# Paramodulation with Non-Monotonic Orderings and Simplification

## Summary (3 min read)

### 1 Introduction

- Another source of the interest in dropping ordering requirements is that, in many cases, it is not clear if a particular ordering will be good (e.g. for reducing the search space) in some given problem.
- The authors show that some redundancy notions w.r.t. (in particular, w.r.t. the reduction ordering r included in ) can be applied in this framework while keeping refutation completeness.
- Now, if the authors want to add redundancy notions, the first natural choice is to define them w.r.t. the ordering that they use in the ordered paramodulation inference rules.

### 2 Preliminaries

- There are some (quasi-)orderings that play a central role in their results (in particular, in the redundancy notions).
- In order to ease the reading, sometimes the authors will denote marked terms by superindexing their marked subterms with variables.
- The authors also define the following equivalence relation.
- Lemma 4 ·>m is stable under substitutions.

### 4 Paramodulation with Equations

- In the following the authors will assume that the marked terms of every equation do share the same substitution (if necessary, the substitutions can be extended).
- Moreover, variables (both marking and non-marking) of each pair of equations will be considered to be disjoint (if necessary, the variables can be renamed).
- The inference system E for E consists of the following single inference rule:.

### Paramodulation:

- Note that since the marking variables introduced by the previous inference rule are fresh, the authors are mantaining the invariant on disjointness of marking and non-marking variables.
- Observe that, as a particular case, if both premises have no marks and the leftmost premise can be oriented with respect to the reduction ordering at hand, then condition 4a is always fulfilled, and hence no marks are introduced in the conclusion, making their inference rule coincide with the usual ordered paramodulation inference rule.
- Here the authors introduce redundancy notions from a static point of view, that is, by first defining the notion of saturated set, regardless of how this saturated set can be obtained.
- R. Notice that universally quantifying the TRS R allows us to capture the particular TRS RE (see Definition 11) defining the model, which cannot be known in advance.
- The authors consider the case where sγθ is reducible (the other one is analogous).

### 5 General Clauses

- Here the authors extend the presented calculus to general first order clauses, and prove it complete.
- The authors consider that in each clause with a non-empty antecedent one of its negative equations, the one that is written underlined, has been selected.
- The authors use the orderings defined in Section 3.2, and assume that all marked terms in the premises of an inference share a unique substitution.
- The authors inference system G consists of the following four inference rules:.

### Equality factoring:

- The following example illustrates how the previous inference system works.
- The authors will show that for every terminating ground TRS R and every ground substitution ρ such that Cρ and Dρ are non-topmost variable irreducible w.r.t.
- In what follows the authors will precisely describe how they can apply forward simplification using marked rewriting.
- First of all, the authors will show how the empty clause can be derived under their inference system (which introduces the convenient marks).

### 6 Knuth-Bendix Completion

- Finding practically useful procedures remained as an open problem for a long time.
- Also, Devie showed that for left- and right linear E (i.e., no variable occurs more than once in a side of an equation) standard Knuth-Bendix completion finds R (Devie (1990)).
- But it was not 36 until in Bofill et al (1999, 2003) that a procedure for the general case, not relying on the enumeration of all equational consequences, was presented.
- From their current result, it directly follows that the method of Bofill et al (2003) can be made compatible (to some amount) with simplification by rewriting.
- Now the ordered paramodulation rule is also applied on top of the small sides, and hence the authors obtain an interreduced ground TRS for the model.

### 7 Experiments

- In order to check if their ideas are feasible at least for small examples the authors have developed a prototype written in Prolog that implements for the equational case the inference system given in Section 4 and the practical notions of redundancy described in Section 5.5.
- The second one defines a non-strict comparison.
- The final west-ordering is obtained as the union of the reduction ordering, the subterm relation and all instances of the given definitions.
- When saturating a set of equations, first of all, the system checks whether some of the equations can be oriented with respect to the reduction ordering, which allows us to make 38 some simplification initially.
- As explained throughout the paper, these marks have the effect of diminishing the amount of redundancy, as marked subterms are treated as variables for redundancy purposes.

### 8 Building West Orderings

- There are some known ways to build west orderings that are not monotonic.
- Still, the authors have monotonicity for some comparisons, and a monotonic subrelation of a given SPO can be extracted.
- This is exploited in the method called the Monotonic Semantic Path Ordering, (Borralleras et al (2000)), MSPO for short, which can be used to define the monotonic part of the SPO, and hence a reduction ordering inside the west ordering.
- Note that case 2 can as well be written the other way round, i.e. with f ∈FP, g /∈FP, preserving well-foundedness.

### 9 Conclusion

- By adapting those refutation complete inference systems so that they are compatible with powerful redundancy elimination techniques such as demodulation and, hence, making them more likely to be used in practice.the authors.
- The authors have proposed some inference systems (for equations and general first order clauses) that work with a pair of orderings: a west ordering , which is used when performing inferences, and a reduction ordering r, included in , which is used for applying simplification by rewriting.
- As stated in Section 5.5, in their notions of redundancy it is necessary that non-marking variables occurring in the skeleton of a marked term are only instantiated with variables when performing redundancy.
- The authors results on Knuth-Bendix completion could also be extended in the line of the results on completion with termination tools in Wehrman et al (2006); Winkler and Middeldorp (2010).
- Note that, in their case fixing in advance the non-monotonic ordering and the reduction ordering included in it (using techniques like the ones given in Section 8) is even harder than in the standard case, where only a reduction ordering is needed.

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##### Citations

34 citations

### Cites background from "Paramodulation with Non-Monotonic O..."

...Also of interest is Bofill and Rubio’s [22] integration of nonmonotonic orders in ordered paramodulation, a precursor of superposition....

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...Also of interest is Bofill and Rubio’s [13] integration of nonmonotonic orders in ordered paramodulation, a precursor of superposition....

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### Cites background from "Paramodulation with Non-Monotonic O..."

...Even without this property, we expect the orders to be usable in a λ-free higher-order generalization of superposition, possibly at the cost of some complications [19]....

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### Cites background from "Paramodulation with Non-Monotonic O..."

...Nonetheless, we expect the order to be usable for λ-free higher-order superposition, at the cost of some complications [13]....

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### "Paramodulation with Non-Monotonic O..." refers background in this paper

..., [2]) or the recursive path ordering [15] with argument filterings ([1, 21]) cannot be used, because they do not fulfill the subterm property....

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