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sav08:instantiation_plus_ground_resolution [2008/04/01 16:20] vkuncak |
sav08:instantiation_plus_ground_resolution [2015/04/21 17:30] (current) |
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| - | ====== Gdound Instantiation plus Ground Resolution as a Proof System ====== | + | ====== Ground Instantiation plus Ground Resolution as a Proof System ====== |
| We introduce this proof system as an intermediate step between the naive enumeration algorithm based on Herbrand's theorem and resolution for first-order logic. | We introduce this proof system as an intermediate step between the naive enumeration algorithm based on Herbrand's theorem and resolution for first-order logic. | ||
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| The proof system has two rules: ground instantiation and ground resolution. | The proof system has two rules: ground instantiation and ground resolution. | ||
| - | **Ground Instantiation Rule** | + | === Ground Instantiation Rule === |
| - | \[ | + | \begin{equation*} |
| \frac{C}{subst(\sigma)(C)} | \frac{C}{subst(\sigma)(C)} | ||
| - | \] | + | \end{equation*} |
| where $C$ is a clause and $\sigma$ is a ground substitution. | where $C$ is a clause and $\sigma$ is a ground substitution. | ||
| Ground substitution is of the form $\{x_1 \mapsto t_1,\ldots,x_n \mapsto t_n\}$ where each $t_i$ is a ground term. | Ground substitution is of the form $\{x_1 \mapsto t_1,\ldots,x_n \mapsto t_n\}$ where each $t_i$ is a ground term. | ||
| - | **Ground Resolution Rule** | + | === Ground Resolution Rule === |
| If $A$ is a ground atom and $C,D$ are ground claues, then | If $A$ is a ground atom and $C,D$ are ground claues, then | ||
| - | \[ | + | \begin{equation*} |
| \frac{C \cup \{\lnot A\}\ \ \ D \cup \{A\}} | \frac{C \cup \{\lnot A\}\ \ \ D \cup \{A\}} | ||
| {C \cup D} | {C \cup D} | ||
| - | \] | + | \end{equation*} |
| Note that this is propositional resolution where propositional variables have "long names" (they are ground atoms). | Note that this is propositional resolution where propositional variables have "long names" (they are ground atoms). | ||
| ==== Example ==== | ==== Example ==== | ||
| + | |||
| + | |||
| ===== Completeness ===== | ===== Completeness ===== | ||
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| The proof is based on [[Herbrand's Expansion Theorem]] (see also the proof of [[Compactness for First-Order Logic]]). | The proof is based on [[Herbrand's Expansion Theorem]] (see also the proof of [[Compactness for First-Order Logic]]). | ||
| - | Suppose a set $S$ of clauses is contradictory. By [[Herbrand's Expansion Theorem]] and [[Compactness Theorem|Compactness Theorem for Propositional Formulas]], there is some finite subset $S_0 \subseteq expand(S)$ is contradictory. Then there exists a derivation of empty clause from $S_0$ viewed as set of propositional formulas, using propositional resolution. In other words, there exists a derivation of empty clause from $S_0$ using ground resolution rule. Each element of $S_0$ can be obtained from an element of $S$ using instantiation rule. This means that there exists a proof tree that starts by a single application of instantiation rule, and then performs ground resolution. | + | Suppose a set $S$ of clauses is contradictory. By [[Herbrand's Expansion Theorem]] and [[Compactness Theorem|Compactness Theorem for Propositional Formulas]], some finite subset $S_0 \subseteq expand(S)$ is contradictory. Then there exists a derivation of empty clause from $S_0$ viewed as set of propositional formulas, using propositional resolution. In other words, there exists a derivation of empty clause from $S_0$ using ground resolution rule. Each element of $S_0$ can be obtained from an element of $S$ using instantiation rule. This means that there exists a proof tree whose leaves are followed by a single application of instantiation rule, and inner nodes contain ground resolution steps. |