Undecidability of Multiplicative Subexponential Logic

Subexponential logic is a variant of linear logic with a family of exponential connectives--called subexponentials--that are indexed and arranged in a pre-order. Each subexponential has or lacks associated structural properties of weakening and contraction. We show that classical propositional multiplicative linear logic extended with one unrestricted and two incomparable linear subexponentials can encode the halting problem for two register Minsky machines, and is hence undecidable.Comment: In Proceedings LINEARITY 2014, arXiv:1502.0441

Modelling Combinatorial Auctions in Linear Logic

We show that linear logic can serve as an expressive framework in which to model a rich variety of combinatorial auction mechanisms. Due to its resource-sensitive nature, linear logic can easily represent bids in combinatorial auctions in which goods may be sold in multiple units, and we show how it naturally generalises several bidding languages familiar from the literature. Moreover, the winner determination problem, i.e., the problem of computing an allocation of goods to bidders producing a certain amount of revenue for the auctioneer, can be modelled as the problem of finding a proof for a particular linear logic sequent

The Computational Complexity of Propositional Cirquent Calculus

Introduced in 2006 by Japaridze, cirquent calculus is a refinement of sequent calculus. The advent of cirquent calculus arose from the need for a deductive system with a more explicit ability to reason about resources. Unlike the more traditional proof-theoretic approaches that manipulate tree-like objects (formulas, sequents, etc.), cirquent calculus is based on circuit-style structures called cirquents, in which different "peer" (sibling, cousin, etc.) substructures may share components. It is this resource sharing mechanism to which cirquent calculus owes its novelty (and its virtues). From its inception, cirquent calculus has been paired with an abstract resource semantics. This semantics allows for reasoning about the interaction between a resource provider and a resource user, where resources are understood in the their most general and intuitive sense. Interpreting resources in a more restricted computational sense has made cirquent calculus instrumental in axiomatizing various fundamental fragments of Computability Logic, a formal theory of (interactive) computability. The so-called "classical" rules of cirquent calculus, in the absence of the particularly troublesome contraction rule, produce a sound and complete system CL5 for Computability Logic. In this paper, we investigate the computational complexity of CL5, showing it is $\Sigma_2^p$-complete. We also show that CL5 without the duplication rule has polynomial size proofs and is NP-complete