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Polybori: A framework for Gröbner-basis computations with Boolean polynomials. (English) Zbl 1186.68571
Summary: This work presents a new framework for Gröbner-basis computations with Boolean polynomials. Boolean polynomials can be modelled in a rather simple way, with both coefficients and degree per variable lying in \(\{0,1\}\). The ring of Boolean polynomials is, however, not a polynomial ring, but rather the quotient ring of the polynomial ring over the field with two elements modulo the field equations \(x^{2}=x\) for each variable \(x\). Therefore, the usual polynomial data structures seem not to be appropriate for fast Gröbner-basis computations. We introduce a specialised data structure for Boolean polynomials based on zero-suppressed binary decision diagrams, which are capable of handling these polynomials more efficiently with respect to memory consumption and also computational speed. Furthermore, we concentrate on high-level algorithmic aspects, taking into account the new data structures as well as structural properties of Boolean polynomials. For example, a new useless-pair criterion for Gröbner-basis computations in Boolean rings is introduced. One of the motivations for our work is the growing importance of formal hardware and software verification based on Boolean expressions, which suffer-besides from the complexity of the problems -from the lack of an adequate treatment of arithmetic components. We are convinced that algebraic methods are more suited and we believe that our preliminary implementation shows that Gröbner-bases on specific data structures can be capable of handling problems of industrial size.

MSC:
68W30 Symbolic computation and algebraic computation
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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