# Paramodulation with Well-founded Orderings

## Summary (3 min read)

### 1 Introduction

- Equations are ubiquitous in mathematics, logic, and many areas of computer science.
- If the authors take the opposite orientation, i.e., ffg(x)→ fgfg(x) the system is already closed, i.e., standard Knuth-Bendix completion would generate only this additional rule.
- Another typical situation is in deduction modulo built-in equational theories E. And, even worse, for many theories E such orderings cannot exist.
- Another (less obvious) potential application of their results may be in goaloriented deduction since, in some cases, a goal-oriented paramodulation proof can only be obtained if the ordering contradicts the subterm property.

### 1.1 About the proof technique

- The authors proof follows the lines of Bachmair and Ganzinger’s model generation proof method of [3] and its variation used in [7] for the non-monotonic case.
- In order to express these blockings the authors use abstracted terms, which have variables in the place of blocked subterms.
- In fact, most of this paper is devoted to the proof of confluence of the generated rewrite system R. Once confluence is proved, the authors can easily show that there exists a B-rewrite proof with R for every logical consequence of the set of equations from which R has been generated.
- In Section 5, using the results obtained for the equational case, the authors prove refutation completeness of ordered paramodulation with well-founded orderings for the case of Horn clauses with equality.

### 2 Basic notions

- The authors use the standard definitions and notation of [10,17].
- Note that constants and variables have size 1 (since n = 0).
- By Newman’s lemma, terminating locally-confluent relations are confluent.
- (It is commonly assumed that lefthand sides of rules are not variables, and that all variables occurring in a righthand side also occur in the corresponding lefthand side.).
- Every well-founded ordering can be totalized [20], and hence every well-founded ordering satisfying the subterm property can be extended to a west-ordering.

### 3 Blocked rewriting

- Here the authors define a special kind of rewrite relation, called blocked rewriting (Brewriting for short), which is an essential ingredient for their completeness proof.
- This relation restricts standard rewriting by mainly forbidding rewrite steps below righthand sides introduced in previous rewrite steps.
- This makes this relation terminating by definition.
- In order to express these blockings, the authors use abstracted terms, which have variables in the place of the blocked subterms.
- Two abstracted terms s·γ and u·δ are called equivalent, denoted by s·γ ≡ t·δ, if s and t have the same variable positions and sγ ≡ uδ.

### 4 Paramodulation with equations

- This section is devoted to the pure equational case.
- The authors will also assume that all equations at hand have disjoint variables (if necessary, they can be renamed).

### 4.1 The inference system

- Note that this situation is possible, together with lθσ ≻ rθσ, as the ordering ≻ is allowed to contradict the subterm property.
- The authors must note that the need of paramodulations from variables into variables at top positions has to do with their proof technique.
- At present, the authors have not found any counterexample showing incompleteness when paramodulation into variables is completely disallowed.
- Let ≻ be the well-founded ordering used in the ordered paramodulation inference rule E.

### 4.2 The rewrite system

- In order to take into account the fact that RE will be used applying Brewriting, and hence the righthand sides will become blocked after each rewrite step, in the construction of RE the authors will not require irreducibility at those positions that are below subterms belonging to Right(RE).
- The main difference is that left- and righthand sides of rules can be reducible at positions below subterms in Right(RE).
- For technical reasons, many of the results are proved for R◦E .

### 4.3 Some relevant properties

- Recall that the generated RE can be overlapping and non-terminating and, hence, confluence does not follow easily.
- This property generalizes the notion in [7] of irreducibility for the rules of the ground TRS that defines the model.
- Altogether, this lets us conclude by the induction hypothesis.
- Then there are no two rules in RE with the same lefthand side.

### 4.4 The Return Property

- This strong property is crucial to prove that their generated RE , in spite of being potentially overlapping and non-terminating, is confluent: after the Return Property is proved, confluence follows from the fact that the lefthand sides only overlap below subterms corresponding to righthand sides.
- The following lemma states that, in a sequence by B-rewriting with a TRS R, if some step has taken place at the border, w.r.t. Right(R), of the initial term, then it is not possible to get back to the initial term until a normal form has been reached.
- Now the authors prove the main lemma in which the Return Property is based.
- Let σ′′ be the substitution of that instance.

### 4.5 Rewrite proofs for the equational case

- The Return Property is the cornerstone for proving local confluence.
- So the diamond closes with a final step with the same rule.
- The authors proceed by induction on size of the terms without counting the variables, i.e., |u1|v + |u2|v. Lemma 30.
- Note that rewriting with yRE also provides a rewrite sequence with RE .

### 6 Conclusion

- By showing that wellfoundedness of the term ordering is sufficient for completeness of ordered paramodulation for Horn clauses with equality.the authors.
- The authors hope that the shorter and easier to read proof that they have presented here can open the door to extend their completeness results to deduction modulo any theory E with a decidable E-unification problem.
- Moreover, like in [7], similar problems concerning the use of demodulation, i.e., simplification by rewriting, arise.
- The problem of providing concrete redundancy notions for paramodulation with non-monotonic orderings has been addressed in [9], where results on restricted forms of demodulation for paramodulation with well-founded orderings fulfilling the subterm property, are given.
- The problem is that, within the model generation proof technique, the completeness for general clauses is usually based on the fact that, at most, one positive equation is satisfied in the generated model.

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##### Frequently Asked Questions (2)

###### Q2. What have the authors stated for future works in "Paramodulation with well-founded orderings" ?

The authors hope that the shorter and easier to read proof that they have presented here can open the door to extend their completeness results to deduction modulo any theory E with a decidable E-unification problem. Although these potential applications are interesting, additional techniques for redundancy elimination, like simplification by rewriting, should be investigated in this framework to make the calculus useful in practice. However, the authors have found no counterexample showing that subsumption can cause incompleteness. B-rewriting is closely related to basic strategies ( see [ 5,15 ] ), which the authors believe can be compatible with their results.