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In [[logic]] and [[mathematics]], '''relation reduction''' and '''relational reducibility''' have to do with the extent to which a given [[relation (mathematics)|relation]] is determined by an [[indexed family]] or a [[sequence]] of other relations, called the ''relation dataset''.  The relation under examination is called the ''reductandum''.  The relation dataset typically consists of a specified relation over sets of relations, called the ''reducer'', the ''method of reduction'', or the ''relational step'', plus a specified set of other relations, simpler in some measure than the reductandum, called the ''reduciens'' or the ''relational base''.

In [[logic]] and [[mathematics]], '''relation reduction''' and '''relational reducibility''' have to do with the extent to which a given [[relation (mathematics)|relation]] is determined by a set of other relations, called the ''relation dataset''.  The relation under examination is called the ''reductandum''.  The relation dataset typically consists of a specified relation over sets of relations, called the ''reducer'', the ''method of reduction'', or the ''relational step'', plus a set of other relations, called the ''reduciens'' or the ''relational base'',  each of which is properly simpler in a specified way than relation under examination.

A question of relation reduction or relational reducibility is sometimes posed as a question of '''relation reconstruction''' or '''relational reconstructibility''', since a useful way of stating the question is to ask whether the reductandum can be reconstructed from the reduciens.  See [[Humpty Dumpty]].

A question of relation reduction or relational reducibility is sometimes posed as a question of '''relation reconstruction''' or '''relational reconstructibility''', since a useful way of stating the question is to ask whether the reductandum can be reconstructed from the reduciens.  See [[Humpty Dumpty]].

==Discussion==

==Discussion==

The main thing that keeps the general problem of relational reducibility from being fully well-defined is that one would have to survey all of the conceivable ways of "getting new relations from old" in order to say precisely what is meant by the claim that the relation ''L'' is reducible to the set of relations {''L''_''j'' : ''j'' in ''J'' }.  This is tantamount to claiming that if one is given a set of "simpler" relations ''L''_''j'' , for indices ''j'' in some set ''J'', that this collection of data would somehow or other fix the original relation ''L'' that one is seeking to analyze, to determine, to specify, or to synthesize.

The main thing that keeps the general problem of relational reducibility from being fully well-defined is that one would have to survey all of the conceivable ways of "getting new relations from old" in order to say precisely what is meant by the claim that the relation <math>L\!</math> is reducible to the set of relations <math>\{ L_j : j \in J \}.</math> This amounts to claiming one can be given a set of ''properly simpler'' relations <math>L_j\!</math> for values <math>j\!</math> in a given index set <math>J\!</math> and that this collection of data would suffice to fix the original relation <math>L\!</math> that one is seeking to analyze, determine, specify, or synthesize.

In practice, however, apposite discussion of a particular application typically settles on either one of two different notions of reducibility as capturing the pertinent issues, namely:

In practice, however, apposite discussion of a particular application typically settles on either one of two different notions of reducibility as capturing the pertinent issues, namely: