Linear Lambda Research Articles

A General Deep Learning Method for Computing Molecular Parameters of a Viscoelastic Constitutive Model by Solving an Inverse Problem.

Prediction of molecular parameters and material functions from the macroscopic viscoelastic properties of complex fluids are of great significance for molecular and formulation design in fundamental research as well as various industrial applications. A general learning method for computing molecular parameters of a viscoelastic constitutive model by solving an inverse problem is proposed. The accuracy, convergence and robustness of a deep neural network (DNN)-based numerical solver have been validated by considering the Rolie-Poly model for modeling the linear and non-linear steady rheometric properties of entangled polymer solutions in a wide range of concentrations. The results show that as long as the DNN could be trained with a sufficiently high accuracy, the DNN-based numerical solver would rapidly converge to its solution in solving an inverse problem. The solution is robust against small white noise disturbances to the input stress data. However, if the input stress significantly deviates from the original stress, the DNN-based solver could readily converge to a different solution. Hence, the resolution of the numerical solver for inversely computing molecular parameters is demonstrated. Moreover, the molecular parameters computed by the DNN-based numerical solver not only reproduce accurately the steady viscoelastic stress of completely monodisperse linear lambda DNA solutions over a wide range of shear rates and various concentrations, but also predict a power law concentration scaling with a nearly same scaling exponent as those estimated from experimental results.

Linear Additives

We introduce LAM, a subsystem of IMALL2 with restricted additive rules able to manage duplication linearly, called linear additive rules. LAM is presented as the type assignment system for a calculus endowed with copy constructors, which deal with substitution in a linear fashion. As opposed to the standard additive rules, the linear additive rules do not affect the complexity of term reduction: typable terms of LAM enjoy linear strong normalization. Moreover, a mildly weakened version of cut-elimination for this system is proven which takes a cubic number of steps. Finally, we define a sound translation from proofs of LAM into linear lambda terms of IMLL2, and we study its complexity.

A Braided Lambda Calculus

We present an untyped linear lambda calculus with braids, the corresponding combinatory logic, and the semantic models given by crossed G-sets.

Deriving Theorems in Implicational Linear Logic, Declaratively

The problem we want to solve is how to generate all theorems of a given size in the implicational fragment of propositional intuitionistic linear logic. We start by filtering for linearity the proof terms associated by our Prolog-based theorem prover for Implicational Intuitionistic Logic. This works, but using for each formula a PSPACE-complete algorithm limits it to very small formulas. We take a few walks back and forth over the bridge between proof terms and theorems, provided by the Curry-Howard isomorphism, and derive step-by-step an efficient algorithm requiring a low polynomial effort per generated theorem. The resulting Prolog program runs in O(N) space for terms of size N and generates in a few hours 7,566,084,686 theorems in the implicational fragment of Linear Intuitionistic Logic together with their proof terms in normal form. As applications, we generate datasets for correctness and scalability testing of linear logic theorem provers and training data for neural networks working on theorem proving challenges. The results in the paper, organized as a literate Prolog program, are fully replicable. Keywords: combinatorial generation of provable formulas of a given size, intuitionistic and linear logic theorem provers, theorems of the implicational fragment of propositional linear intuitionistic logic, Curry-Howard isomorphism, efficient generation of linear lambda terms in normal form, Prolog programs for lambda term generation and theorem proving.

Let-Binding in a Linear Lambda Calculus with First-Class Continuations

In our previous work, we proposed a linear lambda calculus with first-class continuations. In the usual lambda calculus, an argument value can be duplicated and deleted, but the continuation cannot. In our calculus, a value is allowed to be neither duplicated nor deleted, but a continuation is allowed, so it can be considered to be dual with the usual lambda calculus. In this paper, we extend a linear lambda calculus with first-class continuations by adding let-binding. We define a syntax, a call-by-value reduction, and a type system of the calculus. Then, we discuss the let-polymorphism in our calculus.

Linear Numeral Systems

We investigate numeral systems in the lambda calculus; specifically in the linear lambda calculus where terms cannot be copied or erased. Our interest is threefold: representing numbers in the linear calculus, finding constant time arithmetic operations when possible for successor, addition and predecessor, and finally, efficiently encoding subtraction—an operation that is problematic in many numeral systems. This paper defines systems that address these points, and in addition provides a characterisation of linear numeral systems.

Semantics of higher-order quantum computation via geometry of interaction

While much of the current study on quantum computation employs low-level formalisms such as quantum circuits, several high-level languages/calculi have been recently proposed aiming at structured quantum programming. The current work contributes to the semantical study of such languages by providing interaction-based semantics of a functional quantum programming language; the latter is, much like Selinger and Valiron's, based on linear lambda calculus and equipped with features like the ! modality and recursion. The proposed denotational model is the first one that supports the full features of a quantum functional programming language; we prove adequacy of our semantics. The construction of our model is by a series of existing techniques taken from the semantics of classical computation as well as from process theory. The most notable among them is Girard's Geometry of Interaction (GoI), categorically formulated by Abramsky, Haghverdi and Scott. The mathematical genericity of these techniques—largely due to their categorical formulation—is exploited for our move from classical to quantum.

Strong Typed Böhm Theorem and Functional Completeness on the Linear Lambda Calculus

In this paper, we prove a version of the typed B\"ohm theorem on the linear lambda calculus, which says, for any given types A and B, when two different closed terms s1 and s2 of A and any closed terms u1 and u2 of B are given, there is a term t such that t s1 is convertible to u1 and t s2 is convertible to u2. Several years ago, a weaker version of this theorem was proved, but the stronger version was open. As a corollary of this theorem, we prove that if A has two different closed terms s1 and s2, then A is functionally complete with regard to s1 and s2. So far, it was only known that a few types are functionally complete.

Linear lambda terms as invariants of rooted trivalent maps

AbstractThe main aim of the paper is to give a simple and conceptual account for the correspondence (originally described by Bodini, Gardy, and Jacquot) between α-equivalence classes of closed linear lambda terms and isomorphism classes of rooted trivalent maps on compact-oriented surfaces without boundary, as an instance of a more general correspondence between linear lambda terms with a context of free variables and rooted trivalent maps with a boundary of free edges. We begin by recalling a familiar diagrammatic representation for linear lambda terms, while at the same time explaining how such diagrams may be read formally as a notation for endomorphisms of a reflexive object in a symmetric monoidal closed (bi)category. From there, the “easy” direction of the correspondence is a simple forgetful operation which erases annotations on the diagram of a linear lambda term to produce a rooted trivalent map. The other direction views linear lambda terms as complete invariants of their underlying rooted trivalent maps, reconstructing the missing information through a Tutte-style topological recurrence on maps with free edges. As an application in combinatorics, we use this analysis to enumerate bridgeless rooted trivalent maps as linear lambda terms containing no closed proper subterms, and conclude by giving a natural reformulation of the Four Color Theorem as a statement about typing in lambda calculus.

A correspondence between rooted planar maps and normal planar lambda terms

A rooted planar map is a connected graph embedded in the 2-sphere, with one edge marked and assigned an orientation. A term of the pure lambda calculus is said to be linear if every variable is used exactly once, normal if it contains no beta-redexes, and planar if it is linear and the use of variables moreover follows a deterministic stack discipline. We begin by showing that the sequence counting normal planar lambda terms by a natural notion of size coincides with the sequence (originally computed by Tutte) counting rooted planar maps by number of edges. Next, we explain how to apply the machinery of string diagrams to derive a graphical language for normal planar lambda terms, extracted from the semantics of linear lambda calculus in symmetric monoidal closed categories equipped with a linear reflexive object or a linear reflexive pair. Finally, our main result is a size-preserving bijection between rooted planar maps and normal planar lambda terms, which we establish by explaining how Tutte decomposition of rooted planar maps (into vertex maps, maps with an isthmic root, and maps with a non-isthmic root) may be naturally replayed in linear lambda calculus, as certain surgeries on the string diagrams of normal planar lambda terms.

Embedding a full linear Lambda calculus in Haskell

We present an encoding of full linear lambda calculus in Haskell using higher order abstract syntax. By making use of promoted data kinds, multi-parameter type classes and functional dependencies, the encoding allows Haskell to do both linear type checking and linear type inference.

A deep inference system with a self-dual binder which is complete for linear lambda calculus

We recall that SBV, a proof system developed under the methodology of deep inference, extends multiplicative linear logic with the self-dual non-commutative logical operator Seq. We introduce SBVQ that extends SBV by adding the self-dual quantifier Sdq. The system SBVQ is consistent because we prove that (the analogous of) cut elimination holds for it. Its new logical operator Sdq operationally behaves as a binder, in a way that the interplay between Seq, and Sdq can model {\beta}-reduction of linear {\lambda}-calculus inside the cut-free subsystem BVQ of SBVQ. The long term aim is to keep developing a programme whose goal is to give pure logical accounts of computational primitives under the proof-search-as-computation analogy, by means of minimal, and incremental extensions of SBV.

On the π-calculus and Co-intuitionistic Logic. Notes on Logic for Concurrency and λP Systems

We reconsider work by Bellin and Scott in the 1990s on R. Milner and S. Abramsky's encoding of linear logic in the π-calculus and give an account of efforts to establish a tight connection between the structure of proofs and of the cut elimination process in multiplicative linear logic, on one hand, and the input-output behaviour of the processes that represent them, on the other, resulting in a proof-theoretic account of a variant of Chu's construction. But Milner's encoding of the linear lambda calculus suggests consideration of multiplicative co-intuitionistic linear logic: we provide a term assignment for it, a calculus of coroutines which presents features of concurrent and distributed computing. Finally, as a test case of its adequacy as a logic for distributed computation, we represent our term assignment as a λP system. We argue that translations of typed functional languages in concurrent and distributed systems such as π-calculi or λP systems are best typed with co-intuitionistic logic, where some features of computations match the logical properties in a natural way.

Session Types in Abelian Logic

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.

Verification of a decrease in the rigidity of the phage λ DNA polymeric chain in low ionic strength aqueous solutions by testing the polymer-polymer interlink interactions

Changes in the rigidity of the polymetric chain of phage lambda double-strand DNA have been studied by laser correlation spectroscopy. It was shown that, as the ionic strength increases, the effect of the screening of the hydrodynamic interaction of the links of the polymeric chain specific for polymeric coils arises in a DNA solution. It is assumed that the screening occurs when the threshold of the overlapping of DNA coils is achieved. The overlapping of coils is the result of a previously observed significant rise of DNA coil size from abnormally small DNA coils in low ionic strength buffers (about 10(-2) M Na+ or less) to maximum possible large coils in the 5SSC and 5SSC-like buffers. Further analysis of the far interlink interactions in linear lambda phage DNA coils in similar buffers at pH 7 and 4 confirms the earlier proposal about the role of H+ ions in the appearance of abnormally small DNA coils. The abnormal decrease in the DNA coil size in low ionic strength buffers is not a specific feature of lambda phage DNA only.

A calculus for hardware description

AbstractIn efforts to overcome the complexity of the syntax and the lack of formal semantics of conventional hardware description languages, a number of functional hardware description languages have been developed. Like conventional hardware description languages, however, functional hardware description languages eventually convert all source programs into netlists, which describe wire connections in hardware circuits at the lowest level and conceal all high-level descriptions written into source programs. We develop a calculus, calledlλ (linear lambda), which may serve as an intermediate functional language just above netlists in the hierarchy of hardware description languages. In order to support higher-order functions,lλ uses a linear type system, which enforces the linear use of variables of function type. The translation oflλ into structural descriptions of hardware circuits is sound and complete in the sense that it maps expressions only to realizable hardware circuits, and that every realizable hardware circuit has a corresponding expression inlλ. To illustrate the use oflλ as a practical intermediate language for hardware description, we design a simple hardware description language that extendslλ with polymorphism, and use it to implement a fast Fourier transform circuit and a bitonic sorting network.

Gödel’s system [formula omitted] revisited

The linear lambda calculus, where variables are restricted to occur in terms exactly once, has a very weak expressive power: in particular, all functions terminate in linear time. In this paper we consider a simple extension with natural numbers and a restricted iterator: only closed linear functions can be iterated. We show properties of this linear version of Gödel’s T using a closed reduction strategy, and study the class of functions that can be represented. Surprisingly, this linear calculus offers a huge increase in expressive power over previous linear versions of T , which are ‘closed at construction’ rather than ‘closed at reduction’. We show that a linear T with closed reduction is as powerful as T .

Functional netlists

In efforts to overcome the complexity of the syntax and the lack of formal semantics of conventional hardware description languages, a number of functional hardware description languages have been developed. Like conventional hardware description languages, however, functional hardware description languages eventually convert all source programs into netlists, which describe wire connections in hardware circuits at the lowest level and conceal all high-level descriptions written into source programs. We develop a variant of the lambda calculus, called l λ (linear lambda), which may serve as a high-level substitute for netlists. In order to support higher-order functions, l λ uses a linear type system which enforces the linear use of variables of function type. The translation of l λ into structural descriptions of hardware circuits is sound and complete in the sense that it maps expressions only to realizable hardware circuits and that every realizable hardware circuit has a corresponding expression in l λ. To illustrate the use of l λ as a high-level substitute for netlists, we design a simple hardware description language that extends l λ with polymorphism, and use it to implement a Fast Fourier Transform circuit.

Quantitative Analysis of Isolated and Clustered DNA Damage Induced by Gamma-rays, Carbon Ion Beams, and Iron Ion Beams

Ionizing radiation induces multiple damaged sites (clustered damage) together with isolated lesions in DNA. Clustered damage consists of closely spaced lesions within a few helical turns of DNA and is considered to be crucial for understanding the biological consequences of ionizing radiation. In the present study, two types of DNA, supercoiled plasmid DNA and linear lambda DNA, were irradiated with gamma-rays, carbon ion beams, and iron ion beams, and the spectra and yield of isolated DNA damage and bistranded clustered DNA damage were fully analyzed. Despite using different methods for damage analysis, the experiments with plasmid and lambda DNA gave largely consistent results. The spectra of both isolated and clustered damage were essentially independent of the quality of the ionizing radiation used for irradiation. The yields of clustered damage as well as of isolated damage decreased with the different radiation beams in the order gamma> C > Fe, thus exhibiting an inverse correlation with LET [gamma (0.2 keV/microm) < C (13 keV/microm) < Fe (200 keV/microm)]. Consistent with in vitro data, the yield of chromosomal DNA DSBs decreased with increasing LET in Chinese hamster cells irradiated with carbon ion beams with different LETs, suggesting that the decrease in the yield of clustered damage with increasing LET is not peculiar to in vitro irradiation of DNA, but is common for both in vitro and in vivo irradiation. These results suggest that the adverse biological effect of the ionizing radiation is not simply accounted for by the yield of clustered DNA damage, and that the complexity of the clustered damage needs to be considered to understand the biological consequences of ionizing radiation.

Separating Syntax and Combinatorics in Categorial Grammar

This paper argues that modern versions of categorial grammar and in particular multimodal categorial grammar can profit considerably from reintroducing Haskell Curry’s old distinction between what he called phenogrammatics and tectogrammatics. Tectogrammatics is the abstract way in which linguistic signs are built up, while phenogrammatics deals with concrete processes of string formation and the way in which the sign ultimately manifests itself. The distinction will be modeled in a theory called Lambda Grammars in which tectogrammatics is formalized by taking linear lambda terms over a given lexicon. Phenogrammatics can then be formalized with the help of a set of modal operators. The procedure is illustrated by means of an analysis of some aspects of Dutch word order that is based on earlier multimodal work of Oehrle and Moortgat on Dutch.