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--- sav08:unification [2008/04/02 20:30]
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-====== Unification ======
-Unification is the process of solving equations in term algebra, which is the algebra of ground terms, with functions defined as in the [[Ground Terms]] section of proof of Herbrand theorem.
-The solution process is essentially variable elimination, based on two main properties
-  * equation $f(t_1,\ldots,t_n) \doteq f(s_1,\ldots,s_n)$ ++| is equivalent to system $t_1 \doteq s_1,\ldots,t_n \doteq s_n$++
-  * equation $f(t_1,\ldots,t_n) \doteq g(s_1,\ldots,s_m)$ where $f \neq g$ ++| is contradictory++
-  * equation $x \doteq t(x)$ where $t$ is a term containing $x$ but not identical to $x$ ++| is contradictory++
-===== Unification Examples =====
-First-order language:
-  * $R$ - binary relation symbol
-  * $f$ - binary function symbol
-  * $a,b,c$ - constants
-  * $x,y,z,u,v,w$ - variables
-**Example 1**
-\[
-  R(x,f(x,y)) \doteq R(f(a,v),f(f(u,b),f(u,u)))
-\]
-**Example 2**
-\[
-  R(x,f(x,x)) \doteq R(f(a,v),f(f(u,b),f(u,u)))
-\]
-**Example 3**
-\[
-  R(x,f(x,y)) \doteq R(f(u,v),v)
-\]
-**Definition:** Unifier of a set $E$ of syntactic equations is a substitution that makes all equations true.
-**Definition:** Composition $\circ$ of substitutions.
-**Definition:** A //renaming// is a substitution which is (a total function and) a bijection on the set of all variables.
-**Lemma:** If $\sigma$ is a unifier for $E$ and $\tau$ is a substitution, then $\tau \circ \sigma$ is also a unifier.
-A unifier is a solution of equations.  mgu is the most general solution.
-**Definition:** A most general unifier for $E$ is a unifier $\sigma$ such that if $\sigma'$ is another unifier, then there exists a substitution $\tau$ such that $\sigma' = \tau \circ \sigma$.
-===== Algorithm =====
-We next sketch an algorithm for computing mgu.
-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
-\[
-   \{x_1,\ldots,x_n \} \cap (FV(t_1) \cup \ldots FV(t_n)) = \emptyset
-\]
-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.
-=== Orient ===
-Select $t \doteq x$ where t is not x, and replace it with $x \doteq t$.
-=== Delete ===
-Select $x \doteq x$, remove it.
-=== Eliminate ===
-Given $x \doteq t$ where $x$ does not occur in $t$, substitute $x$ with $t$ in all remaining equations.
-=== Occurs Check ===
-Given $x \doteq t$ where $x$ occurs in $t$, report clash.
-=== Decomposition ===
-Given $f(t_1,\ldots,t_n) \doteq f(s_1,\ldots,s_n)$, replace it with $t_1 \doteq s_1,\ldots,t_n \doteq s_n$.
-=== Decomposition Clash ===
-Given $f(t_1,\ldots,t_n) \doteq g(s_1,\ldots,s_m)$ for $f$ not $g$, report clash
-===== References =====
-  * [[http://www.cs.bu.edu/fac/snyder/publications/UnifChapter.pdf|Unification Theory Chapter in Handbook of Automated Reasoning]] (also {{sav08:unification-handbook.pdf|pdf file}}, see pdf page 10, Rule based approach)
-  * [[http://doi.acm.org/10.1145/357162.357169|An Efficient Unification Algorithm]] (also {{sav08:unification-p258-martelli.pdf|pdf file}})
-  * [[http://www.mpi-inf.mpg.de/~uwe/lehre/eqlogic/v5.pdf|Course at MPI]]