we also obtain that Separation Logic, with a separating conjunction but without the magic wand, is strictly weaker than MSO over memory heaps, settling an open question of Brochenin et al.

## References

1. 1.
Brochenin, R., Demri, S., Lozes, É.: On the almighty wand. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 323–338. Springer, Heidelberg (2008)
2. 2.
Cardelli, L., Gardner, P., Ghelli, G.: A spatial logic for querying graphs. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 597–610. Springer, Heidelberg (2002)
3. 3.
Compton, K.J.: A logical approach to asymptotic combinatorics II: Monadic second-order properties. J. Comb. Theory, Ser. A 50(1), 110–131 (1989)
4. 4.
Courcelle, B.: The expression of graph properties and graph transformations in monadic second-order logic. In: Rozenberg, G. (ed.) Handbook of Graph Grammars, pp. 313–400. World Scientific, Singapore (1997)Google Scholar
5. 5.
Dawar, A., Gardner, P., Ghelli, G.: Adjunct elimination through games in static ambient logic(Extended abstract). In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 211–223. Springer, Heidelberg (2004)
6. 6.
Dawar, A., Gardner, P., Ghelli, G.: Expressiveness and complexity of graph logic. Inf. Comput. 205(3), 263–310 (2007)
7. 7.
Ebbinghaus, H.-D., Flum, J.: Finite Model Theory, 2nd edn. Springer, Heidelberg (1999)
8. 8.
Grädel, E., Hirsch, C., Otto, M.: Back and forth between guarded and modal logics. ACM Trans. Comput. Log. 3(3), 418–463 (2002)
9. 9.
Marcinkowski, J.: On the expressive power of graph logic. In: Ésik, Z. (ed.) CSL 2006. LNCS, vol. 4207, pp. 486–500. Springer, Heidelberg (2006)
10. 10.
Reynolds, J.C.: Separation logic: A logic for shared mutable data structures. In: LICS, pp. 55–74. IEEE Computer Society, Los Alamitos (2002)Google Scholar