Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. We show that unlike most other propositional (quantifier-free) logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, which indicates unboundedness of resources, the decision problem becomes pspace-complete. We also establish membership in np for the multiplicative fragment, np-completeness for the multiplicative fragment extended with unrestricted weakening, and undecidability for certain fragments of noncommutative propositional linear logic. 1 Introduction Linear logic, introduced by Girard [14, 18, 17], is a refinement of classical logic which may be derived from a Gentzen-style sequent calculus axiomatization of classical logic in three steps. The resulting sequent system Lincoln@CS.Stanford.EDU Department of Computer Science, Stanford University, Stanford, CA 94305, and the Computer Science Labo...

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The multiplicative fragment of second order propositional linear logic is shown to be undecidable. Introduction Decision problems for propositional (quantifier-free) linear logic were first studied by Lincoln et al. [LMSS]. In referring to linear logic fragments, let M stand for multiplicatives, A for additives, E for exponentials (or modalities), 1 for first order quantifiers, 2 for second order propositional quantifiers, and I for "intuitionistic" version. In [LMSS] it was shown that full propositional linear logic is undecidable and that MALL is PSPACEcomplete. The main problems left open in [LMSS] were the NP-completeness of MLL, the decidability of MELL, and the decidability of various fragments of propositional linear logic without exponentials but extended with second order propositional quantifiers. The decision problem for MELL is still open, but almost all the other problems have been solved: ffl The NP-completeness of MLL has been obtained by Kanovich [K1]. Moreover, Linco...

. Recently, Lincoln, Scedrov and Shankar showed that the multiplicative fragment of second order intuitionistic linear logic is undecidable, using an encoding of second order intuitionistic logic. Their argument applies to the multiplicative-additive fragment, but it does not work in the classical case, because second order classical logic is decidable. Here we show that the multiplicative-additive fragment of second order classical linear logic is also undecidable, using an encoding of two-counter machines originally due to Kanovich. The faithfulness of this encoding is proved by means of the phase semantics. In this paper, we write LL for the full propositional fragment of linear logic, MLL for the multiplicative fragment, MALL for the multiplicative-additive fragment, and MELL for the multiplicative-exponential fragment. Similarly, we write ILL, IMLL, etc. for the fragments of intuitionistic linear logic, LL2, MLL2, etc. for the second order fragments of linear logic, and ILL2, IML...

The utility of including loops in plans has been long recognized by the planning community. Loops in a plan help increase both its applicability and the compactness of representation. However, progress in finding such plans has been limited largely due to lack of methods for reasoning about the correctness and safety properties of loops of actions. We present novel algorithms for determining the applicability and progress made by a general class of loops of actions. These methods can be used for directing the search for plans with loops towards greater applicability while guaranteeing termination, as well as in post-processing of computed plans to precisely characterize their applicability. Experimental results demonstrate the efficiency of these algorithms. 1.

Moody is right that the doctrine of conscious inessentialism (CI) is false. Unfortunately, his zombie-based argument against (CI), once made sufficiently clear to evaluate, is revealed as nothing but legerdemain. The fact is, though Moody has---for reasons I explain---convinced himself otherwise, certain zombies are impenetrable: that they are zombies, and not conscious beings like us, is something beyond the capacity of humans to divine. 1 Moody's Argument Moody's argument is imaginative, but not exactly rigorous: it's painfully difficult to identify his premises, and his inferences therefrom to the conclusion that conscious inessentialism (CI) is false. Charitable exegesis yields the following overarching reasoning: (1) If (CI) is true, then a group of zombies visiting us from a zombie-world would not bear a mark of zombiehood. (2) A group of zombies visiting us from a zombie-world would bear a mark of zombiehood. Therefore: (3) :(CI) I'm indebted to Larry Hauser for many stim...

The multiplicative fragment of second order propositional linear logic is shown to be undecidable. Introduction Decision problems for propositional (quantifier-free) linear logic were first studied by Lincoln et al. [LMSS]. In referring to linear logic fragments, let M stand for multiplicatives, A for additives, E for exponentials (or modalities), 1 for first order quantifiers, 2 for second order propositional quantifiers, and I for "intuitionistic" version. In [LMSS] it was shown that full propositional linear logic is undecidable and that MALL is PSPACEcomplete. The main problems left open in [LMSS] were the NP-completeness of MLL, the decidability of MELL, and the decidability of various fragments of propositional linear logic without exponentials but extended with second order propositional quantifiers. The decision problem for MELL is still open, but almost all the other problems have been solved: ffl The NP-completeness of MLL has been obtained by Kanovich [K1]. Moreover, Linco...

Minimalist grammars (Stabler 1997) capture some essential ideas about the basic operations of sentence construction in the Chomskyian syntactic tradition. Their affinity with the unformalized theories of working linguists makes it easier to implement and thereby to better understand the operations appealed to in neatly accounting for some of the regularities perceived in language. Here we characterize the expressive power of two, apparently quite different, variations on the basic minimalist grammar framework, gotten by: 1. adding a mechanism of ‘feature percolation ’ (Kobele, forthcoming), or 2. instead of adding a central constraint on movement (the ‘specifier island condition’, Stabler 1999), using it to replace another one (the ‘shortest move condition’, Stabler 1997, 1999) (Gärtner and Michaelis 2005). We demonstrate that both variants have equal, unbounded, computing power by showing how each can simulate straightforwardly a 2-counter automaton.

Abstract. The decidability of multiplicative exponential linear logic (MELL) is currently open. I show that two independently interesting refinements of MELL that alter only the syntax of proofs—leaving the underlying truth untouched— are undecidable. The first refinement uses new modal connectives between the linear and the unrestricted judgments, and the second is based on focusing with priority assignments that conforms to a staging discipline. Both refinements can adequately encode the transitions of a two-register Minsky machine. While neither refinement is weak enough to entail the undecidability of MELL, they show that no additive connectives are necessary for undecidability. 1

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