## Topological Incompleteness and Order Incompleteness of the Lambda Calculus (2001)

Venue: | ACM TRANSACTIONS ON COMPUTATIONAL LOGIC |

Citations: | 23 - 15 self |

### BibTeX

@ARTICLE{Salibra01topologicalincompleteness,

author = {Antonino Salibra},

title = {Topological Incompleteness and Order Incompleteness of the Lambda Calculus},

journal = {ACM TRANSACTIONS ON COMPUTATIONAL LOGIC},

year = {2001},

volume = {4},

pages = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

A model of the untyped lambda calculus induces a lambda theory, i.e., a congruence relation on λ-terms closed under ff- and fi-conversion. A semantics (= class of models) of the lambda calculus is incomplete if there exists a lambda theory which is not induced by any model in the semantics. In this paper we introduce 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 by Bastonero-Gouy [6, 7] and by Berline [9]. The main results of the paper are a topological incompleteness theorem and an order incompleteness theorem. In the first one we show the incompleteness of the lambda calculus semantics given in terms of topological models whose topology satisfies a property of connectedness. In the second one we prove the incompleteness of the class of partially ordered models with finitely many connected components w.r.t. the Alexandroff topology. A further result of the paper is a proof of the completeness of the semantics of the lambda calculus given in terms of topological models whose topology is non-trivial and metrizable.

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Citation Context ...ulus, as well as by semantic ones, a lambda theory may be induced by a model of lambda calculus through the kernel congruence relation of the interpretation function (see e.g. [Abramsky and Ong 1993; =-=Barendregt 1984-=-; Berline 2000]). Since the lattice of the lambda theories is a very rich and complex structure, syntactical techniques are usually difficult to use in the study of lambda theories. Therefore, semanti... |

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Citation Context ...of observations or informations) and of computable functions as monotonic functions over these sets. After Scott, mathematical models of the lambda calculus in various categories of domains (see e.g. =-=[1]-=-) were classified into semantics according to the nature of their representable functions (see [2, 3, 4, 9, 18, 26]). Scott's continuous semantics [30] is given in the category whose objects are compl... |

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Citation Context ...alculus that are closed under derivation. They arise by syntactical considerations, a lambda theory may correspond to a possible operational (observational) semantics of the lambda calculus (see e.g. =-=[2, 3]-=-), as well as by semantic ones, a lambda theory may be the theory of a lambda calculus model (see e.g. [3, 9]). Since the lattice of the lambda theories is a very rich and complex structure (see [3]),... |

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Citation Context ...ee e.g. [Abramsky 1991]) were classified into semantics according to the nature of their representable functions (see e.g. [Barendregt 1984; Berline 2000; Plotkin 1993). Scott’s continuous semantics [=-=Scott 1972-=-] is given in the category whose objects are complete partial orders and morphisms are Scott continuous functions. The stable semantics [Berry 1978] and the recent strongly stable semantics [Bucciarel... |

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Citation Context ...aracterized as a minimal subset of A which is both upward closed and downward closed. Given a poset (A, ≤), we can find many T0-topologies τ on A for which ≤ is the specialization ordering of τ (see [=-=Johnstone 1982-=-, Section II.1.8]). The maximal one with this property is the Alexandroff topology, which is constituted by the collection of all upward closed sets of A, i.e., U is an Alexandroff open iff U = U↑. Th... |

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Citation Context ...f the stable semantics. Other more semantic proofs of incompleteness for the continuous, stable and hypercoherence semantics (that is a subclass of the strongly stable semantics introduced by Ehrhard =-=[15]-=-) can be found in [6, 7] and are briefly described in the following. The Park model P was first defined in the framework of continuous semantics. It is a variant of the Scott model D1 , but with a ver... |

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Citation Context ... are complete partial orders and morphisms are continuous functions. The stable semantics introduced by Berry in [10] and the recent strongly stable semantics introduced by Bucciarelli and Ehrhard in =-=[11]-=- are strengthenings 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-dom... |

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Citation Context ...icians since the early 1930's, its model theory developed only much later, following the pioneering model construction made by Dana Scott. The notion of an environment model (the name is due to Meyer =-=[23]) is descr-=-ibed by Meyer as "the natural, most general formulation of what might be meant by mathematical models of the untyped lambda calculus". The drawback of environment models is that they are hig... |

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Citation Context ...s. After Scott, mathematical models of the lambda calculus in various categories of domains (see e.g. [1]) were classified into semantics according to the nature of their representable functions (see =-=[2, 3, 4, 9, 18, 26]-=-). Scott's continuous semantics [30] 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... |

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Citation Context ...ization (up to isomorphism) of environment models as an elementary class of combinatory algebras called -models ([3, Def. 5.2.7]). They were first axiomatized by Meyer [23] and independently by Scott =-=[31]-=-; the axiomatization, while elegant, is not equational. We now define the notion of a -model. Let C be a combinatory algebra and letsc be a new symbol for each c 2 C. Extend the language of the lambda... |

34 |
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Citation Context ...echnique to prove the incompleteness of a wide range of lambda calculus semantics, including the strongly stable one, whose incompleteness had been conjectured by Bastonero-Gouy [6, 7] and by Berline =-=[9]-=-. The main results of the paper are a topological incompleteness theorem and an order incompleteness theorem. In the first one we show the incompleteness of the lambda calculus semantics given in term... |

28 |
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Citation Context ...alculus that are closed under derivation. They arise by syntactical considerations, a lambda theory may correspond to a possible operational (observational) semantics of the lambda calculus (see e.g. =-=[2, 3]-=-), as well as by semantic ones, a lambda theory may be the theory of a lambda calculus model (see e.g. [3, 9]). Since the lattice of the lambda theories is a very rich and complex structure (see [3]),... |

20 | Strong stability and the incompleteness of stable models of λ-calculus - Bastonero, Gouy - 1999 |

20 | On the algebraic models of lambda calculus - Salibra |

19 |
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Citation Context ...ions as morphisms. All these semantics are structurally and equationally rich in the sense that it is possible to build up 2 @ 0 models in each of them inducing pairwise distinct lambda theories (see =-=[21, 22]-=-). The problem of the equational richness is related to the problem of the completeness/incompleteness of a semantics: is the set of lambda theories determined by these semantics equal or strictly inc... |

17 |
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Citation Context ... all open sets U and V , we have that V ∩ U �= ∅. We recall that a space with no disjoint open sets is called hyperconnected, while a space with no disjoint closed sets is called ultraconnected (see [=-=Steen and Seebach 1978-=-, Section 4]). Then we have the following implications: and hyperconnectedness ⇒ co-connectedness ⇒ connectedness ultraconnectedness ⇒ co-connectedness ⇒ connectedness. Then closed-open-connectedness ... |

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15 |
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Citation Context ...ntinuous semantics. They proved, via a hard syntactical proof, that the contextual lambda theory induced by the set of closed terms does not admit a continuous model. Following a similar method, Gouy =-=[16]-=- proved the incompleteness of the stable semantics with a much harder syntactical proof. Semantic and more simple proofs of incompleteness for the continuous and stable semantics can be found in [7]. ... |

12 |
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Citation Context ...s. After Scott, mathematical models of the lambda calculus in various categories of domains (see e.g. [1]) were classified into semantics according to the nature of their representable functions (see =-=[2, 3, 4, 9, 18, 26]-=-). Scott's continuous semantics [30] 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... |

12 | Lambda abstraction algebras: coordinatizing models of lambda calculus - Pigozzi, Salibra - 1998 |

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9 |
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Citation Context ...semantics. In [Salibra 2001a] the author has shown that continuous, stable and strongly stable semantics omit a continuum of lambda theories. In fact, by applying a well-known theorem by Visser (see [=-=Visser 1980-=-], [Barendregt 1984, Thm. 17.1.10]), it is possible to get a continuum of distinct lambda theories satisfying the conditions: Ωxx = Ω; Ω(Ωki)Ω �= Ω. From the proof of Thm. 3.5 it follows that any sema... |

8 |
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Citation Context ...e exist a sequence M 1 ; : : : ; M n 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 =-=[32]-=-) have shown that T 1 is inconsistent, while Plotkin and Selinger (see [32]) obtained the same result for T 2 . It is an open problem whether T n (ns3) can be consistent. We would like to point out th... |

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- 1996
(Show Context)
Citation Context ... the incompleteness problem of the untyped lambda calculus. As already mentioned there are computational motivations for taking ordered structures as models of the lambda calculus. Recently, Selinger =-=[33]-=- asked the following question: (Q5) (Selinger) Is the semantics of the lambda calculus given in terms of non-trivial po-models incomplete? Is there an order-incomplete lambda theory (that is a lambda ... |

8 |
Varieties obeying homotopy laws
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Citation Context ...lts that we need in the subsequent part of the paper. With regard to the lambda calculus we follow the notation and terminology of Barendregt [1984]. The main references for topological algebras are [=-=Taylor 1977-=-; Bentz 1999; Coleman 1996; Coleman 1997]. 2.1 Topology A topological space (A, τ) (we will occasionally avoid explicit mention of τ) is nontrivial if there are nonempty, proper subsets X and Y of A s... |

7 | All compact Hausdorff lambda models are degenerate
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Citation Context ...ndegerate model of the lambda calculus in the category of posets and monotone mappings, and in the category of complete ultrametric spaces and nonexpansive mappings. Hoffman and Mislove have shown in =-=[17]-=- that the category of k-spaces and continuous maps has no nondegenerate, compact T 2 -topological model. A k-space is a topological space in which a subset is open if and only if its intersection with... |

7 |
Games and full abstraction for the lazy λ-calculus
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Citation Context ...· Antonino Salibra calculus, as well as by semantic ones, a lambda theory may be induced by a model of lambda calculus through the kernel congruence relation of the interpretation function (see e.g. [=-=Abramsky and Ong 1993-=-; Barendregt 1984; Berline 2000]). Since the lattice of the lambda theories is a very rich and complex structure, syntactical techniques are usually difficult to use in the study of lambda theories. T... |

5 |
Ronchi Della Rocca S., "An Approximation Theorem for Topological Lambda Models and the Topological Incompleteness of Lambda Calculus
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Citation Context ...s. After Scott, mathematical models of the lambda calculus in various categories of domains (see e.g. [1]) were classified into semantics according to the nature of their representable functions (see =-=[2, 3, 4, 9, 18, 26]-=-). Scott's continuous semantics [30] 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... |

5 |
Continuous lattices", In: Toposes, Algebraic geometry and Logic
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Citation Context ...lculus in various categories of domains (see e.g. [1]) were classified into semantics according to the nature of their representable functions (see [2, 3, 4, 9, 18, 26]). Scott's continuous semantics =-=[30]-=- 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 recent strongly stable semantics in... |

4 |
Ronchi Della Rocca, "Reasoning about Interpretations in Qualitative Lambda Models
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Citation Context ...of continuous semantics. It is a variant of the Scott model D1 , but with a very different equational theory. This model has a stable analogue P s (which was defined by Honsell and Ronchi della Rocca =-=[19]-=-), and a strongly stable analogue P fs (defined by Bastonero and Gouy [7]). It is possible to give a semantic proof of incompleteness for the continuous semantics by using the Park stable model P s . ... |

4 | On a question of H. Friedman
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Citation Context ...ected components, each one containing exactly one -term denotation. Selinger has shown in [32, 33] that the problem of the order-incompleteness is also related to the following question by G. Plotkin =-=[27]-=-: (Q11) (Plotkin) Is there an absolutely unorderable combinatory algebra? A combinatory algebra is absolutely unorderable if it cannot be embedded in any non-trivially partially ordered combinatory al... |

4 |
On subtractive varieties I
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(Show Context)
Citation Context ...be applied in the next section to prove that there exists a lambda theory whose topological models are not co-connected. The notion of subtractivity in Universal Algebra was introduced by Aldo Ursini =-=[37]-=-. An algebra is subtractive if it satisfies the identities s(x; x) = 0; s(x; 0) = x (2) for some binary term s and constant 0. Subtractive algebras abound in classical algebras and in algebraic logic ... |

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 ...r we introduce 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 by Bastonero-Gouy =-=[6, 7]-=- and by Berline [9]. The main results of the paper are a topological incompleteness theorem and an order incompleteness theorem. In the first one we show the incompleteness of the lambda calculus sema... |

3 |
Topological implications in varieties
- Bentz
- 1999
(Show Context)
Citation Context ...he subsequent part of the paper. With regard to the lambda calculus we follow the notation and terminology of Barendregt (see [3]). The main references for topological algebras are Taylor [36], Bentz =-=[8]-=- and Coleman [12, 13]. 3 2.1. Topology A topological space (A; ) (we will occasionally avoid explicit mention ofs) is non-trivial ifsis neither the discrete nor the indiscrete topology (see Steen-Seeb... |

3 |
Separation in topological algebras
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(Show Context)
Citation Context ...rt of the paper. With regard to the lambda calculus we follow the notation and terminology of Barendregt (see [3]). The main references for topological algebras are Taylor [36], Bentz [8] and Coleman =-=[12, 13]-=-. 3 2.1. Topology A topological space (A; ) (we will occasionally avoid explicit mention ofs) is non-trivial ifsis neither the discrete nor the indiscrete topology (see Steen-Seebach [34]). If A is a ... |

3 |
Topological equivalents to n-permutability
- Coleman
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(Show Context)
Citation Context ...rt of the paper. With regard to the lambda calculus we follow the notation and terminology of Barendregt (see [3]). The main references for topological algebras are Taylor [36], Bentz [8] and Coleman =-=[12, 13]-=-. 3 2.1. Topology A topological space (A; ) (we will occasionally avoid explicit mention ofs) is non-trivial ifsis neither the discrete nor the indiscrete topology (see Steen-Seebach [34]). If A is a ... |

3 | Towards lambda calculus order-incompleteness
- Salibra
- 2001
(Show Context)
Citation Context ...nected components w.r.t. the Alexandroff topology, is incomplete. 1 A preliminary version of this result was presented to Workshop on Bohm theorem: applications to Computer Science Theory (BOTH 2001) =-=[28]-=-. 17 Proof: The order incompleteness follows from Thm. 5.1(iii). 2 We now analyse the relationships between the order incompleteness theorem and the topological incompleteness theorem. Let P be the cl... |

3 | Functionality, polymorphism, and concurrency: a mathematical investigation of programming paradigms
- Selinger
- 1997
(Show Context)
Citation Context ...=T y (1 ≤ i < n). Plotkin and Simpson have shown that the above Mal’cev conditions are inconsistent with lambda calculus for n = 1, while Plotkin and Selinger obtained the same result for n = 2 (see [=-=Selinger 1997-=-]). It is an open problem whether n can be greater than or equal to 3. The problem of order-incompleteness can be also characterized as follows: a lambda theory T is order-incomplete if, and only if, ... |

2 |
Varieties obeying homotopy laws", Canad
- Taylor
- 1977
(Show Context)
Citation Context ...we need in the subsequent part of the paper. With regard to the lambda calculus we follow the notation and terminology of Barendregt (see [3]). The main references for topological algebras are Taylor =-=[36]-=-, Bentz [8] and Coleman [12, 13]. 3 2.1. Topology A topological space (A; ) (we will occasionally avoid explicit mention ofs) is non-trivial ifsis neither the discrete nor the indiscrete topology (see... |

2 |
Equational incompleteness and incomparability results for λ-calculus semantics
- Bastonero
- 1998
(Show Context)
Citation Context ...ther more semantic proofs of incompleteness for the continuous, stable and hypercoherence semantics (that is a subclass of the strongly stable semantics introduced by Ehrhard [1993]) can be found in [=-=Bastonero 1998-=-; Bastonero and Gouy 1999] and are briefly described in the following. The Park model P was first defined in the framework of continuous semantics. It is a variant of the Scott model D∞, but with a ve... |

2 | Reasoning about interpretation in qualitative λmodels - Honsell, Rocca, et al. - 1990 |

2 | On the construction of stable models of λ-calculus, Theoretical Computer Science 269 - Kerth - 2001 |

1 |
On the construction of stable models of -calculus", Theoretical Computer Science
- Kerth
(Show Context)
Citation Context ...ions as morphisms. All these semantics are structurally and equationally rich in the sense that it is possible to build up 2 @ 0 models in each of them inducing pairwise distinct lambda theories (see =-=[21, 22]-=-). The problem of the equational richness is related to the problem of the completeness/incompleteness of a semantics: is the set of lambda theories determined by these semantics equal or strictly inc... |

1 |
Topologies in free algebras
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(Show Context)
Citation Context ... o T such that c d = e holds in M o T ; the two equations k = xy:x and s = xyz:xz(yz) are also included. Since M T is a free algebra, then we can apply the following construction due tosSwierczkowski =-=[35]-=- (see also Taylor [36, Thm. 2.1] and Coleman [13, Section 3]). Let (X; d 0 ) be an arbitrary metric space whose universe X is the set of variables of the lambda calculus. By applyingsSwierczkowski's t... |

1 | An approximation theorem for topological λ-models and the topological incompleteness of λ-calculus - Honsell, Rocca, et al. - 1992 |