50,487 research outputs found
Decidability of quantified propositional intuitionistic logic and S4 on trees
Quantified propositional intuitionistic logic is obtained from propositional
intuitionistic logic by adding quantifiers \forall p, \exists p over
propositions. In the context of Kripke semantics, a proposition is a subset of
the worlds in a model structure which is upward closed. Kremer (1997) has shown
that the quantified propositional intuitionistic logic H\pi+ based on the class
of all partial orders is recursively isomorphic to full second-order logic. He
raised the question of whether the logic resulting from restriction to trees is
axiomatizable. It is shown that it is, in fact, decidable. The methods used can
also be used to establish the decidability of modal S4 with propositional
quantification on similar types of Kripke structures.Comment: v2, 9 pages, corrections and additions; v1 8 page
Eigenlogic: Interpretable Quantum Observables with applications to Fuzzy Behavior of Vehicular Robots
This work proposes a formulation of propositional logic, named Eigenlogic,
using quantum observables as propositions. The eigenvalues of these operators
are the truth-values and the associated eigenvectors the interpretations of the
propositional system. Fuzzy logic arises naturally when considering vectors
outside the eigensystem, the fuzzy membership function is obtained by the Born
rule of the logical observable.This approach is then applied in the context of
quantum robots using simple behavioral agents represented by Braitenberg
vehicles. Processing with non-classical logic such as multivalued logic, fuzzy
logic and the quantum Eigenlogic permits to enlarge the behavior possibilities
and the associated decisions of these simple agents
Formal concept analysis and structures underlying quantum logics
A Hilbert space induces a formal context, the Hilbert formal context , whose associated concept lattice is isomorphic to the lattice of closed subspaces of . This set of closed subspaces, denoted , is important in the development of quantum logic and, as an algebraic structure, corresponds to a so-called ``propositional system'', that is, a complete, atomistic, orthomodular lattice which satisfies the covering law.
In this paper, we continue with our study of the Chu construction by introducing the Chu correspondences between Hilbert contexts, and showing that the category of Propositional Systems, PropSys, is equivalent to the category of of Chu correspondences between Hilbert contextsUniversidad de Málaga. Campus de Excelencia Internacional Andalucía Tech
Second-order Propositional Announcement Logic
International audienceIn this paper we introduce Second-order Propositional Announcement Logic (SOPAL): a language to express arbitrary announcements in Public Announcement Logic, by means of propositional quantification. We present SOPAL within a multi-agent context, and show that it is rich enough to express complex notions such as preservation under arbitrary announcements, knowability, and successfulness. We analyse the model theory of SOPAL and prove that it is strictly more expressive than Arbitrary PAL [2], and as expressive as Second-order Propositional Epistemic Logic [4], even though exponentially more succinct than the latter. These results points to a rich logic, with nice computational properties nonetheless, such as a decidable model checking problem and a complete axiomatisation
A minimal classical sequent calculus free of structural rules
Gentzen's classical sequent calculus LK has explicit structural rules for
contraction and weakening. They can be absorbed (in a right-sided formulation)
by replacing the axiom P,(not P) by Gamma,P,(not P) for any context Gamma, and
replacing the original disjunction rule with Gamma,A,B implies Gamma,(A or B).
This paper presents a classical sequent calculus which is also free of
contraction and weakening, but more symmetrically: both contraction and
weakening are absorbed into conjunction, leaving the axiom rule intact. It uses
a blended conjunction rule, combining the standard context-sharing and
context-splitting rules: Gamma,Delta,A and Gamma,Sigma,B implies
Gamma,Delta,Sigma,(A and B). We refer to this system M as minimal sequent
calculus.
We prove a minimality theorem for the propositional fragment Mp: any
propositional sequent calculus S (within a standard class of right-sided
calculi) is complete if and only if S contains Mp (that is, each rule of Mp is
derivable in S). Thus one can view M as a minimal complete core of Gentzen's
LK.Comment: To appear in Annals of Pure and Applied Logic. 15 page
Strong Equivalence of Qualitative Optimization Problems
We introduce the framework of qualitative optimization problems (or, simply, optimization problems) to represent preference theories. The formalism uses separate modules to describe the space of outcomes to be compared (the generator) and the preferences on outcomes (the selector). We consider two types of optimization problems. They differ in the way the generator, which we model by a propositional theory, is interpreted: by the standard propositional logic semantics, and by the equilibrium-model (answer-set) semantics. Under the latter interpretation of generators, optimization problems directly generalize answer-set optimization programs proposed previously. We study strong equivalence of optimization problems, which guarantees their interchangeability within any larger context. We characterize several versions of strong equivalence obtained by restricting the class of optimization problems that can be used as extensions and establish the complexity of associated reasoning tasks. Understanding strong equivalence is essential for modular representation of optimization problems and rewriting techniques to simplify them without changing their inherent properties
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