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There are three sorts in **the** **calculus**: Prop, Set and Type, which represent **the** proposi- tional or logical level **of** **the** language and **the** hierarchical universes **of** types, respectively. **The** sort Type has an enumeration according to **the** type level needed in **the** theory.
**The** identifiers give **the** global definitions **of** **the** language and are included in contexts when doing proofs. **The** identifiers are variables and names for constants, constructors and types. Together with **the** third syntactic category built **the** terms **of** **the** language. **The** computational terms include **the** product @ which is an upper-level abstraction for function **definition** while **the** other terms are **the** usual constructions for abstraction, term application and local definitions (let). **The** elimination form case, goes together with **the** inductive definitions introduced by constants I. Finally, another elimination form is recursion and is performed by a fix-point term.

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Pierre Clairambault and Marc de Visme. 2020. Full Abstraction for **the** Quantum **Lambda**-**Calculus**. Proc. ACM Program. Lang. 4, POPL, Article 63 (January 2020), 28 pages. https://doi.org/10.1145/3371131
1 INTRODUCTION
Quantum computation promises to have a huge impact in computing. Algorithms like Shor’s [ Shor 1997 ] or Grover’s [ Grover 1996 ] challenge our traditional view **of** algorithmics and complexity, and applications exploiting quantum features in cryptography [ Gisin et al. 2002 ] are already deployed. **The** field is moving fast, with large companies investing massively in **the** race for quantum hardware. To accompany this trend, researchers have developed programming languages for quantum computing. **The** quantum λ-**calculus** [ Selinger and Valiron 2006 ] is a paradigmatic such language, marrying quantum computation with classical control. Finding denotational semantics for **the** quantum λ-**calculus** has attracted a lot **of** attention, and over **the** years, **models** were given for various fragments [ Delbecque 2011 ; Hasuo and Hoshino 2017 ; Malherbe 2013 ; Malherbe et al. 2013 ;

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Universit` a Ca’Foscari di Venezia,
Dipartimento di Informatica Via Torino 155, 30172 Venezia, Italy
Abstract. A longstanding open problem is whether there exists a non- syntactical model **of** **the** untyped λ-**calculus** whose theory is exactly **the** least λ-theory λβ. In this paper we investigate **the** more general ques- tion **of** whether **the** equational/order theory **of** a model **of** **the** untyped λ-**calculus** can be recursively enumerable (r.e. for brevity). We intro- duce a notion **of** effective model **of** λ-**calculus**, which covers in particular all **the** **models** individually introduced in **the** literature. We prove that **the** order theory **of** an effective model is never r.e.; from this it follows that its equational theory cannot be λβ, λβη. We then show that no effective model living in **the** stable or strongly stable semantics has an r.e. equational theory. Concerning Scott’s semantics, we investigate **the** class **of** graph **models** and prove that no order theory **of** a graph model can be r.e., and that there exists an effective graph model whose equa- tional/order theory is **the** minimum one. Finally, we show that **the** class **of** graph **models** enjoys a kind **of** downwards L¨ owenheim-Skolem theorem. Keywords: **Lambda** **calculus**, Effective **lambda** **models**, Recursively enu- merable **lambda** theories, Graph **models**, L¨ owenheim-Skolem theorem.

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σ := f : p ⊃ p, x : p ⇒ p); hence, CHURCH is inﬁnite.
If, as for emptiness checking, we disregard **the** ability **of** **the** described method to deal with decontraction, we can see similarities with ﬁniteness checking for context-free grammars (Hopcroft and Ullman 1979, Theorem 6.6). Additionally, Hopcroft and Ullman mention that decisions based on **the** pumping lemma are ‘highly ineﬃcient,’ but those were alluded to in **the** approach by Takahashi et al. to inhabitation based on grammars (Takahashi et al. 1996, Corollary 3.8). **The** algorithm described by Hopcroft and Ullman does not work on a given context-free grammar but on an equivalent one that is in Chomsky normal form and has no useless symbols (Hopcroft and Ullman 1979, p.88); since there is no empty word in λ-**calculus**, we slightly simpliﬁed **the** presentation. In **the** Chomsky normal form, without useless symbols, eﬀective dependency **of** a non- terminal is identiﬁed by looking at **the** non-terminals that appear in **the** right-hand side **of** its productions. Those dependencies constitute a directed graph, and ﬁniteness **of** **the** grammar is equivalent to **the** absence **of** cycles in **the** dependency graph, thanks to **the** absence **of** useless symbols.

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Since its discovery, differential linear logic (DLL) inspired numerous domains. In denotational semantics, categorical **models** **of** DLL are now commune, and **the** simplest one is Rel, **the** category **of** sets and relations. In proof theory this naturally gave birth to differential proof nets that are full and complete for DLL. In turn, these tools can naturally be translated to their intuitionistic counterpart. By taking **the** co-Kleisly category asso- ciated to **the** ! comonad, Rel becomes MRel, a model **of** **the** λ-**calculus** that contains a notion **of** differentiation. Proof nets can be used naturally to extend **the** λ-**calculus** into **the** **lambda** **calculus** with resources, a **calculus** that contains notions **of** linearity and differentiations. **Of** course MRel is a model **of** **the** λ-**calculus** with resources, and it has been proved adequate, but is it fully abstract?

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Categorical semantics for both call-by-name and call-by-value versions **of** Parigot’s λµ **calculus** [ 15 ] with disjunction types was given by Selinger in [ 20 ]. **The** two variants **of** λµ **calculus** are shown to be isomorphic in **the** presence **of** product and disjunction types. Hofmann and Stre- icher presented categorical continuation **models** for call-by-name λµ **calculus** in [ 9 ] and showed **the** completeness. Lengrand gave categorical semantics for typed λµe µ **calculus** and λξ **calculus** (impli- cational fragment **of** **the** classical sequent **calculus** LK) in [ 13 ]. First attempt to give denotational semantics for pure (untyped) λµ **calculus** is presented in Laurent [ 12 ] by defining a type system with intersection and union types.

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1.1. FUNCTIONS AND TERMINATION 19
**The** Decision Problem Hilbert had complemented his consistency program with **the** so-called Entscheidungsproblem (literally, **the** Decision Problem), which consisted in finding an algorithmic procedure that, given a first order formula F and an (effective) set **of** axioms, would output True if F is a syntactic consequence **of** **the** axioms and False if not, in **the** case such a procedure existed. Gödel’s theorems were a shock for many mathematicians, philosophers and logicians, and moreover, they were a strong indication that an algorithm **of** **the** Entscheidungsproblem did not exist as well, which had not been hitherto suspected. However, Gödel’s results did not straightforwardly give this negative answer, because their proof did not address **the** topic **of** computation, which was essential to understand what algorithmic procedures are and how they behave. Thus, **the** notion **of** computation, that had actually been overlooked by mathematicians and by logicians since **the** introduction **of** mathematics, came into light and caused intense reflection on its nature. Several alternative paradigms were proposed to provide a formal and comprehensive **definition** **of** computation. In his proof **of** **the** incompleteness theorems, Gödel had considered some obviously computable functions that are nowadays known as **the** primitive recursive functions. Integrating some remarks from Herbrand, he then defined **the** set **of** (partial) recursive functions, despite **the** fact he did not believe them to capture all possible computations (see [100], chapter 17). Church, who had introduced **the** λ-**calculus** in 1928, was convinced that a function was effectively computable iff it could be encoded by a λ-term, but many researchers, including Gödel, were skeptical. Finally, Turing defined his celebrated abstract machine [104] model, ever since known as **the** Turing machines. Turing explicitly conceived his machines by emulating (i.e. imitating in an abstract way) **the** human mind, seen as a device having a finite number **of** possible states and a reading/writing head interacting with an infinite tape, that is empty at **the** beginning **of** **the** execution (except for finitely many symbols). Very roughly, this captured **the** idea that (1) a human mind (or a cluster thereof) can handle only a finite number **of** data (i.e. what is already written on **the** tape) and this, in finitely many ways (captured by a finite transition function) (2) a human being writes/erases one letter after **the** other. Last, **the** assumption that **the** tape is infinite gives rise to **the** possibility to conduct a computation (or a reasoning) without limitation in space or time (just, **the** computation or **the** reasoning must stop at some point), which is what **the** notions **of** decidability and computability are about.

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F V (λx.M ) = λk.λx.F(M ) k
Fischer’s translation
Figure 2: CBV CPS-translations, from λ V to λ EvalArg
reducing these redexes. **The** precise **definition** **of** which redexes are adminis- trative is crucial, since this choice might or might not make **the** refined Fischer translation a Galois connection or a reflection (**Definition** 2), as we shall see in section 3.4. In Fig. 3 we give **the** refinement for a particular choice **of** ad- ministrative redexes. In this figure, K ranges over λ-terms. We shall see that, for **the** inductive **definition** to work, it is sufficient to restrict **the** range **of** K to particular terms called continuations.

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As can be expected, denotational completeness does not imply opera- tional completeness. Clearly, **the** operational power **of** machines using mas- sive parallelism cannot be matched by sequential machines. For instance, on networks **of** cellular automata, integer multiplication can be done in real time (cf. Atrubin, 1962 [1], see also Knuth, [21] p.394-399), whereas on Tur- ing machines, an Ω(n/ log n) time lower bound is known. Keeping within sequential computation **models**, multitape Turing machines have greater op- erational power than one-tape Turing machines. Again, this is shown using a complexity argument: palindromes recognition can be done in linear time on two-tapes Turing machines, whereas it requires computation time O(n 2 )

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Clearly, this lemma is too weak for defining interesting coin- ductive examples. Indeed, even a simple functor such as F (X) = 1 ⊕ X (**the** type for **the** natural numbers extended with an infi- nite object) is not right-distribute. By proof search it is easy to verify that we cannot derive a closed judgment for §(1 ⊕ X) ( 1 ⊕ §X. Similarly, we cannot derive a closed judgment for **the** type §(A⊗X) ( A⊗§X, or **the** type §(A⊗A⊗X) ( A⊗A⊗§X. So, we cannot encode infinite lists and infinite trees using Theorem 20 . **The** root **of** this problem lies in **the** fact that in LALC, as in Light Linear Logic, **the** modality § does not distribute on **the** right with ⊗ and ⊕. That is, we cannot prove in general §(A ⊗ B) ( §A ⊗ §B or §(A ⊕ B) ( §A ⊕ §B. Without these distributive rules it seems hard to program any interesting coinductive data in LALC. For this reason, **the** next step is to add these rules to our language. 6. LALC with Distributions

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4.4. **Definition** **of** graph **models**
**The** class **of** graph **models** belongs to Scott continuous semantics, it is **the** simplest class **of** **models** **of** **the** untyped λ-**calculus**; nevertheless it is very rich. All known classes **of** webbed λ-**models** can be presented as variations **of** this class (see [ 6]). **The** simplest graph model, is Engeler’s model E (Example 7.15(i)); it is moreover, from far, **the** simplest **of** all non syntactical λ-**models**. Historically, **the** first graph model which has been isolated was Plotkin and Scott’s Pω, and it was followed soon by E . **The** word graph refers to **the** fact that **the** continuous functions are encoded in **the** model via (a sufficient fragment **of**) their graphs, namely their traces, as recalled below. For more details we refer to [ 6], and to [ 7]. **Definition** 4.4 A total pair G is a pair (G, iG ) where G is an infinite set and iG : G ∗ × G → G is an injective total function.

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The free ompletion method in K; G oh ; and Ghy oh : S ott's model D1 and Park's model P1 are the extensional ompletions of the graph models P and P 0 In parti ular D1 is generated by a p[r]

[Copyright notice will appear here once ’preprint’ option is removed.]
1. Introduction
In computer science, probability theory can be used for **models** that enable system abstraction, and also as a way to compute in a set- ting where having access to a source **of** randomness is essential to achieve correctness, as in randomised computation or cryptogra- phy (Goldwasser and Micali 1984). Domains in which probabilis- tic **models** play a key role include robotics (Thrun 2002), linguis- tics (Manning and Sch¨utze 1999), and especially machine learn- ing (Pearl 1988). **The** wealth **of** applications has stimulated **the** de- velopment **of** concrete and abstract programming languages, that most often are extensions **of** their deterministic ancestors. Among **the** many ways probabilistic choice can be captured in program- ming, a simple one consists in endowing **the** language **of** programs with an operator modelling **the** sampling from (one or many) dis- tributions. This renders program evaluation a probabilistic process, and under mild assumptions **the** language becomes universal for probabilistic computation. Particularly fruitful in this sense has been **the** line **of** work in **the** functional paradigm.

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PG itself is a compact closed category, but we are interested in **the** subcategory PG − , where ∼-arenas and strategies are negative (that is, all initial moves are negative), and strategies are moreover well-threaded (meaning that events in S depend on a unique initial move).
Let A and B be objects **of** PG − . Their tensor product A ⊗ B is simply defined as A k B. **The** tensorial unit is **the** empty ∼-arena, and moreover **the** tensor is closed: **the** function space A ( B has events those **of** (k min(B) A ⊥ ) k B with same polarity. **The** causal dependency is induced, with extra causal links {((2, b), (1, (b, a))) | b ∈ min(B), a ∈ A}. **The** function χ : (A ( B) → A ⊥ k B defined as (1, (b, a)) 7→ (1, a) and (2, b) 7→ (2, b) allows us to characterise consistent sets and iso families concisely: Con A(B is defined as **the** largest set making χ a map **of** esps, and an order-isomorphism θ between configurations **of** A ( B is in ^ A ( B iff χθ ∈ ^ A ⊥ k B. PG − also has cartesian products, with A & B defined as A k B, only with consistent sets restricted to those **of** A k ∅ and ∅ k B. **The** rest **of** **the** structure, including symmetry, is induced from A k B by restriction.

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3. Control instruction and continuations
We now extend this machine with a call-by-name control instruction, and with continuations. There are two advantages : first, an obvious utility for programming ; second, in **the** frame **of** realisability theory (see **the** introduction), this allows **the** typing **of** programs in classical logic and no longer only in intuitionistic logic. Indeed, **the** type **of** **the** instruction call/cc is Peirce’s law ((A → B) → A) → A (see [2]). As we did before, we give first an informal description **of** **the** machine, then mathematical definitions.

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101 - 54602 Villers lès Nancy Cedex France Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex France Unité de recherche INRIA Rhône-Alpes : 65[r]

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n ] is a list **of** terms. While **the** structure **of** **the** usual -**calculus** is isomorphic to **the** structure **of** natural deduction, this new
structure is isomorphic to **the** structure **of** Gentzen-style sequent **calculus**. To express **the** basis **of** **the** isomorphism, we consider intuitionistic logic with **the** implication as sole connective. However we do not consider Gentzen's **calculus** LJ, but a **calculus** LJT which leads to restrict **the** notion **of** cut-free proofs in LJ. We need also to explicitly consider, in a simply typed version **of** this -**calculus**, a substitution operator and a

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L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]