Hiroakira Ono on Substructural Logics

Preface
Contents
Editors and Contributors
A Scientific Autobiography
	1 My Early Days
	2 A Start in Academic Life
	3 Going into the Outside World
	4 A New Beginning
	5 Stepping Forward
	6 A Few More Words
Part I Expository and Survey Chapters
Universal Algebraic Methods  for Non-classical Logics
	1 Introduction
	2 Basic Concepts and Results
	3 Terms and Term Operations
	4 Permutability of Congruences
	5 Variants of Distributivity
	6 Abelian Algebras
	7 Filtered Products
	8 Definable Principal Congruences
	9 Controlling Irreducible Algebras
	10 Some Finite Basis Theorems
	11 Lattices of Subvarieties
	12 Maltsev Conditions
	13 Categorical Equivalence
	References
Abstract Algebraic Logic
	1 Introduction
	2 Some Preliminary Issues
	3 Bare Algebraizability
		3.1 The Equational Consequence Relative to a Class  of Algebras
		3.2 Translating Formulas into Equations
		3.3 Translating Equations into Formulas
		3.4 Putting It All Together
	4 The Origins of Algebraizability: The Lindenbaum–Tarski Process
		4.1 The Process for Classical Logic
		4.2 The Process for Algebraizable Logics
		4.3 The Universal Lindenbaum–Tarski Process: matrix semantics
		4.4 Algebraizability and Matrix Semantics: Definability
		4.5 Implicative Logics
	5 Modes of Algebraizability, and Non-algebraizability
	6 Beyond Algebraizability
		6.1 The Leibniz Hierarchy
		6.2 The General Definition of the Algebraic Counterpart of a Logic
		6.3 The Frege Hierarchy
	7 Exploiting Algebraizability: Bridge Theorems and Transfer Theorems
	8 Algebraizability at a More Abstract Level
	References
Topological Duality and Algebraic Completions
	1 Introduction
	2 Completions
		2.1 Join- and Meet-Completions
		2.2 Δ-Completions
	3 Extensions of Maps
		3.1 Extension of Residuated Families of Maps
	4 Canonicity
		4.1 Canonicity for LOs
		4.2 Canonicity of Axioms from Substructural Logic
		4.3 Class Operators and Finitely Generated Varieties
	5 Connections with Topological Duality
	6 Concluding Remarks
	References
An Algebraic Glimpse at Bunched Implications and Separation Logic
	1 Introduction
	2 Logic and Algebra
		2.1 Algebras
		2.2 Congruences
		2.3 Logic
	3 Concrete Models: Standard Models of BI
		3.1 Generalized PPMs
		3.2 Intuitionistic Versus Classical Resource Models
		3.3 Resource Allocation and Generalized Effect Algebras
		3.4 Resource Separation, Memory and the Heap Model
		3.5 Ambient Logic, Trees and Semistructured Data
		3.6 Costs, Logic Programming and Petri Nets
	4 Essentially Noncommutative Models
		4.1 Weakening Relations and Relation Algebras
		4.2 Language Models
	5 Subvarieties of GBI-algebras and InGBI-Algebras
	6 Semantics via Duality
		6.1 Semantics and Duality for Heyting Algebras
		6.2 Semantics and Duality for GBI-Algebras
	7 Decidability Issues
		7.1 Positive Decidability Results
		7.2 Subvarieties with Undecidable Equational Theory
		7.3 Undecidability of Quasi-Equational Theories
	8 A Glimpse at Proof Theory
	9 (B)BI and Separation Logic
		9.1 Basic Ideas of Floyd-Hoare Logic(s)
		9.2 Heap(let)s, Allocation and Separation
		9.3 Local Axioms, Global Specifications and the Frame Rule
	10 Proof Theory and Decidability for Fragments of SL
		10.1 Sketch of a Proof System for SL
		10.2 Decidability Revisited
	11 Bi-Abduction: The Main Issue of SL Proof Theory
		11.1 Abduction and Bi-Abduction Algebraically
		11.2 Bi-Abduction in Separation Logic
	12 Applications and Later Developments
		12.1 Competing Formalisms
		12.2 Tools
		12.3 Concurrency and Algebraic Aspects
	References
Part II Special Topics
Recognizability in Residuated Lattices
	1 Introduction
	2 Background and Motivation
	3 Residuation, Residuated Lattices, and Modules over Residuated Lattices
	4 Recognizable Elements in Residuated Lattices
	5 Regular Elements and Boolean-Recognizability
	References
Finite Embeddability Property  for Residuated Lattices via Regular Languages
	1 Introduction
	2 Finite Embeddability Property
	3 Regular Languages and Syntactic Congruences
	4 Residuated Lattices Induced by a Collection of Languages
	5 Analytic Identities and Corresponding Rules
	6 Finite Embeddability Property
	7 Applications
		7.1 Integral Residuated Lattices
		7.2 Knotted Residuated Lattices
		7.3 Disproving the FEP
	References
Cover Systems for the Modalities  of Linear Logic
	1 Introduction
	2 Modalities on Residuated Lattices
	3 Cover Systems
	4 Residuated Cover Systems
	5 Modal FL-Cover Systems
	6 Representation of Modal FL-Algebras
	7 Kripke-Type Semantics
	8 Negation and Orthogonality
	9 Classical/Grishin Algebras
	References
A Negative Solution to Ono's Problem P52: Existence and Disjunction Properties in Intermediate Predicate Logics
	1 Introduction
	2 Preliminaries
		2.1 Intermediate and Super-Intuitionistic Logics
		2.2 Kripke Semantics
		2.3 Algebraic Semantics
	3 Intermediate Predicate Logics with EP but Lacking DP
	4 Z-normality + EP Imply DP
	5 Concluding Remarks
	References
Conservative Expansions  of Substructural Logics
	1 Foreword
	2 Introduction
	3 Preliminaries
	4 Adding Δ and Other Algebraic Operators
		4.1 Adding Δ
		4.2 Δ-expansions with Special Propositional Operators
	5 Expansions with Δ and with Propositional Quantifiers with Applications to Craig's Interpolation Property
		5.1 Adding Propositional Quantifiers
		5.2 Craig's Interpolation Property
	6 Adding Propositional Quantifiers to Δ-core Fuzzy Logics with the Finite Model Property
		6.1 Algebraizable Expansions of MTL with Propositional Quantifiers
		6.2 Conservative Δ-expansions of Semilinear Logics  and Craig's Interpolation Property
	7 Expansions of QFLΔ
		7.1 Some Definitional Expansions
		7.2 Induction
		7.3 Expansions by Recursion and Łukasiewicz Logic
		7.4 An Alternative Way to Introduce a Product  in MV-Algebras
		7.5 Failure of the Finite Model Property for QFLΔ(oplus,cdot)
	8 Conclusions
	References
Bibliography
A.1  Books
A.2  Edited Special Issues and Books
A.3  Papers