A Note on Negative Tagging for Least Fixed-Point Formulae

¹ Swedish Institute of Computer Science, Box 1263, SE-164 29 Kista, Sweden; dilian@sics.se.
² Department of Computer Science, University of Victoria, Victoria, B.C., Canada V8W 3P6; bmkapron@csc.uvic.ca.

Received: 30 October 1998
Revised: 1 June 1999

Abstract

Proof systems with sequents of the form U ⊢ Φ for proving validity of a propositional modal μ-calculus formula Φ over a set U of states in a given model usually handle fixed-point formulae through unfolding, thus allowing such formulae to reappear in a proof. Tagging is a technique originated by Winskel for annotating fixed-point formulae with information about the proof states at which these are unfolded. This information is used later in the proof to avoid unnecessary unfolding, without having to investigate the history of the proof. Depending on whether tags are used for acceptance or for rejection of a branch in the proof tree, we refer to “positive” or “negative” tagging, respectively. In their simplest form, tags consist of the sets U at which fixed-point formulae are unfolded. In this paper, we generalise results of earlier work by Andersen et al. which, in the case of least fixed-point formulae, are applicable to singleton U sets only.