TY - GEN

T1 - Generalized non-deterministic matrices and (n,k)-ary quantifiers

AU - Avron, Arnon

AU - Zamansky, Anna

PY - 2007

Y1 - 2007

N2 - An (n, k)-ary quantifier is a generalized logical connective, binding k variables and connecting n formulas. Canonical Gentzen-type systems with (n, k)-ary quantifiers are systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of an (n, k)-ary quantifier is introduced. The semantics of such systems for the case of k ∈{0,1} are provided in [16] using two-valued non-deterministic matrices (2Nmatrices). A constructive syntactic coherence criterion for the existence of a 2Nmatrix for which a canonical system is strongly sound and complete, is formulated there. In this paper we extend these results from the case of k ∈ {0,1} to the general case of k > 0. We show that the interpretation of quantifiers in the framework of Nmatrices is not sufficient for the case of k > 1 and introduce generalized Nmatrices which allow for a more complex treatment of quantifiers. Then we show that (i) a canonical calculus G is coherent iff there is a 2GNmatrix, for which G is strongly sound and complete, and (ii) any coherent canonical calculus admits cut-elimination.

AB - An (n, k)-ary quantifier is a generalized logical connective, binding k variables and connecting n formulas. Canonical Gentzen-type systems with (n, k)-ary quantifiers are systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of an (n, k)-ary quantifier is introduced. The semantics of such systems for the case of k ∈{0,1} are provided in [16] using two-valued non-deterministic matrices (2Nmatrices). A constructive syntactic coherence criterion for the existence of a 2Nmatrix for which a canonical system is strongly sound and complete, is formulated there. In this paper we extend these results from the case of k ∈ {0,1} to the general case of k > 0. We show that the interpretation of quantifiers in the framework of Nmatrices is not sufficient for the case of k > 1 and introduce generalized Nmatrices which allow for a more complex treatment of quantifiers. Then we show that (i) a canonical calculus G is coherent iff there is a 2GNmatrix, for which G is strongly sound and complete, and (ii) any coherent canonical calculus admits cut-elimination.

UR - http://www.scopus.com/inward/record.url?scp=35448946513&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-72734-7_3

DO - 10.1007/978-3-540-72734-7_3

M3 - Conference contribution

AN - SCOPUS:35448946513

SN - 3540727329

SN - 9783540727323

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 26

EP - 40

BT - Logical Foundations of Computer Science - International Symposium, LFCS 2007, Proceedings

PB - Springer Verlag

T2 - International Symposium on Logical Foundations of Computer Science, LFCS 2007

Y2 - 4 June 2007 through 7 June 2007

ER -