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NP-complete → NP-hard
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Tom Cornebize
Tom Cornebize
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First, be carefull, it might be that PSPACE ⊄ PSPACE-complete. If you suppose PSPACE ⊆ PSPACE-complete (thus PSPACE = PSPACE-complete since the other inclusion is true), then every problem of PSPACE is PSPACE-complete. Thus, all problems of PSPACE can be reduced to some easy problem of P (which is also in PSPACE). This is very unlikely (but not (yet) proved true or false).

Now, let us consider the reduction you propose.

If I understood well, you want to take a formula of SAT. The reduction would be to take the same formula for QBF (with, I suppose, an existential quantifier for each variable, otherwise it would not be a valid QBF formula).

This is indeed a valid reduction, and the SAT formula would be satisfiable if and only if the QBF formula was true.

By doing so, you reduce SAT to QBF. This shows that QBF is at least as hard as SAT. Since SAT is NP-complete, this shows that QBF is also NP-completehard.

The reduction in the other way (taking a QBF formula and making a SAT formula) is not known. This is why we do not know whether NP=PSPACE or not.

First, be carefull, it might be that PSPACE ⊄ PSPACE-complete. If you suppose PSPACE ⊆ PSPACE-complete (thus PSPACE = PSPACE-complete since the other inclusion is true), then every problem of PSPACE is PSPACE-complete. Thus, all problems of PSPACE can be reduced to some easy problem of P (which is also in PSPACE). This is very unlikely (but not (yet) proved true or false).

Now, let us consider the reduction you propose.

If I understood well, you want to take a formula of SAT. The reduction would be to take the same formula for QBF (with, I suppose, an existential quantifier for each variable, otherwise it would not be a valid QBF formula).

This is indeed a valid reduction, and the SAT formula would be satisfiable if and only if the QBF formula was true.

By doing so, you reduce SAT to QBF. This shows that QBF is at least as hard as SAT. Since SAT is NP-complete, this shows that QBF is also NP-complete.

The reduction in the other way (taking a QBF formula and making a SAT formula) is not known. This is why we do not know whether NP=PSPACE or not.

First, be carefull, it might be that PSPACE ⊄ PSPACE-complete. If you suppose PSPACE ⊆ PSPACE-complete (thus PSPACE = PSPACE-complete since the other inclusion is true), then every problem of PSPACE is PSPACE-complete. Thus, all problems of PSPACE can be reduced to some easy problem of P (which is also in PSPACE). This is very unlikely (but not (yet) proved true or false).

Now, let us consider the reduction you propose.

If I understood well, you want to take a formula of SAT. The reduction would be to take the same formula for QBF (with, I suppose, an existential quantifier for each variable, otherwise it would not be a valid QBF formula).

This is indeed a valid reduction, and the SAT formula would be satisfiable if and only if the QBF formula was true.

By doing so, you reduce SAT to QBF. This shows that QBF is at least as hard as SAT. Since SAT is NP-complete, this shows that QBF is NP-hard.

The reduction in the other way (taking a QBF formula and making a SAT formula) is not known. This is why we do not know whether NP=PSPACE or not.

Source Link
Tom Cornebize
Tom Cornebize
  • 255
  • 3
  • 7

First, be carefull, it might be that PSPACE ⊄ PSPACE-complete. If you suppose PSPACE ⊆ PSPACE-complete (thus PSPACE = PSPACE-complete since the other inclusion is true), then every problem of PSPACE is PSPACE-complete. Thus, all problems of PSPACE can be reduced to some easy problem of P (which is also in PSPACE). This is very unlikely (but not (yet) proved true or false).

Now, let us consider the reduction you propose.

If I understood well, you want to take a formula of SAT. The reduction would be to take the same formula for QBF (with, I suppose, an existential quantifier for each variable, otherwise it would not be a valid QBF formula).

This is indeed a valid reduction, and the SAT formula would be satisfiable if and only if the QBF formula was true.

By doing so, you reduce SAT to QBF. This shows that QBF is at least as hard as SAT. Since SAT is NP-complete, this shows that QBF is also NP-complete.

The reduction in the other way (taking a QBF formula and making a SAT formula) is not known. This is why we do not know whether NP=PSPACE or not.