Case study: Additive linear logic and lattices
We investigate sequent calculus where contexts, called additive contexts, are governed by the operations of a non-distributive lattice. We present a sequent calculus ALL m with multiple antecedents and succedents. ALL m is complete for non-distributive lattices and is equivalent to the additive fragment of linear logic. Weakenings and contractions are postulated for ALL m and cut is redundant. We extend this construction in order to get a sequent calculus for propositional linear logic with both additive and multiplicative contexts.
Then we show that a bottom-up decision procedure based on the cut-free sequent calculi runs in exponential time. We provide a decision algorithm that exploits analytic cuts and whose runtime is polynomial.
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- 1.G. Birkhoff. Lattice Theory, volume XXV. American Mathematical Society, Second edition, 1948.Google Scholar
- 2.H.B. Curry. Foundations of mathematical logic. Dover, 1963 first edition, 1976.Google Scholar
- 3.M. Dunn. Relevance logic and entailment. In D. Gabbay and F.Guenthner, editors, Handbook of philosophical logic, chapter III.3, pages 117–224. D. Reidel Publishing Company, 1986.Google Scholar
- 4.J.-Y. Girard. Linear logic. TCS, 50:1–102, 1987.Google Scholar
- 5.J-Y Girard. Light linear logic. In D. Leivant, editor, LCC'94, pages 145–179. LNCS 960, 1995.Google Scholar
- 6.J.S. Jodas and D. Miller. Logic programming in a fragment of linear logic. Journal of Information and Computation, 110(2):327–365, 1994.Google Scholar
- 7.M. Kanovich. Simulating guarded programs in linear logic. In T. Ito and A. Yonezawa, editors, Int. Work. Theory and Practice of Parallel Programming, pages 45–69. LNCS 907, 1994.Google Scholar
- 8.G. Mints. Resolution calculus for the first order linear logic. Journal of Logic Languages and Information, 2:59–93, 1993.Google Scholar
- 9.T. Tammet. Proof strategies in linear logic. Journal of Automated Reasonning, 12(3):273–304, 1994.Google Scholar
- 10.A. S. Troelstra. Lectures on Linear Logic, volume 29. CSLI, 1992.Google Scholar
- 11.A. Urquhart. Complexity of proofs in classical propositional logic. In Y.N. Moschovakis, editor, Logic from computer science, pages 597–608. Springer-Verlag, 1989.Google Scholar