“Redundant proof”的意思、由来-开放百科全书

词条

Redundant proof

释义

Local redundancy
Global redundancy
Example
Notes

In mathematical logic, a redundant proof is a proof that has a subset that is a shorter proof of the same result. That is, a proof of is considered redundant if there exists another proof of such that (i.e. ) and where is the number of nodes in .^[1]

Local redundancy

A proof containing a subproof of the shapes (here omitted pivots{{explain|date=December 2017}} indicate that the resolvents must be uniquely defined)

is locally redundant.

Indeed, both of these subproofs can be equivalently replaced by the shorter subproof . In the case of local redundancy, the pairs of redundant inferences having the same pivot occur close to each other in the proof. However, redundant inferences can also occur far apart in the proof.

The following definition generalizes local redundancy by considering inferences with the same pivot that occur within different contexts. We write to denote a proof-context with a single placeholder replaced by the subproof .

Global redundancy

A proof

is potentially (globally) redundant. Furthermore, it is (globally) redundant if it can be rewritten to one of the following shorter proofs:

Example

The proof

is locally redundant as it is an instance of the first pattern in the definition

The pattern is

But it is not globally redundant because the replacement terms according to the definition contain in all the cases and does not correspond to a proof. In particular, neither nor can be resolved with , as they do not contain the literal .

The second pattern of potentially globally redundant proofs appearing in global redundancy definition is related to the well-known{{explain|date=December 2017}} notion of regularity{{explain|date=December 2017}}. Informally, a proof is irregular if there is a path from a node to the root of the proof such that a literal is used more than once as a pivot in this path.

Notes

1. ^Fontaine, Pascal; Merz, Stephan; Woltzenlogel Paleo, Bruno. Compression of Propositional Resolution Proofs via Partial Regularization. 23rd International Conference on Automated Deduction, 2011.

1 : Proof theory

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。