- English only

# Lab for Automated Reasoning and Analysis LARA

# Differences

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

sav08:ground_terms [2013/05/10 09:53] vkuncak |
sav08:ground_terms [2015/04/21 17:30] (current) |
||
---|---|---|---|

Line 4: | Line 4: | ||

//Ground term// is a term $t$ without variables, i.e. $FV(t)=\emptyset$, i.e. given by grammar: | //Ground term// is a term $t$ without variables, i.e. $FV(t)=\emptyset$, i.e. given by grammar: | ||

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

GT ::= f(GT,\ldots,GT) | GT ::= f(GT,\ldots,GT) | ||

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

i.e. built from constants using function symbols. | i.e. built from constants using function symbols. | ||

Line 15: | Line 15: | ||

If ${\cal L}$ has no constants then $GT$ is empty. In that case, we add a fresh constant $a_0$ into the language and consider ${\cal L} \cup \{a_0\}$ that has a non-empty $GT$. We call the set $GT$ Herbrand Universe. | If ${\cal L}$ has no constants then $GT$ is empty. In that case, we add a fresh constant $a_0$ into the language and consider ${\cal L} \cup \{a_0\}$ that has a non-empty $GT$. We call the set $GT$ Herbrand Universe. | ||

- | Goal: show that if a formula //without equality// (for now) has a model, then it has a model whose domain is Herbrand universe, that is, a model of the form $I_H = (HU,\alpha_H)$. | + | Goal: show that if a formula //without equality// (for now) has a model, then it has a model whose domain is Herbrand universe, that is, a model of the form $I_H = (GT,\alpha_H)$. |

How to define $\alpha_H$? | How to define $\alpha_H$? | ||

Line 35: | Line 35: | ||

If $R \in {\cal L}$, $ar(R)=n$ and $t_1,\ldots,t_n \in GT$, we call $R(t_1,\ldots,t_n)$ an Herbrand Atom. HA is the set of all Herbrand atoms: | If $R \in {\cal L}$, $ar(R)=n$ and $t_1,\ldots,t_n \in GT$, we call $R(t_1,\ldots,t_n)$ an Herbrand Atom. HA is the set of all Herbrand atoms: | ||

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

HA = \{ R(t_1,\ldots,t_n) \mid R \in {\cal L}\ \land \ t_1,\ldots,t_n \in GT \} | HA = \{ R(t_1,\ldots,t_n) \mid R \in {\cal L}\ \land \ t_1,\ldots,t_n \in GT \} | ||

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

We order elements of $HA$ in sequence (e.g. sorted by length) and establish a bijection $p$ with propositional variables | We order elements of $HA$ in sequence (e.g. sorted by length) and establish a bijection $p$ with propositional variables | ||

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

p : HA \to V | p : HA \to V | ||

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

We will write $p(A)$. | We will write $p(A)$. | ||