We show that any nondeterministic read-once branching program that decides a satisfiable Tseitin formula based on an n × n grid graph has size at least 2^Ω(n). Then using the Excluded Grid Theorem by Robertson and Seymour we show that for an arbitrary graph G(V, E) any nondeterministic read-once branching program that computes a satisfiable Tseitin formula based on G has size at least \(2^{\Omega (\text {tw}(G)^{\delta })}\) for all δ < 1/36, where tw(G) is the treewidth of G (for planar graphs and some other classes of graphs the statement holds for δ = 1). We apply the mentioned results to the analysis of the complexity of derivations in the proof system OBDD(∧,reordering) and show that any OBDD(∧,reordering)-refutation of an unsatisfiable Tseitin formula based on a graph G has size at least \(2^{\Omega (\text {tw}(G)^{\delta })}\). We also show an upper bound O(|E|2^pw(G)) on the size of OBDD representations of a satisfiable Tseitin formula based on G and an upper bound \(O(|E||V| 2^{\text {pw}(G)}+|\text {TS}_{G,c}|^{2})\) on the size of OBDD(∧)-refutation of an unsatisifable Tseitin formula TS_{G, c}, where pw(G) is the pathwidth of G.

我们表明，任何基于n × n网格图确定可满足的Tseitin公式的不确定性一次读取分支程序的大小至少为2Ω ^（n）。然后使用Robertson和Seymour的排除网格定理，我们表明，对于任意图G（V，E），任何基于G来计算可满足的Tseitin公式的不确定性一次读取分支程序的大小至少为\（2 ^ {\ Omega （\ {文本TW}（G）^ {\增量}）} \）对于所有δ <1/36，其中总重量（g ^）是的树宽ģ（对于平面图和某些其他类型的图，该语句适用于δ = 1）。我们将上述结果应用于证明系统OBDD（∧，reordering）的推导复杂度分析，并表明基于图G的不满足Tseitin公式的任何OBDD（∧，reordering）-反推的大小至少为\ （2 ^ {\ Omega（\ text {tw}（G）^ {\\ delta}}} \）。我们还显示了基于G和可满足的Tseitin公式的OBDD表示的大小的上界O（| E | 2 ^pw（G））和上界\（O（| E || V | 2 ^ {\ text {pw}（G）} + | \文本{TS} _ {G，c} | ^ {2}）\）上OBDD的大小（∧）的unsatisifable Tseitin式TS的-refutation _ģ，Ç，其中PW（ģ）是的pathwidth ģ。