Parallel Beta Reduction Is Not Elementary Recursive

AbstractWe analyze the inherent complexity of implementing Lévy's notion of optimal evaluation for the λ-calculus, where similar redexes are contracted in one step via so-called parallel β-reduction. Optimal evaluation was finally realized by Lamping, who introduced a beautiful graph reduction technology for sharing evaluation contexts dual to the sharing of values. His pioneering insights have been modified and improved in subsequent implementations of optimal reduction. We prove that the cost of parallel β-reduction is not bounded by any Kalmár-elementary recursive function. Not only do we establish that the parallel β-step cannot be a unit-cost operation, we demonstrate that the time complexity of implementing a sequence of n parallel β-steps is not bounded as O(2n), O(22n), O(222n), or in general, O(Kl(n)), where Kl(n) is a fixed stack of l 2's with an n on top. A key insight, essential to the establishment of this non-elementary lower bound, is that any simply typed λ-term can be reduced to normal form in a number of parallel β-steps that is only polynomial in the length of the explicitly typed term. The result follows from Statman's theorem that deciding equivalence of typed λ-terms is not elementary recursive. The main theorem gives a lower bound on the work that must be done by any technology that implements Lévy's notion of optimal reduction. However, in the significant case of Lamping's solution, we make some important remarks addressing how work done by β-reduction is translated into equivalent work carried out by his bookkeeping nodes. In particular, we identify the computational paradigms of superposition of values and of higher-order sharing, appealing to compelling analogies with quantum mechanics and SIMD-parallelism

Verification of PCP-Related Computational Reductions in Coq

We formally verify several computational reductions concerning the Post correspondence problem (PCP) using the proof assistant Coq. Our verifications include a reduction of a string rewriting problem generalising the halting problem for Turing machines to PCP, and reductions of PCP to the intersection problem and the palindrome problem for context-free grammars. Interestingly, rigorous correctness proofs for some of the reductions are missing in the literature

Quantitative Models and Implicit Complexity

We give new proofs of soundness (all representable functions on base types lies in certain complexity classes) for Elementary Affine Logic, LFPL (a language for polytime computation close to realistic functional programming introduced by one of us), Light Affine Logic and Soft Affine Logic. The proofs are based on a common semantical framework which is merely instantiated in four different ways. The framework consists of an innovative modification of realizability which allows us to use resource-bounded computations as realisers as opposed to including all Turing computable functions as is usually the case in realizability constructions. For example, all realisers in the model for LFPL are polynomially bounded computations whence soundness holds by construction of the model. The work then lies in being able to interpret all the required constructs in the model. While being the first entirely semantical proof of polytime soundness for light logi cs, our proof also provides a notable simplification of the original already semantical proof of polytime soundness for LFPL. A new result made possible by the semantic framework is the addition of polymorphism and a modality to LFPL thus allowing for an internal definition of inductive datatypes.Comment: 29 page

A Survey on Retrieval of Mathematical Knowledge

We present a short survey of the literature on indexing and retrieval of mathematical knowledge, with pointers to 72 papers and tentative taxonomies of both retrieval problems and recurring techniques.Comment: CICM 2015, 20 page

Heparanase overexpression reduces hepcidin expression, affects iron homeostasis and alters the response to inflammation

Hepcidin is the key regulator of systemic iron availability that acts by controlling the degradation of the iron exporter ferroportin. It is expressed mainly in the liver and regulated by iron, inflammation, erythropoiesis and hypoxia. The various agents that control its expression act mainly via the BMP6/SMAD signaling pathway. Among them are exogenous heparins, which are strong hepcidin repressors with a mechanism of action not fully understood but that may involve the competition with the structurally similar endogenous Heparan Sulfates (HS). To verify this hypothesis, we analyzed how the overexpression of heparanase, the HS degrading enzyme, modified hepcidin expression and iron homeostasis in hepatic cell lines and in transgenic mice. The results showed that transient and stable overexpression of heparanase in HepG2 cells caused a reduction of hepcidin expression and of SMAD5 phosphorylation. Interestingly, the clones showed also altered level of TfR1 and ferritin, indices of a modified iron homeostasis. The heparanase transgenic mice showed a low level of liver hepcidin, an increase of serum and liver iron with a decrease in spleen iron content. The hepcidin expression remained surprisingly low even after treatment with the inflammatory LPS. The finding that modification of HS structure mediated by heparanase overexpression affects hepcidin expression and iron homeostasis supports the hypothesis that HS participate in the mechanisms controlling hepcidin expression

Typing a Core Binary Field Arithmetic in a Light Logic

We design a library for binary field arithmetic and we supply a core API which is completely developed in DLAL, extended with a fix point formula. Since DLAL is a restriction of linear logic where only functional programs with polynomial evaluation cost can be typed, we obtain the core of a functional programming setting for binary field arithmetic with built-in polynomial complexity

An Elementary Affine λ-Calculus with Multithreading and Side Effects

International audienceLinear logic provides a framework to control the complexity of higher-order functional programs. We present an extension of this framework to programs with multithreading and side effects focusing on the case of elementary time. Our main contributions are as follows. First, we introduce a modal call-by-value λ-calculus with multithreading and side effects. Second, we provide a combinatorial proof of termination in elementary time for the language. Third, we introduce an elementary affine type system that guarantees the standard subject reduction and progress properties. Finally, we illustrate the programming of iterative functions with side effects in the presented formalism

System analysis and robustness

Software is increasingly embedded in a variety of physical contexts. This imposes new requirements on tools that support the design and analysis of systems. For instance, modeling embedded and cyberphysical systems needs to blend discrete mathematics, which is suitable for modeling digital components, with continuous mathematics, used for modeling physical components. This blending of continuous and discrete creates challenges that are absent when the discrete or the continuous setting are considered in isolation. We consider robustness, that is, the ability of an analysis of a model to cope with small amounts of imprecision in the model. Formally, we identify analyses with monotonic maps between complete lattices (a mathematical framework used for abstract interpretation and static analysis) and define robustness for monotonic maps between complete lattices of closed subsets of a metric space.</p

Causal categories: relativistically interacting processes

A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This paper is concerned with the encoding of a fixed causal structure within a symmetric monoidal category: causal dependencies will correspond to topological connectedness in the graphical language. We show that correlations, either classical or quantum, force terminality of the tensor unit. We also show that well-definedness of the concept of a global state forces the monoidal product to be only partially defined, which in turn results in a relativistic covariance theorem. Except for these assumptions, at no stage do we assume anything more than purely compositional symmetric-monoidal categorical structure. We cast these two structural results in terms of a mathematical entity, which we call a `causal category'. We provide methods of constructing causal categories, and we study the consequences of these methods for the general framework of categorical quantum mechanics.Comment: 43 pages, lots of figure