Decidability for Residuated Lattices and Substructural Logics

We present a number of results related to the decidability and undecidability of various varieties of residuated lattices and their corresponding substructural logics. The context of this analysis is the extension of residuated lattices by various simple equations, dually, the extension of substructural logics by simple structural rules, with the aim of classifying simple equations by the decidability properties shared by their extensions. We also prove a number of relationships among simple extensions by showing the equational theory of their idempotent semiring reducts coincides with simple extensions of idempotent semirings. On the decidability front, we develop both semantical and syntactical methods for establishing decidability as well as tractability of decision procedures. On the undecidability front, we develop a notion of algebraic machines for which the theory of residuated frames will allow us to encode decision problems within the theories of residuated lattices and their substructural analogues. We prove the undecidability of the word problem for a broad class of simple extensions for both commutative and non-commutative residuated lattices. Furthermore, through a deduction theorem we establish the undecidability of the equational theory for a broad class of simple extensions. Translated in terms of substructural logics, we prove that the undecidability of both provability and deducibility for a multitude of extensions of FLe by simple rules

Topological Foundations of Cognitive Science

A collection of papers presented at the First International Summer Institute in Cognitive Science, University at Buffalo, July 1994, including the following papers: ** Topological Foundations of Cognitive Science, Barry Smith ** The Bounds of Axiomatisation, Graham White ** Rethinking Boundaries, Wojciech Zelaniec ** Sheaf Mereology and Space Cognition, Jean Petitot ** A Mereotopological Definition of 'Point', Carola Eschenbach ** Discreteness, Finiteness, and the Structure of Topological Spaces, Christopher Habel ** Mass Reference and the Geometry of Solids, Almerindo E. Ojeda ** Defining a 'Doughnut' Made Difficult, N .M. Gotts ** A Theory of Spatial Regions with Indeterminate Boundaries, A.G. Cohn and N.M. Gotts ** Mereotopological Construction of Time from Events, Fabio Pianesi and Achille C. Varzi ** Computational Mereology: A Study of Part-of Relations for Multi-media Indexing, Wlodek Zadrozny and Michelle Ki

Most simple extensions of FLe\mathsf{FL_e} are undecidable

All known structural extensions of the substructural logic $\mathsf{FL_e}$, Full Lambek calculus with exchange/commutativity, (corresponding to subvarieties of commutative residuated lattices axiomatized by $\{\vee, \cdot, 1\}$-equations) have decidable theoremhood; in particular all the ones defined by knotted axioms enjoy strong decidability properties (such as the finite embeddability property). We provide infinitely many such extensions that have undecidable theoremhood, by encoding machines with undecidable halting problem. An even bigger class of extensions is shown to have undecidable deducibility problem (the corresponding varieties of residuated lattices have undecidable word problem); actually with very few exceptions, such as the knotted axioms and the other prespinal axioms, we prove that undecidability is ubiquitous. Known undecidability results for non-commutative extensions use an encoding that fails in the presence of commutativity, so and-branching counter machines are employed. Even these machines provide encodings that fail to capture proper extensions of commutativity, therefore we introduce a new variant that works on an exponential scale. The correctness of the encoding is established by employing the theory of residuated frames.Comment: 45 page

Distributive Residuated Frames and Generalized Bunched Implication Algebras

We show that all extensions of the (non-associative) Gentzen system for distributive full Lambek calculus by simple structural rules have the cut elimination property. Also, extensions by such rules that do not increase complexity have the finite model property, hence many subvarieties of the variety of distributive residuated lattices have decidable equational theories. For some other extensions, we prove the finite embeddability property, which implies the decidability of the universal theory, and we show that our results also apply to generalized bunched implication algebras. Our analysis is conducted in the general setting of residuated frames

Multivalued Logic, Neutrosophy and Schrodinger equation

This book was intended to discuss some paradoxes in Quantum Mechanics from the viewpoint of Multi-Valued-logic pioneered by Lukasiewicz, and a recent concept Neutrosophic Logic. Essentially, this new concept offers new insights on the idea of ‘identity’, which too often it has been accepted as given. Neutrosophy itself was developed in attempt to generalize Fuzzy-Logic introduced by L. Zadeh. While some aspects of theoretical foundations of logic are discussed, this book is not intended solely for pure mathematicians, but instead for physicists in the hope that some of ideas presented herein will be found useful. The book is motivated by observation that despite almost eight decades, there is indication that some of those paradoxes known in Quantum Physics are not yet solved. In our knowledge, this is because the solution of those paradoxes requires re-examination of the foundations of logic itself, in particular on the notion of identity and multi-valuedness of entity. The book is also intended for young physicist fellows who think that somewhere there should be a ‘complete’ explanation of these paradoxes in Quantum Mechanics. If this book doesn’t answer all of their questions, it is our hope that at least it offers a new alternative viewpoint for these old questions