Abstract

Linear Logic was introduced by Girard [3] as a resource-sensitive refinement of classical logic. Lincoln, Mitchell, Scedrov, and Shankar [13] have proved the undecidability of full propositional Linear Logic. This implies that Linear Logic is more expressive than traditional classical or intuitionistic logic, even if we consider the modalized versions of those logics. In [9, 10] we prove that standard many-counter Minsky machines [17] can be simulated directly in propositional Linear Logic. Here we are going to present a more transparent and fruitful simulation of many-counter Minsky machines in Linear Logic.

Simulating one system of concepts in terms of another system is known to consist of two procedures: (A) Suggesting an encoding of the first system in terms of the second one, and (B) Proving that the encoding suggested is correct and fair.

Here, based on a computational interpretation of Linear Logic [9, 10], we present: (A) A direct and natural encoding of many-counter Minsky machines in Linear Logic, and (B) Transparent proof of the correctness and fairness of this encoding.

As a corollary, we prove that all partial recursive relations are directly definable in propositional Linear Logic.

Introduction and Summary

Linear Logic was introduced by Girard [3] as a resource-sensitive refinement of classical logic. Lincoln, Mitchell, Scedrov, and Shankar [13] have proved the undecidability of full propositional Linear Logic. In [13] the proof of undecidability of propositional Linear Logic consists of a reduction from the Halting Problem for And-Branching Two Counter Machines Without Zero-Test (specified in the same [13]) to a decision problem in Linear Logic.