## Modular proof systems for partial functions with Evans equality (2006)

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Venue: | Information and Computation |

Citations: | 25 - 13 self |

### BibTeX

@ARTICLE{Ganzinger06modularproof,

author = {Harald Ganzinger and Viorica Sofronie-stokkermans and Uwe Waldmann},

title = {Modular proof systems for partial functions with Evans equality},

journal = {Information and Computation},

year = {2006},

volume = {204},

pages = {1453--1492}

}

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### Abstract

The paper presents a modular superposition calculus for the combination of firstorder theories involving both total and partial functions. The modularity of the calculus is a consequence of the fact that all the inferences are pure – only involving clauses over the alphabet of either one, but not both, of the theories – when refuting goals represented by sets of pure formulae. The calculus is shown to be complete provided that functions that are not in the intersection of the component signatures are declared as partial. This result also means that if the unsatisfiability of a goal modulo the combined theory does not depend on the totality of the functions in the extensions, the inconsistency will be effectively found. Moreover, we consider a constraint superposition calculus for the case of hierarchical theories and show that it has a related modularity property. Finally we identify cases where the partial models can always be made total so that modular superposition is also complete with respect to the standard (total function) semantics of the theories. 1

### Citations

755 |
Introduction to Metamathematics
- Kleene
- 1964
(Show Context)
Citation Context ...′ by id(t) ≈ id(t ′ ). Negative literals are not changed. Resolution calculi for partial functions and partial congruences. An alternative way to dealing with undefinedness, which goes back to Kleene =-=[21]-=-, is to use many-valued logic, with an additional truth value for “undefined”. Kleene’s logic has been used by various authors for giving logical systems for partial functions and for reasoning about ... |

413 | Simplification by cooperating decision procedures
- Nelson, Oppen
- 1979
(Show Context)
Citation Context ...e two signatures. For stably infinite theories where, in addition, Σ1 ∩ Σ2 consists of constants only, pure inference systems exist. This is one of the main consequences of Nelson and Oppen’s results =-=[23]-=- (also see, e.g., Tinelli and Harandi [27] for additional clarification). The results have recently been extended to some situations when the theories T1 and T2 share also non-constant function symbol... |

202 |
Rewrite-based equational theorem proving with selection and simplification
- Bachmair, Ganzinger
- 1994
(Show Context)
Citation Context ...rified clauses. This line of reasearch was pursued e.g. by Hillenbrand [18], who reestablished the correctness of the Nelson-Oppen combination procedure as a consequence of the superposition calculus =-=[3]-=-. Conditions when pure inferences are sufficient for checking unsatisfiability of purified goals in more general combinations of theories were identified by Tinelli and by Ghilardi [26,15]. The presen... |

108 |
Universal Algebra
- Grätzer
- 1968
(Show Context)
Citation Context ...nd compared with the results presented in the paper. Validity of identities in partial algebras. Evans validity is often related to properties of embeddability of partial algebras into total algebras =-=[11,7,17]-=-. This connection allows us to replace equational reasoning for total functions with reasoning about partial functions, or with relational reasoning. Evans validity was also used in (Ganzinger [12]) f... |

77 | A new correctness proof of the Nelson–Oppen combination procedure - Tinelli, Harandi - 1996 |

63 | Automatic recognition of tractability in inference relations
- McAllester
- 1990
(Show Context)
Citation Context ...bols in Σ1\Σ0 are declared as partial, and all constants in Ωc are declared as total. This definition is related to the notion of local equational theory introduced in [12] and of locality in general =-=[16,22]-=-. Example 60 (Sofronie-Stokkermans [25]). The following theory extensions are local: (1) Extensions with free functions: Any extension of a theory T0 with a set of free function symbols is local. (2) ... |

51 |
A Model Theoretic Oriented Approach to Partial Algebras
- Burmeister
- 1986
(Show Context)
Citation Context ...sides Evans validity [11,7,17] there are many other possibilities for defining validity of identities in partial algebras, from which we mention only a few (for further details we refer to Burmeister =-=[6]-=-): – existential validity: (A, β) |= t e ≈ t ′ if and only if (A, β)(t) and (A, β)(t ′ ) are both defined and equal; – strong validity: (A, β) |= t s ≈ t ′ if and only if either both (A, β)(t) and (A,... |

43 | Model theoretic methods in combined constraint satisfiability
- Ghilardi
- 2005
(Show Context)
Citation Context .... g., Tinelli and Harandi [27] for additional clarification). The results have recently been extended to some situations when the theories T1 and T2 share also non-constant function symbols. Ghilardi =-=[14]-=- extended the completeness results for modular inference systems to a more general case of “compatibility” between the component theories Ti. Future work might aim at weakening these compatibility req... |

40 | Hierarchic reasoning in local theory extensions
- Sofronie-Stokkermans
(Show Context)
Citation Context ...eories) also in (Ganzinger [13], 22 Tinelli and Zarba [28]). 5.2 Local extensions of a base theory A more general, but related, case is that of local extensions of a base theory (Sofronie-Stokkermans =-=[25]-=-). The definitions we present here are somewhat more restricted than those in [25], as they refer only to extensions with flat sets of clauses and flat goals. Definition 58. A term is called variable-... |

34 | Combining non-stably infinite theories
- Tinelli, Zarba
(Show Context)
Citation Context ...where n is the size of the input, and k is a constant natural number. This provides an alternative proof of a result established (for arbitrary theories) also in (Ganzinger [13], 22 Tinelli and Zarba =-=[28]-=-). 5.2 Local extensions of a base theory A more general, but related, case is that of local extensions of a base theory (Sofronie-Stokkermans [25]). The definitions we present here are somewhat more r... |

32 | On a rewriting approach to satisfiability procedures: extension, combination of theories and an experimental appraisal
- Armando, Bonacina, et al.
(Show Context)
Citation Context ...ith a theory of elements. This works if the theory of elements is stably infinite. Similar methods are applied for reasoning in certain combinations of theories (Armando, Bonacina, Ranise, and Schulz =-=[1]-=-). Our approach is different. We show that if the extensions only introduce additional partial functions, a superposition calculus for partial functions becomes a complete and modular proof system whe... |

32 |
E-resolution: extension of resolution to include the equality relation
- Morris
- 1969
(Show Context)
Citation Context ...so a model of F . 15 Instead of abstracting out non-variable Σ0-terms eagerly, one can also treat nonvariable Σ0-subterms in the unification algorithm in a similar way as in Morris’s equi-unification =-=[22]-=-, i. e., by turning an appropriate disagreement set into a list of negative literals. We have done this in a previous version [13] of this paper. 26sN |=T0 ⊥ if and only if N |= TG T0 ⊥. Proof. Analog... |

29 | A mechanization of strong Kleene logic for partial functions. SEKI-Report SR-93-20 (SFB), Universität des Saarlandes
- Kerber, Kohlhase
- 1993
(Show Context)
Citation Context ...l functions in a many-valued framework. A resolution calculus for partial functions, where undefinedness is formalized using Kleene’s strong three valued logic, was proposed by Kerber and Kohlhase in =-=[20]-=-. Although we also use a three-valued logic for modeling undefinedness, where the negation is similar to Kleene’s strong negation, the notion of validity used in [20] is different from the one we use,... |

28 | New results on local inference relations - Givan, McAllester - 1992 |

27 | Ordered Chaining Calculi for First-Order Theories of Transitive Relations
- Bachmair, Ganzinger
- 1998
(Show Context)
Citation Context ...ecause of the different notion of validity mentioned and on the other hand because refinements of resolution such as paramodulation or superposition are not considered in [20]. Bachmair and Ganzinger =-=[4]-=- give a version of ordered chaining for partial equivalence and congruence axioms. This calculus is devised for strong or existential validity; consequently, equality resolution is replaced with a rul... |

22 |
The word problem for abstract algebras
- Evans
- 1951
(Show Context)
Citation Context ...idity We consider extensions of a base theory with partial functions. The semantics for partial functions we consider is known as “Evans validity”. It was introduced, in the equational case, by Evans =-=[10,11]-=-, while identifying situations when the uniform word problem in classes of algebras axiomatized by a set E of identities is decidable in PTIME. We briefly present Evans’ method and his motivation for ... |

22 | Relating semantic and proof-theoretic concepts for polynomial time decidability of uniform word problems - Ganzinger - 2001 |

21 |
Basic Paramodulation and Decidable Theories
- Nieuwenhuis
- 1996
(Show Context)
Citation Context ...if T1 \ T0 consists only of shallow clauses. The definition of shallow terms given above is a generalization of the corresponding notion used e.g. by Comon, Haberstrau, and Jouannaud [8], Nieuwenhuis =-=[24]-=-, or Jacquemard, Meyer and Weidenbach [19]. The difference is that we consider terms which are shallow w.r.t. a subset of the function symbols, whose elements are declared to be partial. When defining... |

20 |
Refutational theorem proving for hierarchic first-order theories
- Bachmair, Ganzinger, et al.
- 1994
(Show Context)
Citation Context ...p to redundancy if all inferences between clauses in N are redundant w.r.t. N. A saturated set can be obtained as the limit of a fair theorem proving derivation (see Bachmair, Ganzinger, and Waldmann =-=[5]-=- for the details). We will show that the partial superposition calculus is refutationally complete, that is, that a saturated set of clauses has a model if and only if it does not contain the empty cl... |

19 | Polynomial time uniform word problems
- Burris
- 1995
(Show Context)
Citation Context ...sion of the completion algorithm to first-order clauses. Another reason is that embeddability conditions for partial algebras satisfying (in Evans’ sense) sets of identities or Horn clauses were used =-=[10,11,7]-=- to obtain results on PTIME decidability of (uniform) word problems. We use similar embeddability results in Section 5 to establish a link between extensions with partial and extensions with total fun... |

15 |
Michaël Rusinowitch. A rewriting approach to satisfiability procedures
- Armando, Ranise
(Show Context)
Citation Context ...ary function symbol). 44stensions. 27 Superposition was used for reasoning in combinations of theories over signatures sharing only constants by Hillenbrand [18]. In (Armando, Rusinowitch, and Ranise =-=[2]-=-) superposition with selection is tuned to specific theories (such as lists and arrays, possibly with extensionality) and proved to yield decision procedures with optimal complexity. These results are... |

15 |
Equality-Based binary Resolution
- Digricoli, Harrison
- 1986
(Show Context)
Citation Context ...del of F. 15 Instead of abstracting out non-variable Σ0-terms eagerly, one can also treat nonvariable Σ0-subterms in the unification algorithm in a similar way as in Digricoli and Harrison’s RUE rule =-=[9]-=-, i. e., by turning an appropriate disagreement set into a list of negative literals. We have done this in a previous version [14] of this paper. 26sWe call sorts from S0 base sorts and sorts from S1 ... |

14 | Cooperation of background reasoners in theory reasoning by residue sharing
- Tinelli
- 2003
(Show Context)
Citation Context ...results for modular inference systems to a more general case of “compatibility” between the component theories Ti. Future work might aim at weakening these compatibility requirements even further. In =-=[26]-=-, Tinelli shows that similar modularity results are achieved if the theories share all their function symbols. In this paper we take a different point of departure. We will consider arbitrary theory m... |

11 |
Decidable problems in shallow equational theories (extended abstract
- Comon, Haberstrau, et al.
- 1992
(Show Context)
Citation Context ... ⊆ T1 is shallow if T1 \ T0 consists only of shallow clauses. The definition of shallow terms given above is a generalization of the corresponding notion used e.g. by Comon, Haberstrau, and Jouannaud =-=[8]-=-, Nieuwenhuis [24], or Jacquemard, Meyer and Weidenbach [19]. The difference is that we consider terms which are shallow w.r.t. a subset of the function symbols, whose elements are declared to be part... |

10 | Unification in Extensions of Shallow Equational Theories
- Jacquemard, Meyer, et al.
- 1998
(Show Context)
Citation Context ...s. The definition of shallow terms given above is a generalization of the corresponding notion used e.g. by Comon, Haberstrau, and Jouannaud [8], Nieuwenhuis [24], or Jacquemard, Meyer and Weidenbach =-=[19]-=-. The difference is that we consider terms which are shallow w.r.t. a subset of the function symbols, whose elements are declared to be partial. When defining shallow clauses we require that terms con... |

8 |
Embeddability and the word problem
- Evans, T
- 1953
(Show Context)
Citation Context ...idity We consider extensions of a base theory with partial functions. The semantics for partial functions we consider is known as “Evans validity”. It was introduced, in the equational case, by Evans =-=[10,11]-=-, while identifying situations when the uniform word problem in classes of algebras axiomatized by a set E of identities is decidable in PTIME. We briefly present Evans’ method and his motivation for ... |

8 |
Quantifier elimination and provers integration
- Ghilardi
- 2003
(Show Context)
Citation Context ...e.g., Tinelli and Harandi [27] for additional clarification). The results have recently been extended to some situations when the theories T1 and T2 share also non-constant function symbols. Ghilardi =-=[15]-=- extended the completeness results for modular inference systems to a more general case of “compatibility” between the component theories Ti. Future work might aim at weakening these compatibility req... |

2 |
A superposition view on Nelson-Oppen
- Hillenbrand
- 2004
(Show Context)
Citation Context ...Alternatively, in a “refutational” variant of the Nelson-Oppen procedure, one can analyze all inferences from the set Γ1∪Γ2 of purified clauses. This line of reasearch was pursued e.g. by Hillenbrand =-=[18]-=-, who reestablished the correctness of the Nelson-Oppen combination procedure as a consequence of the superposition calculus [3]. Conditions when pure inferences are sufficient for checking unsatisfia... |