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# Lab for Automated Reasoning and Analysis LARA

# Differences

This shows you the differences between two versions of the page.

sav08:unification [2008/04/02 20:30] vkuncak |
sav08:unification [2015/04/21 17:30] (current) |
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* equation $x \doteq t(x)$ where $t$ is a term containing $x$ but not identical to $x$ ++| is contradictory++ | * equation $x \doteq t(x)$ where $t$ is a term containing $x$ but not identical to $x$ ++| is contradictory++ | ||

- | ===== Unification Examples ===== | + | ===== Examples ===== |

First-order language: | First-order language: | ||

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**Example 1** | **Example 1** | ||

- | \[ | + | \begin{equation*} |

R(x,f(x,y)) \doteq R(f(a,v),f(f(u,b),f(u,u))) | R(x,f(x,y)) \doteq R(f(a,v),f(f(u,b),f(u,u))) | ||

- | \] | + | \end{equation*} |

**Example 2** | **Example 2** | ||

- | \[ | + | \begin{equation*} |

R(x,f(x,x)) \doteq R(f(a,v),f(f(u,b),f(u,u))) | R(x,f(x,x)) \doteq R(f(a,v),f(f(u,b),f(u,u))) | ||

- | \] | + | \end{equation*} |

**Example 3** | **Example 3** | ||

- | \[ | + | \begin{equation*} |

R(x,f(x,y)) \doteq R(f(u,v),v) | R(x,f(x,y)) \doteq R(f(u,v),v) | ||

- | \] | + | \end{equation*} |

**Definition:** Unifier of a set $E$ of syntactic equations is a substitution that makes all equations true. | **Definition:** Unifier of a set $E$ of syntactic equations is a substitution that makes all equations true. | ||

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A set of equations is in //solved form// if it is of the form $\{ x_1 \doteq t_1,\ldots, x_n \doteq t_n \}$ iff variables $x_i$ do not appear in terms $t_j$, that is | A set of equations is in //solved form// if it is of the form $\{ x_1 \doteq t_1,\ldots, x_n \doteq t_n \}$ iff variables $x_i$ do not appear in terms $t_j$, that is | ||

- | \[ | + | \begin{equation*} |

\{x_1,\ldots,x_n \} \cap (FV(t_1) \cup \ldots FV(t_n)) = \emptyset | \{x_1,\ldots,x_n \} \cap (FV(t_1) \cup \ldots FV(t_n)) = \emptyset | ||

- | \] | + | \end{equation*} |

We obtain a solved form in finite time using the non-deterministic algorithm that applies the following rules as long as no clash is reported and as long as the equations are not in solved form. | We obtain a solved form in finite time using the non-deterministic algorithm that applies the following rules as long as no clash is reported and as long as the equations are not in solved form. |