Truth-table Reduction

In computability theory, a truth-table reduction is a reduction from one set of natural numbers to another.
As a "tool", it is weaker than Turing reduction, since not every Turing reduction between sets can be performed by a truth-table reduction, but every truth-table reduction can be performed by a Turing reduction. For the same reason it is said to be a stronger reducibility than Turing reducibility, because it implies Turing reducibility. A weak truth-table reduction is a related type of reduction which is so named because it weakens the constraints placed on a truth-table reduction, and provides a weaker equivalence classification; as such, a "weak truth-table reduction" can actually be more powerful than a truth-table reduction as a "tool", and perform a reduction which is not performable by truth table.

A Turing reduction from a set ''B'' to a set ''A'' computes the membership of a single element in ''B'' by asking questions about the membership of various elements in ''A'' during the computation; it may adaptively determine which questions it asks based upon answers to previous questions. In contrast, a truth-table reduction or a weak truth-table reduction must present all of its (finitely many) oracle queries at the same time. In a truth-table reduction, the reduction also gives a boolean function (a truth table) which, when given the answers to the queries, will produce the final answer of the reduction. In a weak truth-table reduction, the reduction uses the oracle answers as a basis for further computation which may depend on the given answers but may not ask further questions of the oracle.

Equivalently, a weak truth-table reduction is a Turing reduction for which the use of the reduction is bounded by a computable function. For this reason, they are sometimes referred to as bounded Turing (bT) reductions rather than as weak truth-table (wtt) reductions.

Properties

As every truth-table reduction is a Turing reduction, if ''A'' is truth-table reducible to ''B'' (''A'' ≤_tt ''B''), then ''A'' is also Turing reducible to ''B'' (''A'' ≤_T ''B''). Considering also one-one reducibility, many-one reducibility and weak truth-table reducibility,
: $A \leq_1 B \Rightarrow A \leq_m B \Rightarrow A \leq_ B \Rightarrow A \leq_ B \Rightarrow A \leq_T B$,
or in other words, one-one reducibility implies many-one reducibility, which implies truth-table reducibility, which in turn implies weak truth-table reducibility, which in turn implies Turing reducibility.

Furthermore, ''A'' is truth-table reducible to ''B'' iff ''A'' is Turing reducible to ''B'' via a total functional on $2^\omega$. The forward direction is trivial and for the reverse direction suppose $\Gamma$ is a total computable functional. To build the truth-table for computing ''A(n)'' simply search for a number ''m'' such that for all binary strings $\sigma$ of length ''m'' $\Gamma^\sigma(n)$ converges. Such an ''m'' must exist by Kőnig's lemma since $\Gamma$ must be total on all paths through $2^$. Given such an ''m'' it is a simple matter to find the unique truth-table which gives $\Gamma^\sigma(n)$ when applied to $\sigma$. The forward direction fails for weak truth-table reducibility.

References

* H. Rogers, Jr., 1967. ''The Theory of Recursive Functions and Effective Computability'', second edition 1987, MIT Press. (paperback),

Reduction (complexity)

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