It is well known that each resolution refutation $\Pi$ for an unsatisfiable CNF formula $F = C_1\wedge C_2 \wedge ... \wedge C_m$ over variables $X$ can be translated in polynomial time (in the size of $\Pi$) into a deterministic branching program $P$ solving the following search problem:

1) $P$ has one source node and one sink node for each clause $C_i$.

2) For each assingment $\alpha:X\rightarrow \{0,1\}$ there is a consistent path in $P$ from the source node to some sink node associated with a clause that is falsified by $\alpha$.

Question: Is there a proof system strictly stronger than resolution where each proof $\Pi$ can be translated in polynomial time (in the size of $\Pi$) into a not necessarily deterministic branching program $P$ solving the search problem above?

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    $\begingroup$ If I am giving you a CNF $F$ and a non-deterministic BP $P$ with clauses at the sinks, how would you check that $P$ is indeed a proof that $F$ is unsatisfiable? Looks hard: you can construct a read once BP $B_C$ for each clause $C$ computing $\neg C$ with $C$ as its sink and then add a non-deterministic node branching all $B_C$ together. $\endgroup$ – holf Mar 24 '18 at 9:58
  • $\begingroup$ @holf, your argument can be equally applied to deterministic read-$m$ branching programs: For each clause $C_i$ construct a read-once BP $B_{C_i}$ computing $\neg C_i$, and then apply these BPs sequentially. Given an assignment $\alpha$, output the $C_i$ if $B_{C_i}$ is the first BP accepting $\alpha$... The question is not about determining if a given BP is a proof of unsatisfiability. Either in the deterministic or in the non-deterministic case this is guaranteed by the fact that the BP was obtained from a valid refutation. (I have edited the question to make it clearer) $\endgroup$ – verifying Mar 28 '18 at 20:52
  • $\begingroup$ Note also that non-deterministic BPs can be unambiguous, in the sense that each assignment leads to a unique clause. $\endgroup$ – verifying Mar 28 '18 at 20:58

If I understood correctly the question, the so-called Buss-Pudlak game provides a simple transformation from a proof system to such a decision tree (see Buss-Pudlak '94 http://math.cas.cz/%7Epudlak/howtolie.ps). The queries are formulas (not just variables). The tree is also completely deterministic.

Other decision trees that correspond to different propositional proof systems exist: e.g., Linear Decision Trees correspond to Res(lin) refutations (cf., http://www.csc.kth.se/~sokolovd/files/papers/splitting.pdf and http://cs.rhul.ac.uk/home/tzameret/Res-Lin-Two.pdf). But there are many other examples (cf., Tonian Pitassi's work on CP-like proof systems).

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