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sav08:dpll_algorithm_for_sat [2008/03/13 17:45] vkuncak |
sav08:dpll_algorithm_for_sat [2015/04/21 17:30] (current) |
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| Note: for each unit clause and other clause we can perform unit resolution at most once (variable disappears) | Note: for each unit clause and other clause we can perform unit resolution at most once (variable disappears) | ||
| * BCP is polynomial procedure | * BCP is polynomial procedure | ||
| + | |||
| + | We do not need to keep literals that are subset of existing ones: | ||
| + | <code> | ||
| + | def RemoveSubsumed(S : Set[Clause]) : Set[Clause] = | ||
| + | if there are C1,C2 in S such that C1 subset C2 | ||
| + | then RemoveSubsumes(S - {C2}) | ||
| + | else S | ||
| + | </code> | ||
| + | In particular, when we apply unit resolution, the original clause can be deleted. | ||
| ===== DPLL Recursively ===== | ===== DPLL Recursively ===== | ||
| Line 39: | Line 48: | ||
| val P = pick variable from FV(S') | val P = pick variable from FV(S') | ||
| DPLL(F' union {p}) or DPLL(F' union {Not(p)}) | DPLL(F' union {p}) or DPLL(F' union {Not(p)}) | ||
| - | </code> | ||
| - | |||
| - | <code> | ||
| - | def RemoveSubsumed(S : Set[Clause]) : Set[Clause] = | ||
| - | if there are C1,C2 in S such that C1 subset C2 | ||
| - | then RemoveSubsumes(S - {C2}) | ||
| - | else S | ||
| </code> | </code> | ||
| Line 88: | Line 90: | ||
| **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 126: | 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]] | ||