Trivium is a stream cipher candidate of the eStream project. It has successfully moved into phase three of the selection process under the hardware category. No attacks faster than the exhaustive search have so far been reported on Trivium. Bivium-A and Bivium-B are simplified versions of Trivium that are built on the same design principles but with two registers. The simplified design is useful in investigating Trivium type ciphers with a reduced complexity and provides insight into effective attacks which could be extended to Trivium. This paper focuses on an algebraic analysis which uses the boolean satisfiability problem in propositional logic. For reduced variants of the cipher, this analysis recovers the internal state with a minimal amount of keystream observations.