A notion of probabilistic lambda-calculus usually comes with a prescribed reduction strategy, typically call-by-name or call-by-value, as the calculus is non-confluent and these strategies yield different results. This is a break with one of the main advantages of lambda-calculus: confluence, which means results are independent from the choice of strategy. We present a probabilistic lambda-calculus where the probabilistic operator is decomposed into two syntactic constructs: a generator, which represents a probabilistic event; and a consumer, which acts on the term depending on a given event. The resulting calculus, the Probabilistic Event Lambda-Calculus, is confluent, and interprets the call-by-name and call-by-value strategies through different interpretations of the probabilistic operator into our generator and consumer constructs. We present two notions of reduction, one via fine-grained local rewrite steps, and one by generation and consumption of probabilistic events. Simple types for the calculus are essentially standard, and they convey strong normalization. We demonstrate how we can encode call-by-name and call-by-value probabilistic evaluation.
This article is devoted to the presentation of lambda_rex, an explicit substitution calculus with de Bruijn indexes and a simple notation. By being isomorphic to lambda_ex - a recent formalism with variable names -, lambda_rex accomplishes simulation of beta-reduction (Sim), preservation of beta-strong normalization (PSN) and meta-confluence (MC), among other desirable properties. Our calculus is based on a novel presentation of lambda_dB, using a swap notion that was originally devised by de Bruijn. Besides lambda_rex, two other indexed calculi isomorphic to lambda_x and lambda_xgc are presen…
The intrinsic treatment of binding in the lambda calculus makes it an ideal data structure for representing syntactic objects with binding such as formulas, proofs, types, and programs. Supporting such a data structure in an implementation is made difficult by the complexity of the substitution operation relative to lambda terms. In this paper we present the suspension calculus, an explicit treatment of meta level binding in the lambda calculus. We prove properties of this calculus which make it a suitable replacement for the lambda calculus in implementation. Finally, we compare the suspensio…
Particle-style token machines are a way to interpret proofs and programs, when the latter are written following the principles of linear logic. In this paper, we show that token machines also make sense when the programs at hand are those of a simple quantum lambda-calculus with implicit qubits. This, however, requires generalising the concept of a token machine to one in which more than one particle travel around the term at the same time. The presence of multiple tokens is intimately related to entanglement and allows us to give a simple operational semantics to the calculus, coherently with…
In this paper we provide an abstract model theory for the untyped differential lambda-calculus and the resource calculus. In particular we propose a general definition of model of these calculi, namely the notion of linear reflexive object in a Cartesian closed differential category. Examples of models based on relations are provided.
We present two rewriting systems that define labelled explicit substitution lambda-calculi. Our work is motivated by the close correspondence between Levy's labelled lambda-calculus and paths in proof-nets, which played an important role in the understanding of the Geometry of Interaction. The structure of the labels in Levy's labelled lambda-calculus relates to the multiplicative information of paths; the novelty of our work is that we design labelled explicit substitution calculi that also keep track of exponential information present in call-by-value and call-by-name translations of the lam…
The infinitary lambda calculi pioneered by Kennaway et al. extend the basic lambda calculus by metric completion to infinite terms and reductions. Depending on the chosen metric, the resulting infinitary calculi exhibit different notions of strictness. To obtain infinitary normalisation and infinitary confluence properties for these calculi, Kennaway et al. extend $β$-reduction with infinitely many `$\bot$-rules', which contract meaningless terms directly to $\bot$. Three of the resulting Böhm reduction calculi have unique infinitary normal forms corresponding to Böhm-like trees. In this paper…
In this paper we invite the reader to a journey through three lambda calculi with resource control: the lambda calculus, the sequent lambda calculus, and the lambda calculus with explicit substitution. All three calculi enable explicit control of resources due to the presence of weakening and contraction operators. Along this journey, we propose intersection type assignment systems for all three resource control calculi. We recognise the need for three kinds of variables all requiring different kinds of intersection types. Our main contribution is the characterisation of strong normalisation o…
We study an untyped lambda calculus with quantum data and classical control. This work stems from previous proposals by Selinger and Valiron and by Van Tonder. We focus on syntax and expressiveness, rather than (denotational) semantics. We prove subject reduction, confluence and a standardization theorem. Moreover, we prove the computational equivalence of the proposed calculus with a suitable class of quantum circuit families.
On the topic of probabilistic rewriting, there are several works studying both termination and confluence of different systems. While working with a lambda calculus modelling quantum computation, we found a system with probabilistic rewriting rules and strongly normalizing terms. We examine the effect of small modifications in probabilistic rewriting, affine variables, and strategies on the overall confluence in this strongly normalizing probabilistic calculus.
Fitch-style modal deduction, in which modalities are eliminated by opening a subordinate proof, and introduced by shutting one, were investigated in the 1990s as a basis for lambda calculi. We show that such calculi have good computational properties for a variety of intuitionistic modal logics. Semantics are given in cartesian closed categories equipped with an adjunction of endofunctors, with the necessity modality interpreted by the right adjoint. Where this functor is an idempotent comonad, a coherence result on the semantics allows us to present a calculus for intuitionistic S4 that is si…
Fitch-style modal lambda calculi enable programming with necessity modalities in a typed lambda calculus by extending the typing context with a delimiting operator that is denoted by a lock. The addition of locks simplifies the formulation of typing rules for calculi that incorporate different modal axioms, but each variant demands different, tedious and seemingly ad hoc syntactic lemmas to prove normalization. In this work, we take a semantic approach to normalization, called normalization by evaluation (NbE), by leveraging the possible-world semantics of Fitch-style calculi to yield a more m…
We propose to use Church encodings in typed lambda-calculi as the basis for an automata-theoretic counterpart of implicit computational complexity, in the same way that monadic second-order logic provides a counterpart to descriptive complexity. Specifically, we look at transductions i.e. string-to-string (or tree-to-tree) functions - in particular those with superlinear growth, such as polyregular functions, HDT0L transductions and Sénizergues's "k-computable mappings". Our first results towards this aim consist showing the inclusion of some transduction classes in some classes defined by lam…
In this paper we use finite vector spaces (finite dimension, over finite fields) as a non-standard computational model of linear logic. We first define a simple, finite PCF-like lambda-calculus with booleans, and then we discuss two finite models, one based on finite sets and the other on finite vector spaces. The first model is shown to be fully complete with respect to the operational semantics of the language. The second model is not complete, but we develop an algebraic extension of the finite lambda calculus that recovers completeness. The relationship between the two semantics is describ…
Dependently typed lambda calculi such as the Logical Framework (LF) are capable of representing relationships between terms through types. By exploiting the "formulas-as-types" notion, such calculi can also encode the correspondence between formulas and their proofs in typing judgments. As such, these calculi provide a natural yet powerful means for specifying varied formal systems. Such specifications can be transformed into a more direct form that uses predicate formulas over simply typed lambda-terms and that thereby provides the basis for their animation using conventional logic programmin…
Particle-style token machines are a way to interpret proofs and programs, when the latter are defined according to the principles of linear logic. In this paper, we show that token machines also make sense when the programs at hand are those of a simple linear quantum $λ$-calculus. This, however, requires generalizing the concept of a token machine to one in which more than one particle can possibly travel around the term at the same time. This is intimately related to entanglement and allows to give a simple operational semantics to the calculus coherently with the principles of quantum compu…
Algebraic lambda-calculi have been studied in various ways, but their semantics remain mostly untouched. In this paper we propose a semantic analysis of a general simply-typed lambda-calculus endowed with a structure of vector space. We sketch the relation with two established vectorial lambda-calculi. Then we study the problems arising from the addition of a fixed point combinator and how to modify the equational theory to solve them. We sketch an algebraic vectorial PCF and its possible denotational interpretations.
Substructural type systems, such as affine (and linear) type systems, are type systems which impose restrictions on copying (and discarding) of variables, and they have found many applications in computer science, including quantum programming. We describe one linear and one affine type systems and we formulate abstract categorical models for both of them which are sound and computationally adequate. We also show, under basic assumptions, that interpreting lambda abstractions via a monoidal closed structure (a popular method for linear type systems) necessarily leads to degenerate and inadequa…
The intuitionistic fragment of the call-by-name version of Curien and Herbelin's λ\_mu\_{mu}-calculus is isolated and proved strongly normalising by means of an embedding into the simply-typed lambda-calculus. Our embedding is a continuation-and-garbage-passing style translation, the inspiring idea coming from Ikeda and Nakazawa's translation of Parigot's λ\_mu-calculus. The embedding strictly simulates reductions while usual continuation-passing-style transformations erase permutative reduction steps. For our intuitionistic sequent calculus, we even only need "units of garbage" to be passed. …
In each variant of the lambda-calculus, factorization and normalization are two key-properties that show how results are computed. Instead of proving factorization/normalization for the call-by-name (CbN) and call-by-value (CbV) variants separately, we prove them only once, for the bang calculus (an extension of the lambda-calculus inspired by linear logic and subsuming CbN and CbV), and then we transfer the result via translations, obtaining factorization/normalization for CbN and CbV. The approach is robust: it still holds when extending the calculi with operators and extra rules to model so…
For the lambda-calculus with surjective pairing and terminal type, Curien and Di Cosmo were inspired by Knuth-Bendix completion, and introduced a confluent rewriting system that (1) extends the naive rewriting system, and (2) is stable under contexts. The rewriting system has (i) a rule that rewrites term of a terminal type rewrites to a term constant *, unless the term is not *, (ii) rewrite rules for the extensionality of function types and product types, and rewrite rules mediating the rewrite rules (i) and (ii). Curien and Di Cosmo supposed that because of (iii), any reducibility method ca…
This work gives some insights and results on standardisation for call-by-name pattern calculi. More precisely, we define standard reductions for a pattern calculus with constructor-based data terms and patterns. This notion is based on reduction steps that are needed to match an argument with respect to a given pattern. We prove the Standardisation Theorem by using the technique developed by Takahashi and Crary for lambda-calculus. The proof is based on the fact that any development can be specified as a sequence of head steps followed by internal reductions, i.e. reductions in which no head s…
Software security can be ensured by specifying and verifying security properties of software using formal methods with strong theoretical bases. In particular, programs can be modeled in the framework of lambda-calculi, and interesting properties can be expressed formally by contextual equivalence (a.k.a. observational equivalence). Furthermore, imperative features, which exist in most real-life software, can be nicely expressed in the so-called computational lambda-calculus. Contextual equivalence is difficult to prove directly, but we can often use logical relations as a tool to establish it…
The linear-algebraic lambda-calculus and the algebraic lambda-calculus are untyped lambda-calculi extended with arbitrary linear combinations of terms. The former presents the axioms of linear algebra in the form of a rewrite system, while the latter uses equalities. When given by rewrites, algebraic lambda-calculi are not confluent unless further restrictions are added. We provide a type system for the linear-algebraic lambda-calculus enforcing strong normalisation, which gives back confluence. The type system allows an abstract interpretation in System F.
Calculi with control operators have been studied to reason about control in programming languages and to interpret the computational content of classical proofs. To make these calculi into a real programming language, one should also include data types. As a step into that direction, this paper defines a simply typed call-by-value lambda calculus with the control operators catch and throw, a data type of lists, and an operator for primitive recursion (a la Goedel's T). We prove that our system satisfies subject reduction, progress, confluence for untyped terms, and strong normalization for wel…
We study the two Girard's translations of intuitionistic implication into linear logic by exploiting the bang calculus, a paradigmatic functional language with an explicit box-operator that allows both call-by-name and call-by-value lambda-calculi to be encoded in. We investigate how the bang calculus subsumes both call-by-name and call-by-value lambda-calculi from a syntactic and a semantic viewpoint.
Probabilistic applicative bisimulation is a recently introduced coinductive methodology for program equivalence in a probabilistic, higher-order, setting. In this paper, the technique is applied to a typed, call-by-value, lambda-calculus. Surprisingly, the obtained relation coincides with context equivalence, contrary to what happens when call-by-name evaluation is considered. Even more surprisingly, full-abstraction only holds in a symmetric setting.
Logical relations and their generalizations are a fundamental tool in proving properties of lambda-calculi, e.g., yielding sound principles for observational equivalence. We propose a natural notion of logical relations able to deal with the monadic types of Moggi's computational lambda-calculus. The treatment is categorical, and is based on notions of subsconing, mono factorization systems, and monad morphisms. Our approach has a number of interesting applications, including cases for lambda-calculi with non-determinism (where being in logical relation means being bisimilar), dynamic name cre…
Computation can be considered by taking into account two dimensions: extensional versus intensional, and sequential versus concurrent. Traditionally sequential extensional computation can be captured by the lambda-calculus. However, recent work shows that there are more expressive intensional calculi such as SF-calculus. Traditionally process calculi capture computation by encoding the lambda-calculus, such as in the pi-calculus. Following this increased expressiveness via intensionality, other recent work has shown that concurrent pattern calculus is more expressive than pi-calculus. This pap…
A strong confluence result for Q*, a quantum lambda-calculus with measurements, is proved. More precisely, confluence is shown to hold both for finite and infinite computations. The technique used in the confluence proof is syntactical but innovative. This makes Q* different from similar quantum lambda calculi, which are either measurement-free or provided with a reduction strategy.
This paper shows equivalence of several versions of applicative similarity and contextual approximation, and hence also of applicative bisimilarity and contextual equivalence, in LR, the deterministic call-by-need lambda calculus with letrec extended by data constructors, case-expressions and Haskell's seq-operator. LR models an untyped version of the core language of Haskell. The use of bisimilarities simplifies equivalence proofs in calculi and opens a way for more convenient correctness proofs for program transformations. The proof is by a fully abstract and surjective transfer into a call-…
We observe that normalization by evaluation for simply-typed lambda-calculus with weak coproducts can be carried out in a weak bi-cartesian closed category of presheaves equipped with a monad that allows us to perform case distinction on neutral terms of sum type. The placement of the monad influences the normal forms we obtain: for instance, placing the monad on coproducts gives us eta-long beta-pi normal forms where pi refers to permutation of case distinctions out of elimination positions. We further observe that placing the monad on every coproduct is rather wasteful, and an optimal placem…
This paper concerns the explicit treatment of substitutions in the lambda calculus. One of its contributions is the simplification and rationalization of the suspension calculus that embodies such a treatment. The earlier version of this calculus provides a cumbersome encoding of substitution composition, an operation that is important to the efficient realization of reduction. This encoding is simplified here, resulting in a treatment that is easy to use directly in applications. The rationalization consists of the elimination of a practically inconsequential flexibility in the unravelling of…
In reductive proof search, proofs are naturally generalized by solutions, comprising all possibly infinite structures generated by locally correct, bottom-up application of inference rules. We propose an extension of the Curry-Howard paradigm of representation, from proofs to solutions: to represent solutions by possibly infinite terms of a dedicated lambda-calculus. This new, comprehensive approach to proof search is exemplified with the sequent calculus LJT for intuitionistic implication logic. A finitary representation is proposed, comprising lambda-terms extended with a formal greatest fix…
We investigate the relationship between finite terms in lambda-letrec, the lambda calculus with letrec, and the infinite lambda terms they express. As there are easy examples of lambda-terms that, intuitively, are not unfoldings of terms in lambda-letrec, we consider the question: How can those infinite lambda terms be characterised that are lamda-letrec-expressible in the sense that they can be obtained as infinite unfoldings of terms in lambda-letrec? For 'observing' infinite lambda-terms through repeated 'experiments' carried out at the head of the term we introduce two rewrite systems (wit…
There was a PhD student who says "I found a pair of wooden shoes. I put a coin in the left and a key in the right. Next morning, I found those objects in the opposite shoes." We do not claim existence of such shoes, but propose a similar programming abstraction in the context of typed lambda calculi. The result, which we call the Amida calculus, extends Abramsky's linear lambda calculus LF and characterizes Abelian logic.
We define two extensions of the typed linear lambda-calculus that yield minimal Turing-complete systems. The extensions are based on unbounded recursion in one case, and bounded recursion with minimisation in the other. We show that both approaches are compatible with linearity and typeability constraints. Both extensions of the typed linear lambda-calculus are minimal, in the sense that taking out any of the components breaks the universality of the system. We discuss implementation techniques that exploit the linearity of the calculi. Finally, we apply the results to languages with fixpoint …
Factorization -- a simple form of standardization -- is concerned with reduction strategies, i.e. how a result is computed. We present a new technique for proving factorization theorems for compound rewriting systems in a modular way, which is inspired by the Hindley-Rosen technique for confluence. Specifically, our technique is well adapted to deal with extensions of the call-by-name and call-by-value lambda-calculi. The technique is first developed abstractly. We isolate a sufficient condition (called linear swap) for lifting factorization from components to the compound system, and which is…
The framework of psi-calculi extends the pi-calculus with nominal datatypes for data structures and for logical assertions and conditions. These can be transmitted between processes and their names can be statically scoped as in the standard pi-calculus. Psi-calculi can capture the same phenomena as other proposed extensions of the pi-calculus such as the applied pi-calculus, the spi-calculus, the fusion calculus, the concurrent constraint pi-calculus, and calculi with polyadic communication channels or pattern matching. Psi-calculi can be even more general, for example by allowing structured …
In this short overview, we start with the basics of quantum computing, explaining the difference between the quantum and the classical control paradigms. We give an overview of the quantum control line of research within the lambda calculus, ranging from untyped calculi up to categorical and realisability models. This is a summary of the last 10+ years of research in this area, starting from Arrighi and Dowek's seminal work until today.
This paper extends the dual calculus with inductive types and coinductive types. The paper first introduces a non-deterministic dual calculus with inductive and coinductive types. Besides the same duality of the original dual calculus, it has the duality of inductive and coinductive types, that is, the duality of terms and coterms for inductive and coinductive types, and the duality of their reduction rules. Its strong normalization is also proved, which is shown by translating it into a second-order dual calculus. The strong normalization of the second-order dual calculus is proved by transla…
Applied process calculi include advanced programming constructs such as type systems, communication with pattern matching, encryption primitives, concurrent constraints, nondeterminism, process creation, and dynamic connection topologies. Several such formalisms, e.g. the applied pi calculus, are extensions of the the pi-calculus; a growing number is geared towards particular applications or computational paradigms. Our goal is a unified framework to represent different process calculi and notions of computation. To this end, we extend our previous work on psi-calculi with novel abstract patte…
In this paper we present a semantics for a linear algebraic lambda-calculus based on realizability. This semantics characterizes a notion of unitarity in the system, answering a long standing issue. We derive from the semantics a set of typing rules for a simply-typed linear algebraic lambda-calculus, and show how it extends both to classical and quantum lambda-calculi.
We propose a way to unify two approaches of non-cloning in quantum lambda-calculi: logical and algebraic linearities. The first approach is to forbid duplicating variables, while the second is to consider all lambda-terms as algebraic-linear functions. We illustrate this idea by defining a quantum extension of first-order simply-typed lambda-calculus, where the type is linear on superposition, while allows cloning base vectors. In addition, we provide an interpretation of the calculus where superposed types are interpreted as vector spaces and non-superposed types as their basis.
We introduce two extensions of the $λ$-calculus with a probabilistic choice operator, $Λ_\oplus^{cbv}$ and $Λ_\oplus^{cbn}$, modeling respectively call-by-value and call-by-name probabilistic computation. We prove that both enjoys confluence and standardization, in an extended way: we revisit these two fundamental notions to take into account the asymptotic behaviour of terms. The common root of the two calculi is a further calculus based on Linear Logic, $Λ_\oplus^!$, which allows for a fine control of the interaction between choice and copying, and which allows us to develop a unified, modul…
First, we extend Leifer-Milner RPO theory, by giving general conditions to obtain IPO labelled transition systems (and bisimilarities) with a reduced set of transitions, and possibly finitely branching. Moreover, we study the weak variant of Leifer-Milner theory, by giving general conditions under which the weak bisimilarity is a congruence. Then, we apply such extended RPO technique to the lambda-calculus, endowed with lazy and call by value reduction strategies. We show that, contrary to process calculi, one can deal directly with the lambda-calculus syntax and apply Leifer-Milner technique …
Calculi with control operators have been studied as extensions of simple type theory. Real programming languages contain datatypes, so to really understand control operators, one should also include these in the calculus. As a first step in that direction, we introduce lambda-mu-T, a combination of Parigot's lambda-mu-calculus and Gödel's T, to extend a calculus with control operators with a datatype of natural numbers with a primitive recursor. We consider the problem of confluence on raw terms, and that of strong normalization for the well-typed terms. Observing some problems with extending …
