2 research outputs found
Weighted Automata and Monadic Second Order Logic
Let S be a commutative semiring. M. Droste and P. Gastin have introduced in
2005 weighted monadic second order logic WMSOL with weights in S. They use a
syntactic fragment RMSOL of WMSOL to characterize word functions (power series)
recognizable by weighted automata, where the semantics of quantifiers is used
both as arithmetical operations and, in the boolean case, as quantification.
Already in 2001, B. Courcelle, J.Makowsky and U. Rotics have introduced a
formalism for graph parameters definable in Monadic Second order Logic, here
called MSOLEVAL with values in a ring R. Their framework can be easily adapted
to semirings S. This formalism clearly separates the logical part from the
arithmetical part and also applies to word functions.
In this paper we give two proofs that RMSOL and MSOLEVAL with values in S
have the same expressive power over words. One proof shows directly that
MSOLEVAL captures the functions recognizable by weighted automata. The other
proof shows how to translate the formalisms from one into the other.Comment: In Proceedings GandALF 2013, arXiv:1307.416
Connection Matrices and the Definability of Graph Parameters
In this paper we extend and prove in detail the Finite Rank Theorem for
connection matrices of graph parameters definable in Monadic Second Order Logic
with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and
J.A. Makowsky (2009). We demonstrate its vast applicability in simplifying
known and new non-definability results of graph properties and finding new
non-definability results for graph parameters. We also prove a Feferman-Vaught
Theorem for the logic CFOL, First Order Logic with the modular counting
quantifiers