Bisimilarity as a Theory of Functional Programming

We explore functional programming through a 16-week internship at Los Alamos National Laboratory. Functional programming is a branch of computer science that has exploded in popularity over the past decade due to its high-level syntax, ease of parallelization, and abundant applications. First, we summarize functional programming by listing the advantages of functional programming languages over the usual imperative languages, and we introduce the concept of parsing. Second, we discuss the importance of lambda calculus in the theory of functional programming. Lambda calculus was invented by Alonzo Church in the 1930s to formalize the concept of effective computability, and every functional language is essentially some implementation of lambda calculus. Finally, we display the lasting products of the internship: additions to a compiler and runtime system for the pure functional language STG, including both a set of tests that indicate the validity of updates to the compiler and a compiler pass that checks for illegal instances of duplicate names.

The pedagogical desirability of functional programming is effectively swamped by an overwhelmingly imperative‐language‐oriented culture. Because a meaningful subset of the functional paradigm is accessible within imperative languages, it is possible to identify a subset of Modula‐2 that characterizes the functional paradigm subset. A series of list‐processing problems and their solutions in the Modula‐2 subset provides students with ample exposure to the practice and theory of functional programming. Extension to the full functional paradigm and migration to specialized functional languages are managed, in the worst case by initial pseudo‐Modula‐2 programming involving hypothetical extensions supporting higher‐order functions and polymorphism. More powerful solutions to this problem are foreshadowed, in the form of preprocessor implementations of Modula‐2‐ and Ada‐derivative functional languages.

Proceedings of the 13th ACM SIGPLAN international conference on Functional programming

Coinductive data structures, such as streams or infinite trees, have many applications in functional programming and type theory, and are naturally defined using recursive equations. But how do we ensure that such equations make sense, i.e. that they actually generate a productive infinite object? A standard means to achieve productivity is to use Banach's fixed-point theorem, which guarantees the unique existence of solutions to recursive equations on metric spaces under certain conditions. Functions satisfying these conditions are called contractions. In this article, we give a new characterization of contractions on streams in the form of a sound and complete representation theorem, and generalize this result to a wide class of non-well-founded structures, first to infinite binary trees, then to final coalgebras of container functors.These results have important potential applications in functional programming, where coinduction and corecursion are successfully deployed to model continuous reactive systems, dynamic interactivity, signal processing, and other tasks that require flexible manipulation of non-well-founded data. Our representation theorems provide a definition paradigm to compactly compute with such data and easily reason about them.

Metapredicativity is a new general term in proof theory which describes the analysis and study of formal systems whose proof-theoretic strength is beyond the Feferman-Schutte ordinal Γ0 but which are nevertheless amenable to purely predicative methods. Typical examples of formal systems which are apt for scaling the initial part of metapredicativity are the transfinitely iterated fixed point theories IDα whose detailed proof-theoretic analysis is given by Jager, Kahle, Setzer and Strahm in [18]. In this paper we assume familiarity with [18]. For natural extensions of Friedman’s ATR that can be measured against transfinitely iterated fixed point theories the reader is referred to Jager and Strahm [20]. In the mid seventies, Feferman [3, 4] introduced systems of explicit mathematics in order to provide an alternative foundation of constructive mathematics. More precisely, it was the origin of Feferman’s program to give a logical account of Bishop-style constructive mathematics. Right from the beginning, systems of explicit mathematics turned out to be of general interest for proof theory, mainly in connection with the proof-theoretic analysis of subsystems of first and second order arithmetic and set theory, cf. e.g. Jager [15] and Jager and Pohlers [19]. More recently, systems of explicit mathematics have been used to develop a general logical framework for functional programming and type theory, where it is possible to derive correctness and termination properties of functional programs. Important references in this connection are Feferman [6, 7, 9] and Jager [17]. Universes are a frequently studied concept in constructive mathematics at least since the work of Martin-Lof, cf. e.g. Martin-Lof [23] or Palmgren [27] for

In programming, some data acts as a resource (e.g., file handles, channels) subject to usage constraints. This poses a challenge to software correctness as most languages are agnostic to constraints on data. The approach of linear types provides a partial remedy, delineating data into resources to be used but never copied or discarded, and unconstrained values. Bounded Linear Logic provides a more fine-grained approach, quantifying non-linear use via an indexed-family of modalities. Recent work on coeffect types generalises this idea to graded comonads, providing type systems which can capture various program properties. Here, we propose the umbrella notion of graded modal types, encompassing coeffect types and dual notions of type-based effect reasoning via graded monads. In combination with linear and indexed types, we show that graded modal types provide an expressive type theory for quantitative program reasoning, advancing the reach of type systems to capture and verify a broader set of program properties. We demonstrate this approach via a type system embodied in a fully-fledged functional language called Granule, exploring various examples.

We describe experiments in teaching fundamental informatics notions around mathematical structures for formal concepts, and effective algorithms to manipulate them. The major themes of lambda-calculus and type theory served as guides for the effective implementation of functional programming languages and higher-order proof assistants, appropriate for reflecting the theoretical material into effective tools to represent constructively the concepts and formally certify the proofs of their properties. Progressively, a literate programming and proving style replaced informal mathematics in the presentation of the material as executable course notes. The talk will evoke the various stages of (in)completion of the corresponding set of notes along the years, and tell how their elaboration proved to be essential to the discovery of fundamental results.

The research carried out had the objective of analyzing the perspective of students of the Specialization Course in Rural Education in relation to the informal use of mathematics in the classroom. The informal use of mathematics in the classroom was addressed, focusing on a specific case involving a Youth and Adult Education (AYE) class. The qualitative case study was conducted with 8 participating students and took place remotely in June 2022, through the Google Meet platform. As an instrument, an online questionnaire with 6 open questions was used, which explored the participants' opinions about the attitude of a teacher and a student in a situation of disagreement regarding the use of formal mathematics as opposed to practical experience. The students' responses were analyzed and compared. The results indicate that the participants value both theoretical and practical knowledge and recognize the importance of dialogue and interaction between teacher and student. They also emphasize the need to balance theoretical and practical knowledge in the classroom and to value students' prior experience and knowledge. The study highlights the relevance of considering different educational perspectives and adapting the pedagogical approach to student needs and experiences, especially in AYE contexts.

BackgroundRNA interference (RNAi) is a regulatory cellular process that controls post-transcriptional gene silencing. During RNAi double-stranded RNA (dsRNA) induces sequence-specific degradation of homologous mRNA via the generation of smaller dsRNA oligomers of length between 21-23nt (siRNAs). siRNAs are then loaded onto the RNA-Induced Silencing multiprotein Complex (RISC), which uses the siRNA antisense strand to specifically recognize mRNA species which exhibit a complementary sequence. Once the siRNA loaded-RISC binds the target mRNA, the mRNA is cleaved and degraded, and the siRNA loaded-RISC can degrade additional mRNA molecules. Despite the widespread use of siRNAs for gene silencing, and the importance of dosage for its efficiency and to avoid off target effects, none of the numerous mathematical models proposed in literature was validated to quantitatively capture the effects of RNAi on the target mRNA degradation for different concentrations of siRNAs. Here, we address this pressing open problem performing in vitro experiments of RNAi in mammalian cells and testing and comparing different mathematical models fitting experimental data to in-silico generated data. We performed in vitro experiments in human and hamster cell lines constitutively expressing respectively EGFP protein or tTA protein, measuring both mRNA levels, by quantitative Real-Time PCR, and protein levels, by FACS analysis, for a large range of concentrations of siRNA oligomers.ResultsWe tested and validated four different mathematical models of RNA interference by quantitatively fitting models' parameters to best capture the in vitro experimental data. We show that a simple Hill kinetic model is the most efficient way to model RNA interference. Our experimental and modeling findings clearly show that the RNAi-mediated degradation of mRNA is subject to saturation effects.ConclusionsOur model has a simple mathematical form, amenable to analytical investigations and a small set of parameters with an intuitive physical meaning, that makes it a unique and reliable mathematical tool. The findings here presented will be a useful instrument for better understanding RNAi biology and as modelling tool in Systems and Synthetic Biology.

NK cells are innate lymphoid cells that are critical for host defense against infection, and mediate anti-tumor responses. MicroRNAs (miRNAs) are a large family of small noncoding RNAs that target the 3' untranslated region (UTR) of mRNAs, thereby attenuating protein translation. The expression of miRNAs within human peripheral blood and mouse splenic NK cells has been cataloged, with the majority of the miRNA sequence pool represented in the top 60 most abundantly expressed miRNAs. Global miRNA deficiency within NK cells has confirmed their critical role in NK-cell biology, including defects in NK-cell development and altered functionality. Studies using gain- and loss-of-function of individual miRNAs in NK cells have demonstrated the role of specific miRNAs in regulating NK-cell development, maturation, and activation. miRNAs also regulate fundamental NK-cell processes including cytokine production, cytotoxicity, and proliferation. This review provides an update on the intrinsic miRNA regulation of NK cells, including miRNA expression profiles, as well as their impact on NK-cell biology. Additional profiling is needed to better understand miRNA expression within NK-cell developmental intermediates, subsets, tissues, and in the setting of disease. Furthermore, key open questions in the field as well as technical challenges in the study of miRNAs in NK cells are highlighted.

Launchbury and Peyton Jones came up with an ingenious idea for embedding regions of imperative programming in a pure functional language like Haskell. The key idea was based on a simple modification of Hindley-Milner's type system. Our first contribution is to propose a more natural encapsulation construct exploiting higher-order kinds, which achieves the same encapsulation effect, but avoids the ad hoc type parameter of the original proposal. The second contribution is a type safety result for encapsulation of strict state using both the original encapsulation construct and the newly introduced one. We establish this result in a more expressive context than the original proposal, namely in the context of the higher-order lambda-calculus. The third contribution is a type safety result for encapsulation of lazy state in the higher-order lambda-calculus. This result resolves an outstanding open problem on which previous proof attempts failed. In all cases, we formalize the intended implementations as simple big-step operational semantics on untyped terms, which capture interesting implementation details not captured by the reduction semantics proposed previously.

Symbolic automata and transducers extend finite automata and transducers by allowing transitions to carry predicates and functions over rich alphabet theories, such as linear arithmetic. Therefore, these models extend their classic counterparts to operate over infinite alphabets, such as the set of rational numbers. Due to their expressiveness, symbolic automata and transducers have been used to verify functional programs operating over lists and trees, to prove the correctness of complex implementations of BASE64 and UTF encoders, and to expose data parallelism in computations that may otherwise seem inherently sequential. In this paper, we give an overview of what is currently known about symbolic automata and transducers as well as their variants. We discuss what makes these models different from their finite-alphabet counterparts, what kind of applications symbolic models can enable, and what challenges arise when reasoning about these formalisms. Finally, we present a list of open problems and research directions that relate to both the theory and practice of symbolic automata and transducers.