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Communication Dans Un Congrès Année : 2019

Computability of contact logics with measure

Résumé

Contact logics are propositional logics interpreted over Boolean contact algebras. They stem from the point-free approaches of geometry put forward by Whitehead. Their language L(≤, C) includes Boolean terms representing regions. Let X be a set of variables. The set of Boolean terms (s, t, etc) over X being denoted T(X), the set A(X) of atomic formulas over X consists of all expressions of the form s ≤ t ("s is part-of t") and C(s, t) ("s is in contact with t"). The set of all formulas (ϕ, ψ, etc) over X is the least set F(X) containing A(X) and such that for all ϕ, ψ ∈ F(X): ⊥ ∈ F(X), ¬ϕ ∈ F(X) and (ϕ ∨ ψ) ∈ F(X). Of interest are, of course, the sets of all valid formulas determined by the various classes of Boolean contact algebras one may consider. The combination of topological and size information is a fundamental issue for multifarious applications of spatial reasoning. It can be realized by considering Boolean contact algebras with measure, i.e. algebraic structures (A, C, µ) where (A, C) is a Boolean contact algebra and µ is a positive finite measure on A. Contact logics with measure are extensions of contact logics. Their language L(≤, C, ≤m) contains all additional atomic formulas of the form s ≤m t ("the size of s is less or equal than the size of t"). Of interest are, again, the sets of all valid formulas determined by the various classes of Boolean contact algebras with measure one may consider. Using complexity results about linear programming, we show that the set of all valid formulas determined by the class of all Boolean contact algebras with measure is in coNP. Our proof relies on the equivalence between the satisfiability of a given formula ϕ and the consistency of an associated system Sϕ of linear inequalities. It uses the following facts: the computation of Sϕ from ϕ is possible in non-deterministic polynomial time; if a system of k linear inequalities with integer coefficients of length at most n has a non-negative solution then it has a non-negative solution with at most k positive entries of length in O(k.(n + log k)).
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Dates et versions

hal-02411258 , version 1 (08-09-2020)

Identifiants

  • HAL Id : hal-02411258 , version 1

Citer

Philippe Balbiani, Tinko Tinchev. Computability of contact logics with measure. Logic Colloquium 2019, Aug 2019, Prague, Czech Republic. ⟨hal-02411258⟩
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