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Logic for Computer Scientists/Predicate Logic/SATCHMO

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The SATCHMO Theorem Prover was one of the first systems which used model generation, i.e. a bottom-up proof procedure. The prover was given by a small Prolog-program, which implements a tableau proof procedure. One restriction is that it requires range restricted formulae.

Definition 30Edit

A first order clause   is called range restricted if every variable which occurs in the head   occurs in the body   as well.

  1. Convert clauses to range restricted form:
  2. assert range-restricted clauses and dom clauses in Prolog database.
  3. Call satisfiable:
kill satisfiable :-    assume(X) :- asserta(X).     
       (Head <- Body)            assume(X) :-  
       Body, not Head, !,          retract(X), !, fail.  
       component(HLit, Head),      component(E, (E ; _)).      
       assume(HLit),               component(E, (_ ; R)) :-    
       not false,                   !, component(E, R).   
       satisfiable.                component(E, E).  

First-Order completeness via Level-Saturation modification. This proof procedure implements Hyper Tableaux in the ground case.

Hyper Tableau - Ground CaseEdit


All open branches consist of positive literals only Take the following clause set as an example  


Definition 31 (Literal tree, Clausal Tableau)Edit

A literal tree is a pair   consisting of a finite, ordered tree   and a labeling function   that assigns a literal to every non-root node of  .

The successor sequence of a node   in an ordered tree   is the sequence of nodes with immediate predecessor  , in the order given by  .

A (clausal) tableau   of a set of clauses   is a literal tree   in which, for every successor sequence   in   labeled with literals  , respectively, there is a substitution   and a clause   with   for every  .   is called a tableau clause and the elements of a tableau clause are called tableau literals.

Definition 32 (Branch, Open and Closed Tableau, Selection Function)Edit

A branch of a tableau   is a sequence   ( ) of nodes in   such that   is the root of  ,   is the immediate predecessor of   for  , and   is a leaf of  . We say branch   is a prefix of branch  , written as   or  , iff   for some nodes  ,  . The branch literals of branch   are the set  . We find it convenient to use a branch in place where a literal set is required, and mean its branch literals. For instance, we will write expressions like   instead of  .

In order to memorize the fact that a branch contains a contradiction, we allow to label a branch as either open or em closed. A tableau is closed if each of its branches is closed, otherwise it is open.

A selection function is a total function   which maps an open tableau to one of its open branches. If   we also say that   is selected in   by  .

Note that branches are always finite, as tableaux are finite. Fortunately, there is no restriction on which selection function to use. For instance, one can use a selection function which always selects the "leftmost" branch.

Definition 33 (Hyper Tableau - Ground Case)Edit

Let   be a finite set of clauses and   be a selection function. Hyper tableaux for   are inductively defined as follows:
Initialization step: A one node literal tree is a hyper tableau for  . Its single branch is marked as "open".

Hyper extension step: If

  1.   is an open hyper tableau for  ,   (i.e.   is selected in   by  ) with open leaf node  , and
  2.   is a clause from   ( ,  ), called extending clause in this context, and
  3. such that   (referred to as hyper condition)

then the literal tree   is a hyper tableau for  , where   is obtained from   by attaching   child nodes   to   with respective labels


and marking every new branch   with positive leaf as "open", and marking every new branch   with negative leaf as "closed".

Minimal Model ReasoningEdit

The clause set   obviously has two different models:   and  . Under set inclusion these models can be compared and there are some tasks where it is appropriate to compute the (or in general a) smallest one. This is for example the case with

  • Knowledge Representation, Circumscription
  • Basis for default negation (GCWA)
  • Applications: Deductive database updates, Diagnosis

There are basically two different methods to compute minimal models.

Minimal Model Reasoning – Niemel¨a’s ApproachEdit

Given a set of ground clauses   the methods applies a model generating procedure, e.g. hyper tableau, which is able to generate all models.

Lemma 1: For every minimal model   for   there is a branch with literals  .

Assume that   is the set of atoms, which occur in the head of a clause from  , than the following Lemma holds.

Lemma 2:   is a minimal model for   iff  

This offers a general method: Generate model candidates, and test with Lemma 2.

  is not a minimal model in our example from above, because   iff   is unsatisfiable, which is not the case, hence   does not correspond to a minimal model and hence the branch is closed.

  is minimal because   iff   is unsatisfiable. This is the case and hence   is minimal and the branch remains open.

Properties: Soundness (by Lemma 2) Completeness (by Lemma 1), space efficiency.

Minimal Model Reasoning – Bry& Yayha‘s ApproachEdit

As an example we have the set  


Lemma: With complement splitting, the leftmost open branch is a minimal model for  .

General method: Repeat: generate minimal model  , add   to  . Properties: Soundness (by Lemma) Completeness as before, possibly exponentially many new clauses  .