Resolution inference

In propositional logic, a resolution inference is an instance of the following rule:^[1]

{\frac {\Gamma _{1}\cup \left\{\ell \right\}\,\,\,\,\Gamma _{2}\cup \left\{{\overline {\ell }}\right\}}{\Gamma _{1}\cup \Gamma _{2}}}|\ell |

We call:

The clauses $\Gamma _{1}\cup \left\{\ell \right\}$ and $\Gamma _{2}\cup \left\{{\overline {\ell }}\right\}$ are the inference’s premises
$\Gamma _{1}\cup \Gamma _{2}$ (the resolvent of the premises) is its conclusion.
The literal $\ell$ is the left resolved literal,
The literal $\overline {\ell }$ is the right resolved literal,
$|\ell |$ is the resolved atom or pivot.

This rule can be generalized to first-order logic to:^[2]

{\frac {\Gamma _{1}\cup \left\{L_{1}\right\}\,\,\,\,\Gamma _{2}\cup \left\{L_{2}\right\}}{(\Gamma _{1}\cup \Gamma _{2})\phi }}\phi

where $\phi$ is a most general unifier of $L_{1}$ and $\overline {L_{2}}$ and $\Gamma _{1}$ and $\Gamma _{2}$ have no common variables.

Example

The clauses $P(x),Q(x)$ and $\neg P(b)$ can apply this rule with $[b/x]$ as unifier.

Here x is a variable and b is a constant.

{\frac {P(x),Q(x)\,\,\,\,\neg P(b)}{Q(b)}}[b/x]

Here we see that

↑ Fontaine, Pascal; Merz, Stephan; Woltzenlogel Paleo, Bruno. Compression of Propositional Resolution Proofs via Partial Regularization. 23rd International Conference on Automated Deduction, 2011.
↑ Enrique P. Arís, Juan L. González y Fernando M. Rubio, Lógica Computacional, Thomson, (2005).

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