## Towards Lambda Calculus Order-Incompleteness (0)

Venue: | Workshop on Böhm theorem: applications to Computer Science Theory (BOTH 2001) Electronics Notes in Theoretical Computer Science |

Citations: | 3 - 3 self |

### BibTeX

@INPROCEEDINGS{Salibra_towardslambda,

author = {Antonino Salibra},

title = {Towards Lambda Calculus Order-Incompleteness},

booktitle = {Workshop on Böhm theorem: applications to Computer Science Theory (BOTH 2001) Electronics Notes in Theoretical Computer Science},

year = {},

pages = {147--160},

publisher = {Elsevier Science Publishing Company}

}

### OpenURL

### Abstract

After Scott, mathematical models of the type-free lambda calculus are constructed by order theoretic methods and classified into semantics according to the nature of their representable functions. Selinger [47] asked if there is a lambda theory that is not induced by any non-trivially partially ordered model (order-incompleteness problem). In terms of Alexandroff topology (the strongest topology whose specialization order is the order of the considered model) the problem of order incompleteness can be also characterized as follows: a lambda theory T is order-incomplete if, and only if, every partially ordered model of T is partitioned by the Alexandroff topology in an infinite number of connected components (= minimal upper and lower sets), each one containing exactly one element of the model. Towards an answer to the order-incompleteness problem, we give a topological proof of the following result: there exists a lambda theory whose partially ordered models are partitioned by the Alexandroff topology in an infinite number of connected components, each one containing at most one -term denotation. This result implies the incompleteness of every semantics of lambda calculus given in terms of partially ordered models whose Alexandroff topology has a finite number of connected components (e.g. the Alexandroff topology of the models of the continuous, stable and strongly stable semantics is connected).

### Citations

324 | A course in universal algebra
- Burris, Sankappanavar
- 1981
(Show Context)
Citation Context ...ee [3]). For the general theory of lambda calculus the reader may consult Barendregt [3] and Krivine [28]. For the general theory of universal algebras the reader may consult Burris and Sankappanavar =-=[12]-=- Gratzer [21] and McKenzie, McNulty and Taylor [29]. The main references for topological algebras are Taylor [48] [49], Gumm [22], Bentz [8] and Coleman [13] [14]. 2.1 Lambda theoriessdenotes the set ... |

228 |
Data types as lattices
- Scott
- 1976
(Show Context)
Citation Context ...artially ordered models whose specialization order is a complete partial ordering and the representable functions are all the continuous ones w.r.t. the Scott topology. The graph model semantics (see =-=[43]-=- [18] [33] [34] [9, Section 5.5]) is a subclass of the K-semantics isolated by Krivine (see [28] [9, Section 5.6.2]) within the continuous semantics. The filter model semantics was defined by Coppo, D... |

159 | Semantic domains
- Gunter, Scott
- 1990
(Show Context)
Citation Context ...ons over these sets. After Scott, a large number of mathematical models for the lambda calculus, arising from syntax-free constructions, have been introduced in various categories of domains (see [1] =-=[45]-=-) 1 Email: salibra@dsi.unive.it c fl2001 Published by Elsevier Science B. V. Salibra and classified into semantics according to the nature of their representable functions (see [2] [3] [4] [9] [15] [1... |

151 |
The System F of variable types, fifteen years later
- Girard
- 1986
(Show Context)
Citation Context ...5]) 1 Email: salibra@dsi.unive.it c fl2001 Published by Elsevier Science B. V. Salibra and classified into semantics according to the nature of their representable functions (see [2] [3] [4] [9] [15] =-=[19]-=- [24]). Scott's continuous semantics [42] is given in the category whose objects are complete partial orders and morphisms are continuous functions. The stable semantics introduced by Berry in [10] an... |

129 |
types et modeles
- Lambda-calcul
(Show Context)
Citation Context ...e paper. With regard to the lambda calculus we follow the notation and terminology of Barendregt (see [3]). For the general theory of lambda calculus the reader may consult Barendregt [3] and Krivine =-=[28]-=-. For the general theory of universal algebras the reader may consult Burris and Sankappanavar [12] Gratzer [21] and McKenzie, McNulty and Taylor [29]. The main references for topological algebras are... |

79 | spaces - Stone - 1982 |

72 |
Stable models of typed lambda-calculi
- Berry
- 1978
(Show Context)
Citation Context ...5] [19] [24]). Scott's continuous semantics [42] is given in the category whose objects are complete partial orders and morphisms are continuous functions. The stable semantics introduced by Berry in =-=[10]-=- and the strongly stable semantics introduced by Bucciarelli and Ehrhard in [11] are strengthening of the continuous semantics. The stable semantics is given in the category of DI-domains with stable ... |

59 | Hypercoherence: A strongly stable model of linear logic
- Ehrhard
- 1995
(Show Context)
Citation Context ... partially ordered models whose specialization order is a DI-domain with coherence and the representable functions are all the strongly stable ones. The hypercoherence semantics introduced by Ehrhard =-=[17]-=- is a subclass of the strongly stable semantics. A class C of models of the lambda calculus represents a lambda theory T if there is a model in C whose theory is exactly T . A class of models is incom... |

57 |
Sequentiality and strong stability
- Bucciarelli, Ehrhard
- 1991
(Show Context)
Citation Context ...objects are complete partial orders and morphisms are continuous functions. The stable semantics introduced by Berry in [10] and the strongly stable semantics introduced by Bucciarelli and Ehrhard in =-=[11]-=- are strengthening of the continuous semantics. The stable semantics is given in the category of DI-domains with stable functions as morphisms, while the strongly stable one in the category of DI-doma... |

48 |
What is a model of the lambda calculus
- Meyer
- 1982
(Show Context)
Citation Context ... is that, if C is an arbitrary combinatory algebra, it may happen that map f is not representable. The axioms of a subclass of combinatory algebras, called -models or models of lambda calculus (Meyer =-=[30]-=-, Scott [44], [3, Def. 5.2.7]), were expressly chosen to make coherent the previous definition of interpretation. For every -model C, the set Th(C) = ft = u : t; u 2so ; C j= t = ug constitutes a lamb... |

47 | Set-theoretical and other elementary models of the λ-calculus
- Plotkin
- 1993
(Show Context)
Citation Context ...d models whose specialization order is a complete partial ordering and the representable functions are all the continuous ones w.r.t. the Scott topology. The graph model semantics (see [43] [18] [33] =-=[34]-=- [9, Section 5.5]) is a subclass of the K-semantics isolated by Krivine (see [28] [9, Section 5.6.2]) within the continuous semantics. The filter model semantics was defined by Coppo, Dezani, Honsell ... |

38 |
From computation to foundations via functions and application: The λ-calculus and its webbed models
- Berline
(Show Context)
Citation Context ...ee [1] [45]) 1 Email: salibra@dsi.unive.it c fl2001 Published by Elsevier Science B. V. Salibra and classified into semantics according to the nature of their representable functions (see [2] [3] [4] =-=[9]-=- [15] [19] [24]). Scott's continuous semantics [42] is given in the category whose objects are complete partial orders and morphisms are continuous functions. The stable semantics introduced by Berry ... |

35 |
Lambda calculus: some models, some philosophy, The Kleene Symposium
- Scott
- 1980
(Show Context)
Citation Context ... C is an arbitrary combinatory algebra, it may happen that map f is not representable. The axioms of a subclass of combinatory algebras, called -models or models of lambda calculus (Meyer [30], Scott =-=[44]-=-, [3, Def. 5.2.7]), were expressly chosen to make coherent the previous definition of interpretation. For every -model C, the set Th(C) = ft = u : t; u 2so ; C j= t = ug constitutes a lambda theory. C... |

29 |
Domain theory in logical form, Annals of Pure and Applied Logic 51
- Abramsky
- 1991
(Show Context)
Citation Context ...nctions over these sets. After Scott, a large number of mathematical models for the lambda calculus, arising from syntax-free constructions, have been introduced in various categories of domains (see =-=[1]-=- [45]) 1 Email: salibra@dsi.unive.it c fl2001 Published by Elsevier Science B. V. Salibra and classified into semantics according to the nature of their representable functions (see [2] [3] [4] [9] [1... |

28 |
Full abstraction in the lazy -calculus
- Abramsky, Ong
- 1989
(Show Context)
Citation Context ...f domains (see [1] [45]) 1 Email: salibra@dsi.unive.it c fl2001 Published by Elsevier Science B. V. Salibra and classified into semantics according to the nature of their representable functions (see =-=[2]-=- [3] [4] [9] [15] [19] [24]). Scott's continuous semantics [42] is given in the category whose objects are complete partial orders and morphisms are continuous functions. The stable semantics introduc... |

27 |
della Rocca. An approximation theorem for topological lambda models and the topological incompleteness of lambda calculus
- Honsell, Ronchi
(Show Context)
Citation Context ... Email: salibra@dsi.unive.it c fl2001 Published by Elsevier Science B. V. Salibra and classified into semantics according to the nature of their representable functions (see [2] [3] [4] [9] [15] [19] =-=[24]-=-). Scott's continuous semantics [42] is given in the category whose objects are complete partial orders and morphisms are continuous functions. The stable semantics introduced by Berry in [10] and the... |

22 | Strong stability and the incompleteness of stable models of λ-calculus
- Bastonero, Gouy
- 1999
(Show Context)
Citation Context ... Gouy [20] proved the incompleteness of the stable semantics with a much harder syntactical proof. Other more semantic proofs of incompleteness for the continuous and stable semantics can be found in =-=[7]-=-. Bastonero [6] provides an incompleteness result for the hypercoherence semantics. Recently, the author has introduced in [36] a new technique to prove the incompleteness of a wide range of lambda ca... |

22 | Isomorphism and equational equivalence of continuous lambda models - Kerth - 1998 |

18 | A continuum of theories of lambda calculus without semantics
- Salibra
- 2001
(Show Context)
Citation Context ...of incompleteness for the continuous and stable semantics can be found in [7]. Bastonero [6] provides an incompleteness result for the hypercoherence semantics. Recently, the author has introduced in =-=[36]-=- a new technique to prove the incompleteness of a wide range of lambda calculus semantics (including the strongly stable one, whose incompleteness had been conjectured). Roughly, the technique used in... |

17 |
Algebras and combinators. Algebra Universalis 13(3
- Engeler
- 1981
(Show Context)
Citation Context ...lly ordered models whose specialization order is a complete partial ordering and the representable functions are all the continuous ones w.r.t. the Scott topology. The graph model semantics (see [43] =-=[18]-=- [33] [34] [9, Section 5.5]) is a subclass of the K-semantics isolated by Krivine (see [28] [9, Section 5.6.2]) within the continuous semantics. The filter model semantics was defined by Coppo, Dezani... |

17 |
Some ordered sets in computer science
- Scott
- 1982
(Show Context)
Citation Context ...ected). 1 Introduction Many familiar models of the type-free lambda calculus are constructed by order theoretic methods. Computational motivations and intuitions justified Scott's view of models (see =-=[41]-=- [42]) as partially ordered sets and of functions as monotonic functions over these sets. After Scott, a large number of mathematical models for the lambda calculus, arising from syntax-free construct... |

16 |
Etude des théories équationnelles et des propriétés algébriques des modéles stables du λcalcul. Thèse, Université de Paris 7
- Gouy
- 1995
(Show Context)
Citation Context ...luded within the set of consistent lambda theories? The first incompleteness result was obtained by Honsell and Ronchi della Rocca [24] for the continuous semantics via a hard syntactical proof. Gouy =-=[20]-=- proved the incompleteness of the stable semantics with a much harder syntactical proof. Other more semantic proofs of incompleteness for the continuous and stable semantics can be found in [7]. Basto... |

14 |
A finite equational axiomatization of the functional algebras for the lambda calculus
- Salibra, Goldblatt
- 1999
(Show Context)
Citation Context ...ists an algebraic approach to the model theory of lambda calculus, alternative to combinatory logic, that allows to keep the lambda notation and all the functional intuitions (see [31] [32] [37] [38] =-=[39]). 2.-=-3 Topology If (A;s) is a topological space (we will occasionally avoid explicit mention ofs) then the closure of a subset U of A will be denoted by U . Recall that a 2 U if U " V 6= ; for every o... |

12 |
Lambda abstraction algebras: coordinatizing models of lambda calculus
- Pigozzi, Salibra
- 1998
(Show Context)
Citation Context ...e that there exists an algebraic approach to the model theory of lambda calculus, alternative to combinatory logic, that allows to keep the lambda notation and all the functional intuitions (see [31] =-=[32] [37]-=- [38] [39]). 2.3 Topology If (A;s) is a topological space (we will occasionally avoid explicit mention ofs) then the closure of a subset U of A will be denoted by U . Recall that a 2 U if U " V 6... |

12 |
On a question of
- Plotkin
- 1995
(Show Context)
Citation Context ..., while Plotkin and Selinger (see [46]) obtained the same result for T 2 . It is an open problem whether T n (ns3) can be consistent. Order-incompleteness is also related to Plotkin's conjecture (see =-=[35]-=- [46] [47]) about the existence of absolutely unorderable combinatory algebras, where a combinatory algebra is absolutely unorderable if it cannot be embedded in any non-trivially partially ordered co... |

11 |
Lambda abstraction algebras: representation theorems
- Pigozzi, Salibra
- 1995
(Show Context)
Citation Context ...t here that there exists an algebraic approach to the model theory of lambda calculus, alternative to combinatory logic, that allows to keep the lambda notation and all the functional intuitions (see =-=[31]-=- [32] [37] [38] [39]). 2.3 Topology If (A;s) is a topological space (we will occasionally avoid explicit mention ofs) then the closure of a subset U of A will be denoted by U . Recall that a 2 U if U ... |

8 | Order-incompleteness and finite lambda models, extended abstract
- Selinger
- 1996
(Show Context)
Citation Context ...Scott, mathematical models of the type-free lambda calculus are constructed by order theoretic methods and classified into semantics according to the nature of their representable functions. Selinger =-=[47]-=- asked if there is a lambda theory that is not induced by any non-trivially partially ordered model (order-incompleteness problem). In terms of Alexandroff topology (the strongest topology whose speci... |

6 |
The Lambda Calculus: its syntax and semantics," Number 103
- Barendregt
- 1994
(Show Context)
Citation Context ...mains (see [1] [45]) 1 Email: salibra@dsi.unive.it c fl2001 Published by Elsevier Science B. V. Salibra and classified into semantics according to the nature of their representable functions (see [2] =-=[3]-=- [4] [9] [15] [19] [24]). Scott's continuous semantics [42] is given in the category whose objects are complete partial orders and morphisms are continuous functions. The stable semantics introduced b... |

6 |
Varieties obeying homotopy laws, Canad
- Taylor
- 1977
(Show Context)
Citation Context ...eral theory of universal algebras the reader may consult Burris and Sankappanavar [12] Gratzer [21] and McKenzie, McNulty and Taylor [29]. The main references for topological algebras are Taylor [48] =-=[49]-=-, Gumm [22], Bentz [8] and Coleman [13] [14]. 2.1 Lambda theoriessdenotes the set of -terms, whileso denotes the set of closed -terms, where a -term is closed if it does not admit free occurrences of ... |

5 |
Varieties of topological algebras
- Taylor
- 1977
(Show Context)
Citation Context ...e general theory of universal algebras the reader may consult Burris and Sankappanavar [12] Gratzer [21] and McKenzie, McNulty and Taylor [29]. The main references for topological algebras are Taylor =-=[48]-=- [49], Gumm [22], Bentz [8] and Coleman [13] [14]. 2.1 Lambda theoriessdenotes the set of -terms, whileso denotes the set of closed -terms, where a -term is closed if it does not admit free occurrence... |

3 | Modeles fortement stables du -calcul et resultats d'incompletude, These, Universite de Paris 7 - Bastonero - 1996 |

3 |
Equational incompleteness and incomparability results for -calculus semantics, manuscript
- Bastonero
(Show Context)
Citation Context ...ed the incompleteness of the stable semantics with a much harder syntactical proof. Other more semantic proofs of incompleteness for the continuous and stable semantics can be found in [7]. Bastonero =-=[6]-=- provides an incompleteness result for the hypercoherence semantics. Recently, the author has introduced in [36] a new technique to prove the incompleteness of a wide range of lambda calculus semantic... |

3 |
Topological implications in varieties, Algebra Universalis 42
- Bentz
- 1999
(Show Context)
Citation Context ...l algebras the reader may consult Burris and Sankappanavar [12] Gratzer [21] and McKenzie, McNulty and Taylor [29]. The main references for topological algebras are Taylor [48] [49], Gumm [22], Bentz =-=[8]-=- and Coleman [13] [14]. 2.1 Lambda theoriessdenotes the set of -terms, whileso denotes the set of closed -terms, where a -term is closed if it does not admit free occurrences of variables. Lambda theo... |

3 |
Extended type structures and filter -models
- Coppo, Dezani-Ciancaglini, et al.
- 1984
(Show Context)
Citation Context ...1] [45]) 1 Email: salibra@dsi.unive.it c fl2001 Published by Elsevier Science B. V. Salibra and classified into semantics according to the nature of their representable functions (see [2] [3] [4] [9] =-=[15]-=- [19] [24]). Scott's continuous semantics [42] is given in the category whose objects are complete partial orders and morphisms are continuous functions. The stable semantics introduced by Berry in [1... |

3 |
Algebras, lattices, and varieties: Volume i
- McKenzie, McNulty, et al.
- 1987
(Show Context)
Citation Context ...the reader may consult Barendregt [3] and Krivine [28]. For the general theory of universal algebras the reader may consult Burris and Sankappanavar [12] Gratzer [21] and McKenzie, McNulty and Taylor =-=[29]-=-. The main references for topological algebras are Taylor [48] [49], Gumm [22], Bentz [8] and Coleman [13] [14]. 2.1 Lambda theoriessdenotes the set of -terms, whileso denotes the set of closed -terms... |

3 | Functionality, polymorphism, and concurrency: a mathematical investigation of programming paradigms
- Selinger
- 1997
(Show Context)
Citation Context ...sequence M 1 ; : : : ; M n 2 Salibra of closed -terms such that the lambda theory T n , axiomatized by x = M 1 xyy; M i xxy = M i+1 xyy; M n xxy = y (1si ! n); is consistent? Plotkin and Simpson (see =-=[46]-=-) have shown that T 1 is inconsistent, while Plotkin and Selinger (see [46]) obtained the same result for T 2 . It is an open problem whether T n (ns3) can be consistent. Order-incompleteness is also ... |

2 |
Separation in topological algebras, Algebra Universalis 35
- Coleman
- 1996
(Show Context)
Citation Context ...eader may consult Burris and Sankappanavar [12] Gratzer [21] and McKenzie, McNulty and Taylor [29]. The main references for topological algebras are Taylor [48] [49], Gumm [22], Bentz [8] and Coleman =-=[13]-=- [14]. 2.1 Lambda theoriessdenotes the set of -terms, whileso denotes the set of closed -terms, where a -term is closed if it does not admit free occurrences of variables. Lambda theories are consiste... |

2 |
Topological equivalents to n-permutability, Algebra Universalis 38
- Coleman
- 1997
(Show Context)
Citation Context ... may consult Burris and Sankappanavar [12] Gratzer [21] and McKenzie, McNulty and Taylor [29]. The main references for topological algebras are Taylor [48] [49], Gumm [22], Bentz [8] and Coleman [13] =-=[14]-=-. 2.1 Lambda theoriessdenotes the set of -terms, whileso denotes the set of closed -terms, where a -term is closed if it does not admit free occurrences of variables. Lambda theories are consistent ex... |

2 |
Topological implications in n-permutable varieties, Algebra Universalis 19
- Gumm
- 1984
(Show Context)
Citation Context ... of universal algebras the reader may consult Burris and Sankappanavar [12] Gratzer [21] and McKenzie, McNulty and Taylor [29]. The main references for topological algebras are Taylor [48] [49], Gumm =-=[22]-=-, Bentz [8] and Coleman [13] [14]. 2.1 Lambda theoriessdenotes the set of -terms, whileso denotes the set of closed -terms, where a -term is closed if it does not admit free occurrences of variables. ... |

2 |
On the algebraic models of lambda calculus, Theoret
- Salibra
- 2000
(Show Context)
Citation Context ...re exists an algebraic approach to the model theory of lambda calculus, alternative to combinatory logic, that allows to keep the lambda notation and all the functional intuitions (see [31] [32] [37] =-=[38] [39]-=-). 2.3 Topology If (A;s) is a topological space (we will occasionally avoid explicit mention ofs) then the closure of a subset U of A will be denoted by U . Recall that a 2 U if U " V 6= ; for ev... |

1 |
A filter model and the completeness of type assignmen
- Barendregt, Coppo, et al.
- 1983
(Show Context)
Citation Context ...s (see [1] [45]) 1 Email: salibra@dsi.unive.it c fl2001 Published by Elsevier Science B. V. Salibra and classified into semantics according to the nature of their representable functions (see [2] [3] =-=[4]-=- [9] [15] [19] [24]). Scott's continuous semantics [42] is given in the category whose objects are complete partial orders and morphisms are continuous functions. The stable semantics introduced by Be... |

1 |
Combinatory Logic", Vol
- Curry, Feys
- 1958
(Show Context)
Citation Context ... t = u stands fors+ T ` t = u. 2.2 Combinatory algebras and -models An algebra C = (C; \Delta; k; s), where \Delta is a binary operation and k; s are constants, is called a combinatory algebra (Curry =-=[16]-=-, Schonfinkel [40]) if it satisfies the following identities (as usual the symbol \Delta is omitted, and association is to the left): kxy = x; sxyz = xz(yz). In the equational language of combinatory ... |

1 |
Universal Algebra", Second edition
- Gratzer
- 1979
(Show Context)
Citation Context ...the general theory of lambda calculus the reader may consult Barendregt [3] and Krivine [28]. For the general theory of universal algebras the reader may consult Burris and Sankappanavar [12] Gratzer =-=[21]-=- and McKenzie, McNulty and Taylor [29]. The main references for topological algebras are Taylor [48] [49], Gumm [22], Bentz [8] and Coleman [13] [14]. 2.1 Lambda theoriessdenotes the set of -terms, wh... |

1 |
Final semantics for untyped -calculus, LNCS 902
- Honsell, Lenisa
- 1995
(Show Context)
Citation Context ...ns of the lambda calculus that includesfi-conversion. They arise by syntactical considerations, a lambda theory may correspond to a possible operational semantics of lambda calculus (see e.g. [2] [3] =-=[23]-=-), as well as by semantic ones, a lambda theory may be the theory of a model of lambda calculus (see e.g. [3] [9]). The problem of the completeness/incompleteness of a semantics can be stated as follo... |

1 | On the construction of stable models of -calculus, Theoretical Computer Science - Kerth |

1 |
A set theoretic definition of application, Memorandum MIP-R-95
- Plotkin
- 1972
(Show Context)
Citation Context ...rdered models whose specialization order is a complete partial ordering and the representable functions are all the continuous ones w.r.t. the Scott topology. The graph model semantics (see [43] [18] =-=[33]-=- [34] [9, Section 5.5]) is a subclass of the K-semantics isolated by Krivine (see [28] [9, Section 5.6.2]) within the continuous semantics. The filter model semantics was defined by Coppo, Dezani, Hon... |

1 |
Nonmodularity results for lambda calculus, Fundamenta Informaticae
- Salibra
(Show Context)
Citation Context ...t there exists an algebraic approach to the model theory of lambda calculus, alternative to combinatory logic, that allows to keep the lambda notation and all the functional intuitions (see [31] [32] =-=[37] [38]-=- [39]). 2.3 Topology If (A;s) is a topological space (we will occasionally avoid explicit mention ofs) then the closure of a subset U of A will be denoted by U . Recall that a 2 U if U " V 6= ; f... |