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Given some set of formulae {$p_i$}, what can we say about the "literals that can be derived from this set of statements"?

Conjoin the $p_i$ and put the result in minimal disjunctive normal form (using a standard technique like the Quine-McCluskey algorithm). The will produce some $r = r_1 \vee \ldots \vee r_n$, where each $r_i$ is a conjunction of literals.

1. Suppose the set {$p_i$} is consistent. Then a literal $l$ will be derivable from the {$p_i$} just in case $l$ appears in each disjunct $r_i$. Let the set of such literals be {$l_i$}.

2. Suppose {$p_i$} is inconsistent. Then in each disjunct $r_i$ you will find some literals $A$ literal and $\neg A$ both of which appear in each disjunct in $r$. its negation. In this case the set of literals every literal will be derivable from the {$p_i$} is just $p_i$}. Let {$l_i$} then be the set of all literals. Call these the {$l_i$} as well.

What can we now say about disjunctions of such literals the {$l_i$} that are non-redundant in your sense? Each $l_i$ will trivially belong to that set. But $l_i \vee q$ will be redundant for any $q$. So the set you want is just {$l_i$}.

Perhaps I've misunderstood your question, but it any case it sounds like a circuit-minimization problem, and you're not going to find a solution that's better than exponential in the number of atomic variables unless you add some constraints to the problem. You should look at the aforementioned Quine-McCluskey algorithm, as well as the "Espresso" algorithm.

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Given some set of formulae {$p_i$}, what can we say about the "literals that can be derived from this set of statements"?

Conjoin the $p_i$ and reduce put the result to a minimal in disjunctive normal form (using a standard technique like the Quine-McCluskey algorithm). The result will be produce some $r = r_1 \vee \ldots \vee r_n$, where each $r_i$ is a conjunction of literals.

1. Suppose the set {$p_i$} is consistent. Then a literal $l$ will be derivable from the {$p_i$} just in case $l$ appears in each disjunct $r_i$. Let the set of such literals be {$l_i$}.

2. Suppose {$p_i$} is inconsistent. Then you will find some literals $A$ and $\neg A$ both of which appear in each disjunct in $r$. In this case the set of literals derivable from the {$p_i$} is just the set of all literals. Call these the {$l_i$} as well.

What can we now say about disjunctions of such literals that are non-redundant in your sense? Each $l_i$ will trivially belong to that set. But $l_i \vee q$ will be redundant for any $q$. So the set you want is just {$l_i$}.

Perhaps I've misunderstood your question, but it any case it sounds like a circuit-minimization problem, and you're not going to find a solution that's better than exponential in the number of atomic variables unless you add some constraints to the problem. You should look at the aforementioned Quine-McCluskey algorithm, as well as the "Espresso" algorithm.