## Normal Forms in Combinatory Logic (1992)

Venue: | Wesleyan University |

Citations: | 1 - 1 self |

### BibTeX

@TECHREPORT{Johann92normalforms,

author = {Patricia Johann},

title = {Normal Forms in Combinatory Logic},

institution = {Wesleyan University},

year = {1992}

}

### OpenURL

### Abstract

Abstract Let R be a convergent term rewriting system, and let CR-equality on (simply typed) combinatory logic terms be the equality induced by βηRequality on terms of the (simply typed) lambda calculus under any of the standard translations between these two frameworks for higher-order reasoning. We generalize the classical notion of strong reduction to a reduction relation which generates CR-equality and whose irreducibles are exactly the translates of long βR-normal forms. The classical notion of strong normal form in combinatory logic is also generalized, yielding yet another description of these translates. Their resulting tripartite characterization extends to the combined first-order algebraic and higher-order setting the classical combinatory logic descriptions of the translates of long β-normal forms in the lambda calculus. As a consequence, the translates of long βR-normal forms are easily seen to serve as canonical representatives for CR-equivalence classes of combinatory logic terms for nonempty, as well as for empty, R. 573

### Citations

1135 | The Lambda-Calculus, its syntax and semantics - Barendregt - 1984 |

249 | Combinatory Reduction Systems - Klop - 1980 |

117 |
Introduction to Combinators and λCalculus
- Hindley, Seldin
- 1986
(Show Context)
Citation Context ... Huet [16] and Gallier and Snyder [10]). Reflecting βη-equality, C-equality has been well-studied in the literature (see Curry and Feys [5], Hindley [11], Hindley and Lercher [13], Hindley and Seldin =-=[15]-=-, Lercher [20] and [21], and Mezghiche [22]). Since long β-normal forms are fundamentally important in the investigation of βη-equality, it is natural to look for a reduction relation on CL capturing ... |

116 |
A unification algorithm for typed λ-calculus
- Huet
- 1975
(Show Context)
Citation Context ...forms can also be taken as a class of canonical representatives for these purposes. Long β-normal forms have proven quite useful in the study of higher-order unification methods (see for example Huet =-=[16]-=- and Gallier and Snyder [10]). Reflecting βη-equality, C-equality has been well-studied in the literature (see Curry and Feys [5], Hindley [11], Hindley and Lercher [13], Hindley and Seldin [15], Lerc... |

53 |
Higher-order unification revisited: Complete sets of transformations
- Gallier, Snyder
- 1989
(Show Context)
Citation Context ... class of canonical representatives for these purposes. Long β-normal forms have proven quite useful in the study of higher-order unification methods (see for example Huet [16] and Gallier and Snyder =-=[10]-=-). Reflecting βη-equality, C-equality has been well-studied in the literature (see Curry and Feys [5], Hindley [11], Hindley and Lercher [13], Hindley and Seldin [15], Lercher [20] and [21], and Mezgh... |

52 |
Polymorphic Rewriting Conserves Algebraic Strong Normalization, Theoretical Computer Science 83
- Breazu-Tannen, Gallier
- 1991
(Show Context)
Citation Context ...ic terms for nonempty, as well as for empty, R. 573 1Introduction The interaction between higher-order and first-order algebraic reasoning has recently received much attention (see Breazu-Tannen [2], =-=[3]-=-, and [4], and Dougherty [8]), particularly the situation in which the equational theory in question admits presentation as a convergent (confluent and terminating) term rewriting system R. Reasoning ... |

22 | Adding Algebraic Rewriting to the Untyped Lambda Calculus
- Dougherty
- 1992
(Show Context)
Citation Context ...ll as for empty, R. 573 1Introduction The interaction between higher-order and first-order algebraic reasoning has recently received much attention (see Breazu-Tannen [2], [3], and [4], and Dougherty =-=[8]-=-), particularly the situation in which the equational theory in question admits presentation as a convergent (confluent and terminating) term rewriting system R. Reasoning in theories combining rewrit... |

19 | Improving Basic Narrowing Techniques - Réty - 1987 |

18 |
Polymorphic Rewriting Conserves Algebraic Confluence
- Breazu-Tannen, Gallier
(Show Context)
Citation Context ...for nonempty, as well as for empty, R. 573 1Introduction The interaction between higher-order and first-order algebraic reasoning has recently received much attention (see Breazu-Tannen [2], [3], and =-=[4]-=-, and Dougherty [8]), particularly the situation in which the equational theory in question admits presentation as a convergent (confluent and terminating) term rewriting system R. Reasoning in theori... |

16 | Introduction to combinatory logic - Hindley, Lercher, et al. - 1972 |

9 | A Combinatory Logic Approach to Higher-order E-unification
- Dougherty, Johann
- 1992
(Show Context)
Citation Context ...closed under subterm extraction. Indeed, all three of these properties of CR-normal forms are required for the work which originally motivated the investigation reported here: in Dougherty and Johann =-=[9]-=-, the observation that the class of CR-normal forms satisfies them if R is convergent is used to prove that, under that hypothesis, certain transformations for deciding CR-equality on CL-terms can be ... |

6 |
Axioms for strong reduction in combinatory logic Journal of symbolic logic 32-2
- Hindley
- 1967
(Show Context)
Citation Context ...f higher-order unification methods (see for example Huet [16] and Gallier and Snyder [10]). Reflecting βη-equality, C-equality has been well-studied in the literature (see Curry and Feys [5], Hindley =-=[11]-=-, Hindley and Lercher [13], Hindley and Seldin [15], Lercher [20] and [21], and Mezghiche [22]). Since long β-normal forms are fundamentally important in the investigation of βη-equality, it is natura... |

5 | Combinatory reductions and lambda reductions compared, Zeit - Hindley - 1977 |

5 |
Strong reduction and normal forms in combinatory logic
- Lercher
- 1967
(Show Context)
Citation Context ...and Snyder [10]). Reflecting βη-equality, C-equality has been well-studied in the literature (see Curry and Feys [5], Hindley [11], Hindley and Lercher [13], Hindley and Seldin [15], Lercher [20] and =-=[21]-=-, and Mezghiche [22]). Since long β-normal forms are fundamentally important in the investigation of βη-equality, it is natural to look for a reduction relation on CL capturing C-equality in hopes of ... |

4 | Complete Sets of Transformations for Unification Problems - Johann - 1991 |

2 |
A short proof of Curry's normal form theorem
- Hindley, Lercher
- 1970
(Show Context)
Citation Context ... methods (see for example Huet [16] and Gallier and Snyder [10]). Reflecting βη-equality, C-equality has been well-studied in the literature (see Curry and Feys [5], Hindley [11], Hindley and Lercher =-=[13]-=-, Hindley and Seldin [15], Lercher [20] and [21], and Mezghiche [22]). Since long β-normal forms are fundamentally important in the investigation of βη-equality, it is natural to look for a reduction ... |

2 | On pseudo-cβ normal form in combinatory logic - Mezghiche - 1989 |

1 |
Combining algebra and higher-order types,” pp. 82–90
- Breazu-Tannen
- 1988
(Show Context)
Citation Context ...y logic terms for nonempty, as well as for empty, R. 573 1Introduction The interaction between higher-order and first-order algebraic reasoning has recently received much attention (see Breazu-Tannen =-=[2]-=-, [3], and [4], and Dougherty [8]), particularly the situation in which the equational theory in question admits presentation as a convergent (confluent and terminating) term rewriting system R. Reaso... |

1 |
Rewrite systems,” pp. 243–320
- Dershowitz, Jouannaud
- 1990
(Show Context)
Citation Context ...f the simply typed lambda calculus and first-order algebraic rewriting. For definitions and notations regarding rewriting not given explicitly here, the reader is referred to Dershowitz and Jouannaud =-=[7]-=-. The types are formed by closing a set of base types under the operation (α1 → α2) for types α1 and α2. Fix an infinite set, Vars, oftyped variables and an infinite set of typed constants. Certain co... |

1 |
The decidability of Hindley’s axioms for strong reduction
- Lercher
- 1967
(Show Context)
Citation Context ... Gallier and Snyder [10]). Reflecting βη-equality, C-equality has been well-studied in the literature (see Curry and Feys [5], Hindley [11], Hindley and Lercher [13], Hindley and Seldin [15], Lercher =-=[20]-=- and [21], and Mezghiche [22]). Since long β-normal forms are fundamentally important in the investigation of βη-equality, it is natural to look for a reduction relation on CL capturing C-equality in ... |