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sav08:instantiation_plus_ground_resolution [2009/05/14 13:40] vkuncak |
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- | ====== Ground Instantiation plus Ground Resolution as a Proof System ====== | ||
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- | 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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- | Our goal is to avoid explicit enumeration and apply proof system where we do as little search (guessing) as possible, while still being complete. | ||
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- | ===== Definition ====== | ||
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- | The proof system has two rules: ground instantiation and ground resolution. | ||
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- | === Ground Instantiation Rule === | ||
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- | \[ | ||
- | \frac{C}{subst(\sigma)(C)} | ||
- | \] | ||
- | where $C$ is a clause and $\sigma$ is a ground substitution. | ||
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- | 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. | ||
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- | === Ground Resolution Rule === | ||
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- | If $A$ is a ground atom and $C,D$ are ground claues, then | ||
- | \[ | ||
- | \frac{C \cup \{\lnot A\}\ \ \ D \cup \{A\}} | ||
- | {C \cup D} | ||
- | \] | ||
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- | Note that this is propositional resolution where propositional variables have "long names" (they are ground atoms). | ||
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- | ==== Example ==== | ||
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- | ===== Completeness ===== | ||
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- | The proof is based on [[Herbrand's Expansion Theorem]] (see also the proof of [[Compactness for First-Order Logic]]). | ||
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- | 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. | ||
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