An Overview of Residuated Kleene Algebras and Lattices

1. Residuated Lattices with iteration 2. Background: Semirings and Kleene algebras 3. A Gentzen system for Residuated Kleene Lattices and some reducts 4. Interpreting Kleene algebras with tests 1. Residuated Lattices with iteration This talk is mostly about Residuated Kleene Lattices, which are defined as noncommutative residuated 0,1-lattices expanded with a unary operation * that satisfies x * ≤ (x ∨ y) * (* is order preserving) e ∨ x ∨ x * x * = x * (x\x) * = x\x The element x * is intended to represent the reflexive transitive closure of x, also called the Kleene-* of x. Other well-known residuated 0,1-lattices with additional unary operations: Residuated Boolean monoids = Residuated 0,1-lattices with Boolean negation ⊇ Sequential algebras ⊇ Relation algebras Intuitionistic Linear Logic = Commutative residuated 0,1-lattices with storage For a good perspective on residuated Kleene lattices, we need to back up a bit. is a commutative monoid and x(y ∨ z) = xy ∨ xz, (y ∨ z)x = yx ∨ zx, and x0 = 0 = 0x. Here we are writing x · y as xy, and consider this operation to have priority over ∨. Semirings are common generalizations of rings (where (A, ∨, 0) is an abelian group) and bounded distributive lattices (where · is commutative, and x(x∨y) = x = x ∨ xy). 1 A semiring is called idempotent if x ∨ x = x. In this case (A, ∨, 0) is a lower-bounded semilattice, and as usual one defines a partial order x ≤ y by x ∨ y = y. It follows from the distributivity that · is order preserving. We will consider expanding idempotent semirings with one of the following: 1. (* 2) yx ≤ y =⇒ yx * = y MR and RISR are varieties, while KA is only a quasivariety. 2 KA and many related classes were studied by Kleene, Conway, Kozen and others, since it is an algebraic framework for regular languages (sets of strings accepted by automata) and for sequential programs: for programs p, q, pq means running p followed by q, p ∨ q means running p or q, p * means running p repeatedly 0 or more times. RKAs and RKLs have also been called action algebras (Pratt [1991]) and action lattices (Kozen[1992]) respectively, and they are the algebraic version of action logic. In this context, elements represent actions. Pratt illustrates this …