c © TÜBİTAK doi:10.3906/mat-0910-92 Secrecy logic: S-secrecy structures

Let S = 〈L, S 〉 be a deductive system. An S-secrecy logic is a quadruple K = 〈FmL(V),K,B, S 〉, where FmL(V) is the algebra of L-formulas, K,B are S-theories, with B ⊆ K and S ⊆ K such that S ∩ B = ∅. K corresponds to information deducible from a knowledge base, B to information deducible from the publicly accessible (or browsable) part of the knowledge base and S is a secret set, a set of sensitive or private information that the knowledge base aims at concealing from its users. To provide models for this context, the notion of an S-secrecy structure is introduced. It is a quadruple A = 〈A,KA, BA, SA 〉, consisting of an L-algebra A, two S-filters KA, BA on A, with BA ⊆ KA, and a subset SA ⊆ KA, such that SA ∩ BA = ∅. Several model theoretic/universal algebraic and categorical properties of the class of S-secrecy structures, endowed with secrecy homomorphisms, are studied relating to various universal algebraic and categorical constructs. Key Words: Secrecy-preserving reasoning, abstract algebraic logic, logical matrices, protoalgebraic logics, first-order structures, homomorphism theorems, regular categories, subdirect products, subdirectly irre