# Haut PDF On the definition of lambda-calculus models

### On the definition of lambda-calculus models

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### Incrementality and effect simulation in the simply typed lambda calculus

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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### Full abstraction for the quantum lambda-calculus

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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### Lambda theories of effective lambda models

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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### Inhabitation in simply typed lambda-calculus through a lambda-calculus for proof search

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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### On the discriminating power of tests in resource lambda-calculus

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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### Understanding untyped $\overline{\lambda}\mu\widetilde{\mu}$ calculus

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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### Non-idempotent typing operators, beyond the lambda-calculus

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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### Call-by-Value Lambda-calculus and LJQ

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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### ASMs and Operational Algorithmic Completeness of Lambda Calculus

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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### Algebras and Coalgebras in the Light Affine Lambda Calculus

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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### Effective lambda-models vs recursively enumerable lambda-theories

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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### Graph models of lambda-calculus at work, and variations

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]

### A lambda-calculus foundation for universal probabilistic programming

[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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### Fully Abstract Models of the Probabilistic λ-calculus

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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### A call-by-name lambda-calculus machine

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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### Metatheoretic Results for a Modal Lambda Calculus

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### A Lambda-calculus Structure Isomorphic to Gentzen-style Sequent Calculus Structure

::: ; u 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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### A Proof of Weak Termination of the Simply-Typed {$\lambda\sigma$}-Calculus

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### Towards a lambda-calculus for concurrent and communicating systems

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