The decidability of a fragment of BB'IW-logic

Authors:
Sabine Broda;Luís Damas;Marcelo Finger;Paulo Silva e Silva
Affiliations:
DCC-FC, Universidade do Porto, Rua do Campo Alegre 823, 4150 Porto, Portugal;DCC-FC, Universidade do Porto, Rua do Campo Alegre 823, 4150 Porto, Portugal;Departamento de Ciência da Computação, Instituto de Matemática e Estatística, Universidade de São Paulo, Brazil;Departamento de Ciência da Computação, Instituto de Matemática e Estatística, Universidade de São Paulo, Brazil
Venue:
Theoretical Computer Science - Logic, semantics and theory of programming
Year:
2004

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Abstract

Despite its simple formulation, the decidability of the logic BB'IW has remained an open problem. We present here a decision procedure for a fragment of it, called the arity-1 formulas. The decidability proof is based on a representation of formulas called formula-trees, which is coupled with a proof method that computes long normal λ-terms that inhabit a formula. A rewriting-system is associated with such λ-terms, and we show that a formula admits a BB'IW-λ-term if and only if the associated rewriting-system terminates. The fact that termination is decidable is proved using a result on the finiteness of non-ascending sequences of n-tuples in Nn, which is equivalent to Kripke's Lemma.