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3] D. M. GABBAY, Labelleddeductive systems, vol. 1, Clarendon Press, Oxford, 1996. [4] J. HUDELMAIER, Improved decision procedures for the modal logics K, T and S4, in Proceedings of CSL '95 (H. Kleine Biining, editor), Lecture Notes in Computer Science, no. 1092, Springer-Verlag,Berlin, 1996, pp. 320-334. [5] L. D. thesis, Universitat des Saarlandes, Saarbriicken. Germany, 1997. S. BELLANTONI, Towards a new 'Ifeasible" arithmetic. Fields Institute for Research in Mathematical Science, University of Toronto, Toronto M5S 1A4, Canada.

Where - (X ti -- t) := ( X A t ) V ( i x A i t ) . t] = x: and m, ( ~ x=) Tm, ( x ) Then m, is a bijection: m, is its own inverse: [(x t ) We now employ m, to Induce a new Boolean structure on the set B. e.. to construct the Boolean algebra 13, = (B. L,,n. u . where 7 ) x L J : u m,(x) 5 m,(y): x n J := m, (m, ( x ) A m, (J )): x u y := m,(m,(x) V m t ( y ) ) . 98 1998 ASSOCIATION FOR SYMBOLIC LOGIC EIJROPEAN SUMMER MEETING (Thecomplement operation of BI is the same as that of B,since m, ( ~ m( x, ) ) = mi (m,( l x ) ) = l x .

A. J . Luxemburg, editor), Holt, Rinehart and Winston, New York, 1969. DAVID BASIN, SEAN MATTHEWS, AND LUCA V I G A N ~Modal , logics K, T, K4, S4: labelled proof systems and new complexity results. Institut fur Informatik, Universitat Freiburg, Am Flughafen 17,79110 Freiburg, Germany. Max-Planck-Institut fur Informatik, Im Stadtwald, 66123 Saarbriicken, Germany. Institut fur Informatik, Universitat Freiburg, Am Flughafen 17, 79 110 Freiburg, Germany. http://www. de/~{basin,luca). de/-seam In previous work [ l , 51, we developed a framework for giving modular and uniform presentations of modal and other non-classical logics: we present logics as labelled (natural deduction or sequent) proof systems, in which we pair logical formulae with labels in order to formalize the consequence relations of the different logics, and which are sound and complete with respect to the appropriate Kripke semantics.

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