Abstract. Most of the logics commonly used in verification, such as LTL, CTL, CTL∗, and PDL can be embedded into the two-variable fragment of the µ-calculus. It is also known that properties occurring at arbitrarily high levels of the alternation hierarchy can be formalised using only two variables. This raises the question of whether the number of fixed-point variables in µ-formulae can be bounded in general. We answer this question negatively and prove that the variable-hierarchy of the µ-calculus is semantically strict. For any k, we provide examples of formulae with k variables that are not equivalent to any formula with fewer variables. In particular, this implies that Parikh’s Game Logic is less expressive than the µ-calculus, thus resolving an open issue raised by Parikh in 1983.

The variable hierarchy of the $mu$-calculus is strict.

LENZI, Giacomo
2007-01-01

Abstract

Abstract. Most of the logics commonly used in verification, such as LTL, CTL, CTL∗, and PDL can be embedded into the two-variable fragment of the µ-calculus. It is also known that properties occurring at arbitrarily high levels of the alternation hierarchy can be formalised using only two variables. This raises the question of whether the number of fixed-point variables in µ-formulae can be bounded in general. We answer this question negatively and prove that the variable-hierarchy of the µ-calculus is semantically strict. For any k, we provide examples of formulae with k variables that are not equivalent to any formula with fewer variables. In particular, this implies that Parikh’s Game Logic is less expressive than the µ-calculus, thus resolving an open issue raised by Parikh in 1983.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/1870442
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 17
  • ???jsp.display-item.citation.isi??? 11
social impact