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sav08:dpll_algorithm_for_sat [2008/03/13 17:46] vkuncak |
sav08:dpll_algorithm_for_sat [2015/04/21 17:30] (current) |
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**Decision step** (picking variable value): from proofs for | **Decision step** (picking variable value): from proofs for | ||
- | \[ | + | \begin{equation*} |
S' \cup \{p\} \vdash {\it false} | S' \cup \{p\} \vdash {\it false} | ||
- | \] | + | \end{equation*} |
- | \[ | + | \begin{equation*} |
S' \cup \{\lnot p\} \vdash {\it false} | S' \cup \{\lnot p\} \vdash {\it false} | ||
- | \] | + | \end{equation*} |
we would like to construct the proof for | we would like to construct the proof for | ||
- | \[ | + | \begin{equation*} |
S' \vdash {\it false} | S' \vdash {\it false} | ||
- | \] | + | \end{equation*} |
- | ++++Idea:| | + | |
From | From | ||
- | \[ | + | \begin{equation*} |
S' \cup \{p\} \vdash {\it false} | S' \cup \{p\} \vdash {\it false} | ||
- | \] | + | \end{equation*} |
derive proof tree for | derive proof tree for | ||
- | \[ | + | \begin{equation*} |
S' \vdash \lnot p | S' \vdash \lnot p | ||
- | \] | + | \end{equation*} |
and from | and from | ||
- | \[ | + | \begin{equation*} |
S' \cup \{\lnot p\} \vdash {\it false} | S' \cup \{\lnot p\} \vdash {\it false} | ||
- | \] | + | \end{equation*} |
derive proof tree for | derive proof tree for | ||
- | \[ | + | \begin{equation*} |
S' \vdash p | S' \vdash p | ||
- | \] | + | \end{equation*} |
Then combine these two trees with resolution on $\{p\}$ and $\{\lnot p\}$ to get ${\it false}$. | Then combine these two trees with resolution on $\{p\}$ and $\{\lnot p\}$ to get ${\it false}$. | ||
Why can we modify resolution proof to move $p$ from assumption and put its negation to conclusion? | Why can we modify resolution proof to move $p$ from assumption and put its negation to conclusion? | ||
- | ++++ | + | |
=== Lower Bounds on Running Time === | === Lower Bounds on Running Time === | ||
Line 128: | Line 127: | ||
Theorem: for some formulas, shortest resolution proofs are exponential. | Theorem: for some formulas, shortest resolution proofs are exponential. | ||
- | This does not contradict P vs NP question, because there may be "better" proof systems than resolution. | + | This does not contradict that P vs NP question is open, because there may be "better" proof systems than resolution. |
Lower bounds for both resolution and a stronger system are shown here by proving that interpolants can be exponential, and that interpolants are polynomial in proof size (see [[Interpolants from Resolution Proofs]]): | Lower bounds for both resolution and a stronger system are shown here by proving that interpolants can be exponential, and that interpolants are polynomial in proof size (see [[Interpolants from Resolution Proofs]]): | ||
* Pavel Pudlák: [[http://citeseer.ist.psu.edu/36219.html|Lower Bounds for Resolution and Cutting Plane Proofs and Monotone Computations]] | * Pavel Pudlák: [[http://citeseer.ist.psu.edu/36219.html|Lower Bounds for Resolution and Cutting Plane Proofs and Monotone Computations]] |