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We present foundational work on standard bases over rings and on BooleanGroebner bases in the framework of Boolean functions. The research wasmotivated by our collaboration with electrical engineers and computerscientists on problems arising from formal verification of digital circuits. Infact, algebraic modelling of formal verification problems is developed on theword-level as well as on the bit-level. The word-level model leads to Groebnerbasis in the polynomial ring over Z/2n while the bit-level model leads toBoolean Groebner bases. In addition to the theoretical foundations of bothapproaches, the algorithms have been implemented. Using these implementationswe show that special data structures and the exploitation of symmetries makeGroebner bases competitive to state-of-the-art tools from formal verificationbut having the advantage of being systematic and more flexible.

New developments in the theory of Groebner bases and applications to formal verification