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### The formal strong completeness of partial monoidal Boolean BI

Dominique Larchey-Wendling
2014 Journal of Logic and Computation
This paper presents a self-contained proof of the strong completeness of the labeled tableaux method for partial monoidal Boolean BI: if a formula has no tableau proof then there exists a counter-model  ...  In this paper we will focus on Boolean BI (denoted BBI), more precisely on partial monoidal BBI.  ...  Conclusion and perspectives In this paper, we provide a detailed standalone proof of the strong completeness of a labeled tableaux system for partial monoidal Boolean BI.  ...

### Looking at Separation Algebras with Boolean BI-eyes [chapter]

Dominique Larchey-Wendling, Didier Galmiche
2014 Lecture Notes in Computer Science
In this paper, we show that the formulae of Boolean BI cannot distinguish between some of the different notions of separation algebra found in the literature: partial commutative monoids, either cancellative  ...  We obtain this result by the careful study of the specific properties of the counter-models that are generated by tableaux proof-search in Boolean BI.  ...  Boolean BI.  ...

### On Model Checking Boolean BI [chapter]

Heng Guo, Hanpin Wang, Zhongyuan Xu, Yongzhi Cao
2009 Lecture Notes in Computer Science
Boolean BI (BBI) denotes BI with classical interpretation of additives and its model is the commutative monoid.  ...  In the case of finitely related monoid and generator propositions, the model checking problem is EXPSPACE-complete.  ...  But the additive connectives can be interpreted classically in the monoid models, in which the partial order becomes an equivalence relation. This version of BI is called Boolean BI (BBI).  ...

### Possible worlds and resources: the semantics of BI

David J. Pym, Peter W. O'Hearn, Hongseok Yang
2004 Theoretical Computer Science
We discuss in detail the question of completeness, explaining the essential distinction between BI with and without ⊥ (the unit of ∨).  ...  On the one hand, from proof-theoretic or categorical concerns and, on the other, from a possible-worlds semantics based on preordered (commutative) monoids.  ...  The soundness and completeness of the partial monoid semantics for BI with ⊥ was is shown in  .  ...

### Convolution and Concurrency [article]

James Cranch and Simon Doherty and Georg Struth
2020 arXiv   pre-print
We develop a correspondence theory between relational properties in X and algebraic properties in Q and Q^X in the sense of modal and substructural logics, and boolean algebras with operators.  ...  The elements of Q can be understood as weights; the case Q= corresponds to a powerset lifting.  ...  The second and third author have been partially supported by EPSRC grant EP/R032351/1.  ...

### The Undecidability of Boolean BI through Phase Semantics

Dominique Larchey-Wendling, Didier Galmiche
2010 2010 25th Annual IEEE Symposium on Logic in Computer Science
We solve the open problem of the decidability of Boolean BI logic (BBI), which can be considered as the core of separation and spatial logics.  ...  We single out a fragment of ILL which is both undecidable and complete for trivial phase semantics. Therefore, we obtain the undecidability of BBI.  ...  The authors dedicate this work and the resulting paper to the memory of Pr. Noëlle Carbonell.  ...

### A Classical Propositional Logic for Reasoning About Reversible Logic Circuits [chapter]

Holger Bock Axelsen, Robert Glück, Robin Kaarsgaard
2016 Lecture Notes in Computer Science
We show that all strong equivalences of reversible logic circuits are provable in the system, derive an equivalent equational theory, and describe its main applications in the verication of both reversible  ...  demonstrated equivalent to Boolean algebras, and extended categorically to form a sound and complete semantics for this system.  ...  The authors acknowledge support from the Danish Council for Independent Research | Natural Sciences under the Foundations of Reversible Computing project, and partial support from COST Action IC1405 Reversible  ...

### Varieties of Boolean inverse semigroups [article]

Friedrich Wehrung
2016 arXiv   pre-print
In an earlier work, the author observed that Boolean inverse semi-groups, with semigroup homomorphisms preserving finite orthogonal joins, form a congruence-permutable variety of algebras, called biases  ...  strong sense defined in terms of wreath products by finite symmetric groups.  ...  By definition, Typ S is the universal monoid of the partial monoid Int S of all D-classes of elements of S (which we call the type interval of S), endowed with the partial addition defined by x/D + y/D  ...

### Varieties of Boolean inverse semigroups

Friedrich Wehrung
2018 Journal of Algebra
In an earlier work, the author observed that Boolean inverse semigroups, with semigroup homomorphisms preserving finite orthogonal joins, form a congruence-permutable variety of algebras, called biases  ...  strong sense defined in terms of wreath products by finite symmetric groups.  ...  By definition, Typ S is the universal monoid of the partial monoid Int S of all D-classes of elements of S (which we call the type interval of S), endowed with the partial addition defined by x/D + y/D  ...

### The semantics of BI and resource tableaux

D. GALMICHE, D. MÉRY, D. PYM
2005 Mathematical Structures in Computer Science
Then, from these results, we can define a new resource semantics of BI, based on partially defined monoids, and prove that this semantics is complete.  ...  As consequences of the relationships between semantics of BI and resource tableaux, we prove two strong new results for propositional BI: its decidability and the finite model property with respect to  ...  We are grateful to Hongseok Yang, Peter O'Hearn, and the anonymous referees for their help with finding errors in, and with the presentation of, this long and complex paper.  ...

### Nondeterministic Phase Semantics and the Undecidability of Boolean BI

Dominique Larchey-Wendling, Didier Galmiche
2013 ACM Transactions on Computational Logic
We solve the open problem of the decidability of Boolean BI logic (BBI), which can be considered as the core of Separation and Spatial Logics.  ...  We complete the pictures with additional results of undecidability on the models based on the free monoid N × N and the models based on the partial monoid P f (N) (which is also the simplest heap monoid  ...  THE SEMANTICS OF BOOLEAN BI Boolean BI (denoted BBI) is the variant of intuitionistic BI [O' Hearn and Pym 1999] where the additive connectives are interpreted as Boolean connectives, contrary to (intuitionistic  ...

### On the Completeness of the Traced Monoidal Category Axioms in (Rel,+)

Miklós Bartha
2017 Acta Cybernetica
The result is derived from a theorem saying that already the structure of finite partial injections as a traced monoidal category is complete for the given axioms.  ...  It is shown that the traced monoidal category of finite sets and relations with coproduct as tensor is complete for the extension of the traced symmetric monoidal axioms by two simple axioms, which capture  ...  One may even extend (Bi, +) to the traced monoidal category of partial injections.  ...

### Membership problems for regular and context-free trace languages

A. Bertoni, G. Mauri, N. Sabadini
1989 Information and Computation
They can be defined as subsets of a free partially commutative monoid and a theory of trace languages can be developed, generalizing the usual formal languages theory.  ...  Trace languages have been introduced in order to describe the behaviour of concurrent systems in the same way as usual formal languages do for sequential system.  ...  The free partially commutative monoid (fpcm, for short) generated by a concurrent alphabet (Z, C) is the initial object F(C, C) of the category of monoids generated by the elements of Z and satisfying,  ...

### A non-commutative generalization of Stone duality [article]

Mark V Lawson
2009 arXiv   pre-print
As an instance of this duality, we show that the boolean inverse monoid associated with the Cuntz groupoid is the strong orthogonal completion of the polycyclic (or Cuntz) monoid and so its group of units  ...  We prove that the category of boolean inverse monoids is dually equivalent to the category of boolean groupoids.  ...  The boolean inverse monoid C n associated with the Cuntz groupoid G n is the strong orthogonal completion of the polycyclic (or Cuntz) monoid P n .  ...

### An Alternative Direct Simulation of Minsky Machines into Classical Bunched Logics via Group Semantics

Dominique Larchey-Wendling
2010 Electronical Notes in Theoretical Computer Science
) partial monoidal Kripke semantics.  ...  From a proof-theoretical perspective, Boolean BI (or simply BBI) can be considered as the first investigated variant of BI which contained a negation: BBI combines intuitionistic multiplicatives with Boolean  ...  Introduction The logic of bunched implications of Pym and O'Hearn  contains two important families of logics: Boolean BI (BBI) and Classical BI (CBI).  ...
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