Generalising KAT to Verify Weighted Computations

• Main Author: HASLab INESC TEC,Universidade do Minho,R. da Universidade,4710-057 Braga,Portugal
• Publication Date: 2019
• Other Authors: CIDMA,Universidade de Aveiro,Campus Universitario de Santiago,3810-193 Aveiro,Portugal, Luís Soares Barbosa, Madeira,A, Leandro Rafael Gomes, Universidade do Minho,R. da Universidade,4710-057 Braga,Portugal & Quantum Software Engineering Group,INL
• Format: Other
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: http://repositorio.inesctec.pt/handle/123456789/11234; http://dx.doi.org/10.7561/sacs.2019.2.141

Kleene algebra with tests (KAT) was introduced as an algebraic structure to model and reason about classic imperative programs, i.e. sequences of discrete transitions guarded by Boolean tests. This paper introduces two generalisations of this structure able to express programs as weighted transitions and tests with outcomes in non necessarily bivalent truth spaces: graded Kleene algebra with tests (GKAT) and a variant where tests are also idempotent (I-GKAT). In this context, and in analogy to Kozen's encoding of Propositional Hoare Logic (PHL) in KAT we discuss the encoding of a graded PHL in I-GKAT and of its while-free fragment in GKAT. Moreover, to establish semantics for these structures four new algebras are defined: FSET(T), FREL(K,T) and FLANG(K,T) over complete residuated lattices K and T, and M (n, A) over a GKAT or I-GKAT A. As a final exercise, the paper discusses some program equivalence proofs in a graded context.