10 research outputs found
Embedding Kozen-Tiuryn Logic into Residuated One-Sorted Kleene Algebra with Tests
Kozen and Tiuryn have introduced the substructural logic for
reasoning about correctness of while programs (ACM TOCL, 2003). The logic
distinguishes between tests and partial correctness assertions,
representing the latter by special implicational formulas. Kozen and Tiuryn's
logic extends Kleene altebra with tests, where partial correctness assertions
are represented by equations, not terms. Kleene algebra with codomain,
, is a one-sorted alternative to Kleene algebra with tests that
expands Kleene algebra with an operator that allows to construct a Boolean
subalgebra of tests. In this paper we show that Kozen and Tiuryn's logic embeds
into the equational theory of the expansion of with residuals of
Kleene algebra multiplication and the upper adjoint of the codomain operator
One-sorted Program Algebras
Kleene algebra with tests, KAT, provides a simple two-sorted algebraic
framework for verifying properties of propositional while programs. Kleene
algebra with domain, KAD, is a one-sorted alternative to KAT. The equational
theory of KAT embeds into KAD, but KAD lacks some natural properties of KAT.
For instance, not each Kleene algebra expands to a KAD, and the subalgebra of
tests in each KAD is forced to be the maximal Boolean subalgebra of the
negative cone. In this paper we propose a generalization of KAD that avoids
these features while still embedding the equational theory of KAT. We show that
several natural properties of the domain operator of KAD can be added to the
generalized framework without affecting the results. We consider a variant of
the framework where test complementation is defined using a residual of the
Kleene algebra multiplication
Knowledge is a Diamond
In the standard epistemic logic, the knowledge operator is represented as a box operator, a universal quantifier over a set of possible worlds. There is an alternative approach to the semantics of knowledge, according to which an agent a knows a proposition iff a has a reliable (e.g. sensory) evidence that supports the proposition. In this interpretation, knowledge is viewed rather as an existential, i.e. a diamond modality. In this paper, we will propose a formal semantics for substructural logics that allows to model knowledge on the basis of this intuition. The framework is strongly motivated by a similar semantics introduced by (Bílková, Majer, Peliš, 2016). However, as we will argue, our framework overcomes some unintuitive features of the semantics from (Bílková, Majer, Peliš, 2016). Most importantly, knowledge does not distribute over disjunction in our logic