De Morgan Dual Nominal Quantifiers Modelling Private Names in Non-Commutative Logic

This paper explores the proof theory necessary for recommending an expressive but decidable first-order system, named MAV1, featuring a de Morgan dual pair of nominal quantifiers. These nominal quantifiers called `new' and `wen' are distinct from the self-dual Gabbay-Pitts and Miller-Tiu nominal quantifiers. The novelty of these nominal quantifiers is they are polarised in the sense that `new' distributes over positive operators while `wen' distributes over negative operators. This greater control of bookkeeping enables private names to be modelled in processes embedded as formulae in MAV1. The technical challenge is to establish a cut elimination result, from which essential properties including the transitivity of implication follow. Since the system is defined using the calculus of structures, a generalisation of the sequent calculus, novel techniques are employed. The proof relies on an intricately designed multiset-based measure of the size of a proof, which is used to guide a normalisation technique called splitting. The presence of equivariance, which swaps successive quantifiers, induces complex inter-dependencies between nominal quantifiers, additive conjunction and multiplicative operators in the proof of splitting. Every rule is justified by an example demonstrating why the rule is necessary for soundly embedding processes and ensuring that cut elimination holds.Comment: Submitted for review 18/2/2016; accepted CONCUR 2016; extended version submitted to journal 27/11/201

From Proof Nets to the Free *-Autonomous Category

In the first part of this paper we present a theory of proof nets for full multiplicative linear logic, including the two units. It naturally extends the well-known theory of unit-free multiplicative proof nets. A linking is no longer a set of axiom links but a tree in which the axiom links are subtrees. These trees will be identified according to an equivalence relation based on a simple form of graph rewriting. We show the standard results of sequentialization and strong normalization of cut elimination. In the second part of the paper we show that the identifications enforced on proofs are such that the class of two-conclusion proof nets defines the free *-autonomous category.Comment: LaTeX, 44 pages, final version for LMCS; v2: updated bibliograph

Reasoning about Knowledge in Linear Logic: Modalities and Complexity

In a recent paper, Jean-Yves Girard commented that ”it has been a long time since philosophy has stopped intereacting with logic”[17]. Actually, it has no

Relevant Logics Obeying Component Homogeneity

This paper discusses three relevant logics that obey Component Homogeneity - a principle that Goddard and Routley introduce in their project of a logic of significance. The paper establishes two main results. First, it establishes a general characterization result for two families of logic that obey Component Homogeneity - that is, we provide a set of necessary and sufficient conditions for their consequence relations. From this, we derive characterization results for S*fde, dS*fde, crossS*fde. Second, the paper establishes complete sequent calculi for S*fde, dS*fde, crossS*fde. Among the other accomplishments of the paper, we generalize the semantics from Bochvar, Hallden, Deutsch and Daniels, we provide a general recipe to define containment logics, we explore the single-premise/single-conclusion fragment of S*fde, dS*fde, crossS*fdeand the connections between crossS*fde and the logic Eq of equality by Epstein. Also, we present S*fde as a relevant logic of meaninglessness that follows the main philosophical tenets of Goddard and Routley, and we briefly examine three further systems that are closely related to our main logics. Finally, we discuss Routley's criticism to containment logic in light of our results, and overview some open issues

Proof search and counter-model construction for bi-intuitionistic propositional logic with labelled sequents

Bi-intuitionistic logic is a conservative extension of intuitionistic logic with a connective dual to implication, called exclusion. We present a sound and complete cut-free labelled sequent calculus for bi-intuitionistic propositional logic, BiInt, following S. Negri's general method for devising sequent calculi for normal modal logics. Although it arises as a natural formalization of the Kripke semantics, it is does not directly support proof search. To describe a proof search procedure, we develop a more algorithmic version that also allows for counter-model extraction from a failed proof attempt.Estonian Science Foundation - grants no. 5567; 6940Fundação para a Ciência e a Tecnologia (FCT)RESCUE - no. PTDC/EIA/65862/2006TYPES - FP6 ISTCentro de matemática da Universidade do Minh

Basic Logic and Quantum Entanglement

As it is well known, quantum entanglement is one of the most important features of quantum computing, as it leads to massive quantum parallelism, hence to exponential computational speed-up. In a sense, quantum entanglement is considered as an implicit property of quantum computation itself. But...can it be made explicit? In other words, is it possible to find the connective "entanglement" in a logical sequent calculus for the machine language? And also, is it possible to "teach" the quantum computer to "mimic" the EPR "paradox"? The answer is in the affirmative, if the logical sequent calculus is that of the weakest possible logic, namely Basic logic. A weak logic has few structural rules. But in logic, a weak structure leaves more room for connectives (for example the connective "entanglement"). Furthermore, the absence in Basic logic of the two structural rules of contraction and weakening corresponds to the validity of the no-cloning and no-erase theorems, respectively, in quantum computing.Comment: 10 pages, 1 figure,LaTeX. Shorter version for proceedings requirements. Contributed paper at DICE2006, Piombino, Ital