Facets of the linear ordering polytope: A unification for the fence family through weighted graphs

The binary choice polytope appeared in the investigation of the binary choice problem formulated by Guilbaud and Block and Marschak. It is nowadays known to be the same as the linear ordering polytope from operations research (as studied by Grötschel, Jünger and Reinelt).The central problem is to find facet-defining linear inequalities for the polytope. Fence inequalities constitute a prominent class of such inequalities (Cohen and Falmagne; Grötschel, Jünger and Reinelt). Two different generalizations exist for this class: the reinforced fence inequalities of Leung and Lee, and independently Suck, and the stability-critical fence inequalities of Koppen. Together with the fence inequalities, these inequalities form the fence family. Building on previous work on the biorder polytope by Christophe, Doignon and Fiorini, we provide a new class of inequalities which unifies all inequalities from the fence family. The proof is based on a projection of polytopes. The new class of facet-defining inequalities is related to a specific class of weighted graphs, whose definition relies on a curious extension of the stability number. We investigate this class of weighted graphs which generalize the stability-critical graphs.     

MV-Algebras and Quantum Computation

We introduce a generalization of MV algebras motivated by the investigations into the structure of quantum logical gates. After laying down the foundations of the structure theory for such quasi-MV algebras, we show that every quasi-MV algebra is embeddable into the direct product of an MV algebra and a “flat” quasi-MV algebra, and prove a completeness result w.r.t. a standard quasi-MV algebra over the complex numbers. Presented by Heinrich Wansing     

Monotonicity and Reasoning with Exceptions

A proposal by Ferguson [2003, Argumentation 17, 335–346] for a fully monotonic argument form allowing for the expression of defeasible generalizations is critically examined and rejected as a general solution. It is argued that (i) his proposal reaches less than the default-logician’s solution allows, e.g., the monotonously derived conclusion is one-sided and itself not defeasible. (ii) when applied to a suitable example, his proposal derives the wrong conclusion. Unsuccessful remedies are discussed.     

Not Everything is Possible

This paper makes a point about the interpretation of the simplestquantified modal logic, that is, quantified modal logic witha single domain. It is commonly assumed that the domain in questionis to be understood as the set of all possibile objects. Thepoint of the paper is that this assumption is misguided.     

First-Order Theories of Orthogonality Structures

We investigate certain aspects of the first-order theory oforthogonality structures - structures consisting of a domainof lines subject to a binary orthogonality relation. In particular,we establish definitions of various geometric and algebraicnotions in terms of orthogonality, describe the constructionof extremal subspaces using orthogonality, and show that thefirst-order theory of line orthogonality in the Euclidean n-spaceis not ₀-categorical for n 3.     

Many Concepts and Two Logics of Algorithmic Reduction

Within the program of finding axiomatizations for various parts of computability logic, it was proven earlier that the logic of interactive Turing reduction is exactly the implicative fragment of Heyting’s intuitionistic calculus. That sort of reduction permits unlimited reusage of the computational resource represented by the antecedent. An at least equally basic and natural sort of algorithmic reduction, however, is the one that does not allow such reusage. The present article shows that turning the logic of the first sort of reduction into the logic of the second sort of reduction takes nothing more than just deleting the contraction rule from its Gentzen-style axiomatization. The first (Turing) sort of interactive reduction is also shown to come in three natural versions. While those three versions are very different from each other, their logical behaviors (in isolation) turn out to be indistinguishable, with that common behavior being precisely captured by implicative intuitionistic logic. Among the other contributions of the present article is an informal introduction of a series of new — finite and bounded — versions of recurrence operations and the associated reduction operations. Presented by Robert Goldblatt     

Predicate Logical Extensions of some Subintuitionistic Logics

The paper presents predicate logical extensions of some subintuitionistic logics. Subintuitionistic logics result if conditions of the accessibility relation in Kripke models for intuitionistic logic are dropped. The accessibility relation which interprets implication in models for the propositional base subintuitionistic logic considered here is neither persistent on atoms, nor reflexive, nor transitive. Strongly complete predicate logical extensions are modeled with a second accessibility relation, which is a partial order, for the interpretation of the universal quantifier. Presented by Melvin Fitting     

On Conservative Extensions in Logics with Infinitary Predicates

If the language is extended by new individual variables, in classical first order logic, then the deduction system obtained is a conservative extension of the original one. This fails to be true for the logics with infinitary predicates. But it is shown that restricting the commutativity of quantifiers and the equality axioms in the extended system and supposing the merry-go-round property in the original system, the foregoing extension is already conservative. It is shown that these restrictions are crucial for an extension to be conservative. The origin of the results is algebraic logic. Presented by Daniele Mundici Supported by grant OTKA T43242.     

Reasoning about cooperation,actions and preferences

In this paper, a logic for reasoning about coalitional power is developed which explicitly represents agents’ preferences and the actions by which the agents can achieve certain results. A complete axiomatization is given and its satisfiability problem is shown to be decidable and EXPTIME-hard.     

Combinatory Logic and the Semantics of Substructural Logics

The results of this paper extend some of the intimate relations that are known to obtain between combinatory logic and certain substructural logics to establish a general characterization theorem that applies to a very broad family of such logics. In particular, I demonstrate that, for every combinator X, if LX is the logic that results by adding the set of types assigned to X (in an appropriate type assignment system, TAS) as axioms to the basic positive relevant logic B∘T, then LX is sound and complete with respect to the class of frames in the Routley-Meyer relational semantics for relevant and substructural logics that meet a first-order condition that corresponds in a very direct way to the structure of the combinator X itself. Presented by Rob Goldblatt