## A Mechanization of Strong Kleene Logic for Partial Functions (1994)

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Venue: | PROCEEDINGS OF THE 12TH CADE |

Citations: | 29 - 11 self |

### BibTeX

@INPROCEEDINGS{Kerber94amechanization,

author = {Manfred Kerber and Michael Kohlhase},

title = {A Mechanization of Strong Kleene Logic for Partial Functions},

booktitle = {PROCEEDINGS OF THE 12TH CADE},

year = {1994},

pages = {371--385},

publisher = {Springer Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using three-valued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of many-valued truth-functional logics. However, strong Kleene logic, where quantification is restricted and therefore not truth-functional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a resolution calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi.

### Citations

753 |
Introduction to Metamathematics
- Kleene
- 1952
(Show Context)
Citation Context ...”. This phenomenon of “truth value gaps” is studied in various systems of free logic [22, 10, 15]. A related approach that seems more available for mechanization has first been advocated by Kleene in =-=[14]-=-. He introduces an additional individual ⊥ denoting meaningless individuals and a third truth value u, standing for the “undefined” truth value. At first glance this seems to be a great deviation from... |

275 |
Foundation of Constructive Mathematics
- Beeson
- 1985
(Show Context)
Citation Context ...systems for partial functions. Both approaches offer unsorted operationalizations of the systems in sequent calculi by providing special subcalculi for reasoning about definedness. Other authors (cf. =-=[5, 8, 19, 24]-=-) have avoided the problems that accompany treating a third truth value, and simply consider all atomic expressions containing a meaningless term as false. This has the advantage that partial function... |

213 |
Outline of a mathematical theory of computation
- Scott
- 1970
(Show Context)
Citation Context ...his section we will define the three valued semantics for SKL by postulating an “undefined individual” ⊥ in the universe of discourse. Note that this is similar to the classical flat CPO construction =-=[20]-=-, but Kleene’s interpretation of truth values does not make u minimal. Since we are not interested in least fix-points, monotonicity does not play a role in this paper. Definition 23 (Strict Σ-Algebra... |

211 |
On denoting
- Russell
- 1905
(Show Context)
Citation Context ... standard way to deal with this situation in classical mathematics is to reject expressions like 00 as “meaningless”. This phenomenon of “truth value gaps” is studied in various systems of free logic =-=[22, 10, 15]-=-. A related approach that seems more available for mechanization has first been advocated by Kleene in [14]. He introduces an additional individual ⊥ denoting meaningless individuals and a third truth... |

83 | IMPS: An interactive mathematical proof system - Farmer, Guttman, et al. - 1993 |

75 |
A partial functions version of Church’s simple theory of types
- Farmer
- 1990
(Show Context)
Citation Context ...systems for partial functions. Both approaches offer unsorted operationalizations of the systems in sequent calculi by providing special subcalculi for reasoning about definedness. Other authors (cf. =-=[5, 8, 19, 24]-=-) have avoided the problems that accompany treating a third truth value, and simply consider all atomic expressions containing a meaningless term as false. This has the advantage that partial function... |

53 | Systematization of the finite many-valued logics througn the method of tableaux - Carnielli - 1987 |

53 |
A many-sorted calculus based on resolution and paramodulation
- Walther
- 1987
(Show Context)
Citation Context ... ∀x D(x)⇒!IR∗(x) 3 Resolution In this section we present a resolution calculus RPF with dynamic sorts that is a generalization of Weidenbach’s work [24, 25] with ideas from [2, 12]. In the literature =-=[23, 7, 18, 11]-=- there are various calculi for sorted logics that vary in deductive power but have in common that the sort information available to sort reasoning remains static over the length of a proof. These meth... |

50 | The substitutional framework for sorted deduction: Fundamental results on hybrid reasoning
- Frisch
- 1991
(Show Context)
Citation Context ... ∀x D(x)⇒!IR∗(x) 3 Resolution In this section we present a resolution calculus RPF with dynamic sorts that is a generalization of Weidenbach’s work [24, 25] with ideas from [2, 12]. In the literature =-=[23, 7, 18, 11]-=- there are various calculi for sorted logics that vary in deductive power but have in common that the sort information available to sort reasoning remains static over the length of a proof. These meth... |

50 |
A more expressive formulation of many sorted logics
- Cohn
- 1987
(Show Context)
Citation Context ... ∀x D(x)⇒!IR∗(x) 3 Resolution In this section we present a resolution calculus RPF with dynamic sorts that is a generalization of Weidenbach’s work [24, 25] with ideas from [2, 12]. In the literature =-=[23, 7, 18, 11]-=- there are various calculi for sorted logics that vary in deductive power but have in common that the sort information available to sort reasoning remains static over the length of a proof. These meth... |

41 |
Automated Deduction in Multiple-Valued Logics
- Hähnle
- 1993
(Show Context)
Citation Context ...imply labels situations that would be rejected in mathematical practice anyway. In recent years, methods for the operationalization of many-valued logics have been developed by Carnielli [6], Hähnle =-=[12]-=-, Baaz and Fermüller [2]. All of these logics have in common that they are truth-functional , that is, composed formulae obtain their truth values from their components and (for quantifiers) from all... |

38 |
A linear format for resolution with merging and a new technique for establishing completeness
- Anderson, Bledsoe
(Show Context)
Citation Context ...Let Φ be an unsatisfiable set of ground clauses, then there exists a RPF derivation of the empty clause from Φ. Proof: The proof is analogous to the standard k-parameter proof of Anderson and Bledsoe =-=[1]-=-. We show be induction on k := ∑ C∈Φ(card(C) − 1) that there exists a refutation for Φ. If k = 0 then Φ is a set of ground unit clauses. Therefore by Corollary 318 and the assumed unsatisfiability the... |

38 | Computational Aspects of an Order-Sorted Logic with Term Declarations - Schmidt-Schau - 1989 |

25 |
Resolution for many-valued logics
- Baaz, Fermüller
- 1992
(Show Context)
Citation Context ...at would be rejected in mathematical practice anyway. In recent years, methods for the operationalization of many-valued logics have been developed by Carnielli [6], Hähnle [12], Baaz and Fermüller =-=[2]-=-. All of these logics have in common that they are truth-functional , that is, composed formulae obtain their truth values from their components and (for quantifiers) from all instances of the scope. ... |

20 |
On sequents and tableaux for many-valued logics
- Carnielli
- 1991
(Show Context)
Citation Context ...truth value simply labels situations that would be rejected in mathematical practice anyway. In recent years, methods for the operationalization of many-valued logics have been developed by Carnielli =-=[6]-=-, Hähnle [12], Baaz and Fermüller [2]. All of these logics have in common that they are truth-functional , that is, composed formulae obtain their truth values from their components and (for quantif... |

19 |
Computational Aspects of an Order-Sorted Logic with Term Declarations
- Schmidt-Schauß
- 1989
(Show Context)
Citation Context |

15 |
A first order logic for partial functions
- Lucio-Carrrasco, Gavilanes-Franco
- 1989
(Show Context)
Citation Context ...ethods is impossible for Kleene logic, since his quantifiers only range over defined values, that is, not over ⊥. Kleene’s approach has been utilized by Tichy [21], LucioCarrasco and Gavilanes-Franco =-=[16]-=- to give logical systems for partial functions. Both approaches offer unsorted operationalizations of the systems in sequent calculi by providing special subcalculi for reasoning about definedness. Ot... |

13 |
Fraassen. Singular terms, truth-value gaps, and free logic
- van
- 1966
(Show Context)
Citation Context ... standard way to deal with this situation in classical mathematics is to reject expressions like 00 as “meaningless”. This phenomenon of “truth value gaps” is studied in various systems of free logic =-=[22, 10, 15]-=-. A related approach that seems more available for mechanization has first been advocated by Kleene in [14]. He introduces an additional individual ⊥ denoting meaningless individuals and a third truth... |

9 | Dual systems of sequents and tableaux for many-valued logics - Baaz, Fermüller, et al. - 1993 |

9 | A sorted logic using dynamic sorts - Weidenbach - 1991 |

8 |
Non-clausal resolution and superposition with selection and redundancy criteria
- Bachmair, Ganzinger
- 1992
(Show Context)
Citation Context ...r to be feasible for practical mathematics. In particular this calculus does not address the question of the efficient mechanization of equality, here paramodulation (cf. [17]) or even superposition (=-=[4]-=-) methods would be interesting to study. However, we believe that this endeavor will mainly involve the development of the sort aspects for these calculi, because we think that the aspects of three-va... |

8 |
Foundations of partial type theory
- Tichy
- 1982
(Show Context)
Citation Context ...Therefore a direct utilization of these methods is impossible for Kleene logic, since his quantifiers only range over defined values, that is, not over ⊥. Kleene’s approach has been utilized by Tichy =-=[21]-=-, LucioCarrasco and Gavilanes-Franco [16] to give logical systems for partial functions. Both approaches offer unsorted operationalizations of the systems in sequent calculi by providing special subca... |

7 |
A resolution calculus with dynamic sort structures and partial functions. SEKI-Report SR-89-23, Fachbereich Informatik, Universität
- Weidenbach
- 1989
(Show Context)
Citation Context ...systems for partial functions. Both approaches offer unsorted operationalizations of the systems in sequent calculi by providing special subcalculi for reasoning about definedness. Other authors (cf. =-=[5, 8, 19, 24]-=-) have avoided the problems that accompany treating a third truth value, and simply consider all atomic expressions containing a meaningless term as false. This has the advantage that partial function... |

6 | Automated Theorem Proving in Multiple Valued Logics - Hähnle - 1994 |

4 |
Logics without Existence Assumptions
- Schock
- 1968
(Show Context)
Citation Context |

3 |
Paramodulation and TP in first order theories with equality
- Robinson, Wos
- 1969
(Show Context)
Citation Context ...have to be considered in order to be feasible for practical mathematics. In particular this calculus does not address the question of the efficient mechanization of equality, here paramodulation (cf. =-=[17]-=-) or even superposition ([4]) methods would be interesting to study. However, we believe that this endeavor will mainly involve the development of the sort aspects for these calculi, because we think ... |

3 |
A sorted logic using dynamic sorts', Technical Report MPI-I-91-218, Max-Planck-Institut f ur Informatik, Im Stadtwald, Saarbr ucken
- Weidenbach
- 1991
(Show Context)
Citation Context ...ividuals. Our calculus can be seen as an extension of classical logic that combines methods from many-valued logics (cf. [2, 12]) for a correct treatment of the undefined and order-sorted logics (see =-=[24, 25]-=-) for an adequate treatment of the defined. It differs from the sequent calculus in [16] in that the use of dynamic sort techniques greatly simplifies the calculus. In particular in the version with s... |

2 |
Completeness theorems for some presupposition-free logics
- Leblanc, Thomason
- 1968
(Show Context)
Citation Context ... standard way to deal with this situation in classical mathematics is to reject expressions like 00 as “meaningless”. This phenomenon of “truth value gaps” is studied in various systems of free logic =-=[22, 10, 15]-=-. A related approach that seems more available for mechanization has first been advocated by Kleene in [14]. He introduces an additional individual ⊥ denoting meaningless individuals and a third truth... |

1 | Logics without Existence Assumptions. Almquist - Schock - 1968 |