Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider sets $\Gamma(n,s,k)$ of narrow clauses expressing that no definition of a size $s$ circuit with $n$ inputs is refutable in resolution R in $k$ steps.  show that every CNF shortly refutable in Extended R, ER, can be easily reduced to an instance of $\Gamma(0,s,k)$ (with $s,k$ depending on the size of the ER-refutation) and, in particular, that $\Gamma(0,s,k)$ when interpreted as a relativized NP search problem is complete among all such problems provably total in bounded arithmetic theory $V^1_1$. #R##N# use the ideas of implicit proofs to define from $\Gamma(0,s,k)$ a non-relativized NP search problem $i\Gamma$ and  show that it is complete among all such problems provably total in bounded arithmetic theory $V^1_2$. The reductions are definable in $S^1_2$. #R##N# indicate how similar results can be proved for some other propositional proof systems and bounded arithmetic theories and how the construction can be used to define specific random unsatisfiable formulas, and  formulate two open problems about them.
['Jan Krajíček']
I am writing a research paper on a idea Consistency of circuit evaluation, extended resolution and total NP search problems
[{"from": "human", "value": "I am writing a research paper on a idea Consistency of circuit evaluation, extended resolution and total NP search problems"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider sets $\\Gamma(n,s,k)$ of narrow clauses expressing that no definition of a size $s$ circuit with $n$ inputs is refutable in resolution R in $k$ steps.  show that every CNF shortly refutable in Extended R, ER, can be easily reduced to an instance of $\\Gamma(0,s,k)$ (with $s,k$ depending on the size of the ER-refutation) and, in particular, that $\\Gamma(0,s,k)$ when interpreted as a relativized NP search problem is complete among all such problems provably total in bounded arithmetic theory $V^1_1$. #R##N# use the ideas of implicit proofs to define from $\\Gamma(0,s,k)$ a non-relativized NP search problem $i\\Gamma$ and  show that it is complete among all such problems provably total in bounded arithmetic theory $V^1_2$. The reductions are definable in $S^1_2$. #R##N# indicate how similar results can be proved for some other propositional proof systems and bounded arithmetic theories and how the construction can be used to define specific random unsatisfiable formulas, and  formulate two open problems about them."}]