In 2004 Atserias, Kolaitis, and Vardi proposed $\text {OBDD}$-based propositional proof systems that prove unsatisfiability of a CNF formula by deduction of an identically false $\text {OBDD}$ from $\text {OBDD}$s representing clauses of the initial formula. All $\text {OBDD}$s in such proofs have the same order of variables. We initiate the study of $\text {OBDD}$ based proof systems that additionally contain a rule that allows changing the order in $\text {OBDD}$s. At first we consider a proof system $\text {OBDD}(\land , \text{reordering})$ that uses the conjunction (join) rule and the rule that allows changing the order. We exponentially separate this proof system from $\text {OBDD}(\land )$ proof system that uses only the conjunction rule. We prove exponential lower bounds on the size of $\text {OBDD}(\land , \text{reordering})$ refutations of Tseitin formulas and the pigeonhole principle. The first lower bound was previously unknown even for $\text {OBDD}(\land )$ proofs and the second one extends the result of Tveretina et al. from $\text {OBDD}(\land )$ to $\text {OBDD}(\land , \text{reordering})$.

In 2001 Aguirre and Vardi proposed an approach to the propositional satisfiability problem based on $\text {OBDD}$s and symbolic quantifier elimination (we denote algorithms based on this approach as $\text {OBDD}(\land , \exists )$ algorithms). We augment these algorithms with the operation of reordering of variables and call the new scheme $\text {OBDD}(\land , \exists , \text{reordering})$ algorithms. We notice that there exists an $\text {OBDD}(\land , \exists )$ algorithm that solves satisfiable and unsatisfiable Tseitin formulas in polynomial time (a standard example of a hard system of linear equations over $\mathbb {F}_2$), but we show that there are formulas representing systems of linear equations over $\mathbb {F}_2$ that are hard for $\text {OBDD}(\land , \exists , \text{reordering})$ algorithms. Our hard instances are satisfiable formulas representing systems of linear equations over $\mathbb {F}_2$ that correspond to checksum matrices of error correcting codes.

We prove an exponential lower bound on the size of proofs in the proof system operating with ordered binary decision diagrams introduced by Atserias, Kolaitis and Vardi [2]. In fact, the lower bound applies to semantic derivations operating with sets defined by OBDDs. We do not assume any particular format of proofs or ordering of variables, the hard formulas are in CNF. We utilize (somewhat indirectly) feasible interpolation.

We define a proof system combining resolution and the OBDD proof system.

We investigate the structure of “worst-case” quasi reduced ordered decision diagrams and Boolean functions whose truth tables are associated to: we suggest different ways to count and enumerate them. We, then, introduce a notion of complexity which leads to the concept of “hard” Boolean functions as functions whose QROBDD are “worst-case” ones. So we exhibit the relation between hard functions and the Storage Access function (also known as Multiplexer).

Ordered binary decision diagrams (OBDDs) and several more general BDD models have turned out to be representations of Boolean functions which are useful in applications like verification, timing analysis, test pattern generation or combinatorial optimization.The hidden weighted bit function (HWB) is of particular interest, since it seems to be the simplest function with exponential OBDD size. The complexity of this function with respect to different circuit models, formulas, and various BDD models is discussed.