Second-order polynomials generalize classical (=first-order) ones in allowing for additional variables that range over functions rather than values. We are motivated by their applications in higher-order computational complexity theory, extending for instance discrete classes (like P/FP or PSPACE/FPSPACE) to operators in Analysis [http://doi.org/10.1137/S0097539794263452], [http://doi.org/10.1145/2189778.2189780]. The degree subclassifies ordinary polynomial growth into linear, quadratic, cubic, etc. To similarly classify second-order polynomials, we (well-)define their degree by structural induction as an "arctic" first-order polynomial: a term/expression over integer variable D and operations + and ⋅ and binary max(). This generalized degree turns out to transform nicely under (now two kinds of) polynomial composition. As examples, we collect and determine the degrees of previous and new asymptotic analyses of algorithms and operators receiving function/oracle arguments. Then we motivate and introduce third-order polynomials and their degrees as arctic second-order polynomials, along with their transformations under three kinds of composition. Proceeding to fourth order and beyond yields a hierarchy, with characterization in Simply Typed Lambda Calculus.

When encoding real numbers as (necessarily infinite) bit-strings, the naïve binary/decimal expansion is well-known [ doi:10.1112/plms/s2-43.6.544 ] computably “ un reasonable”, rendering, for example, tripling qualitatively discontinuous on Cantor’s sequence space. Encoding reals as sequences of (finite integer numerators and denominators, in binary, of) rational approximations does make common operations qualitatively computable, yet admits no bounds on their computational complexity/quantitative continuity. Dyadic approximations, on the other hand, are known polynomially, and signed binary expansions even linearly, “reasonable” in a rigorous sense recalled in the introduction of this work. But how to distinguish between un/suitable encodings of spaces common in Calculus beyond the reals, such as Banach or Sobolev? With respect to qualitative computability/continuity on topological spaces, the technical condition of admissibility had been identified [ doi:10.1016/0304-3975(85)90208-7 ] for an encoding over Cantor space (historically called a representation ) to be “reasonable” [ doi:10.1007/978-3-030-59234-9_9 ] . Roughly speaking, admissibility requires the representation to be (i) continuous, and to be (ii) maximal with respect to continuous reduction. Admissible representations exist for a large class of spaces. And for (precisely) these does the Kreitz–Weihrauch—sometimes aka Main —Theorem of Computable Analysis hold, which characterizes continuity of functions by continuity of mappings translating codes, so-called realizers . We refine qualitative computability/continuity on topological spaces to quantitative continuity/complexity on metric spaces by proposing a notion, and investigating the properties, of polynomially/linearly admissible representations. Roughly speaking, these are (i) close to “optimally” continuous, namely linearly/polynomially relative to the space’s entropy, and they are (ii) maximal with respect to relative linear/polynomial quantitatively continuous reductions defined in the main text. Quantitatively admissible representations are closed under composition over generalized ground spaces beyond Cantor’s. Such representations exhibit a quantitative strengthening of the qualitative Main Theorem , namely now characterizing quantitative continuity of functions by quantitative continuity of realizers. A large class of compact metric spaces is shown to admit polynomially admissible representations over compact ultra metric spaces, and some even a generalization of the linearly admissible signed binary encoding. Quantitative admissibility thus provides the desired criterion for complexity-theoretically “reasonable” encodings.

We propose a simple imperative programming language, ERC, that features arbitrary real numbers as primitive data type, exactly. Equipped with a denotational semantics, ERC provides a formal programming language-theoretic foundation to the algorithmic processing of real numbers. In order to capture multi-valuedness, which is well-known to be essential to real number computation, we use a Plotkin powerdomain and make our programming language semantics computable and complete: all and only real functions computable in computable analysis can be realized in ERC. The base programming language supports real arithmetic as well as implicit limits; expansions support additional primitive operations (such as a user-defined exponential function). By restricting integers to Presburger arithmetic and real coercion to the `precision' embedding $\mathbb{Z}\ni p\mapsto 2^p\in\mathbb{R}$, we arrive at a first-order theory which we prove to be decidable and model-complete. Based on said logic as specification language for preconditions and postconditions, we extend Hoare logic to a sound (w.r.t. the denotational semantics) and expressive system for deriving correct total correctness specifications. Various examples demonstrate the practicality and convenience of our language and the extended Hoare logic.

According to Haar's Theorem, every compact group $G$ admits a unique (regular, right and) left-invariant Borel probability measure $μ_G$. Let the Haar integral (of $G$) denote the functional $\int_G:\mathcal{C}(G)\ni f\mapsto \int f\,dμ_G$ integrating any continuous function $f:G\to\mathbb{R}$ with respect to $μ_G$. This generalizes, and recovers for the additive group $G=[0;1)\mod 1$, the usual Riemann integral: computable (cmp. Weihrauch 2000, Theorem 6.4.1), and of computational cost characterizing complexity class #P$_1$ (cmp. Ko 1991, Theorem 5.32). We establish that in fact every computably compact computable metric group renders the Haar integral computable: once asserting computability using an elegant synthetic argument, exploiting uniqueness in a computably compact space of probability measures; and once presenting and analyzing an explicit, imperative algorithm based on 'maximum packings' with rigorous error bounds and guaranteed convergence. Regarding computational complexity, for the groups $\mathcal{SO}(3)$ and $\mathcal{SU}(2)$ we reduce the Haar integral to and from Euclidean/Riemann integration. In particular both also characterize #P$_1$. Implementation and empirical evaluation using the iRRAM C++ library for exact real computation confirms the (thus necessary) exponential runtime.

We apply average-case complexity theory to physical problems modeled by continuous-time dynamical systems. The computational complexity when simulating such systems for a bounded time-frame mainly stems from trajectories coming close to complex singularities of the system. We show that if for most initial values the trajectories do not come close to singularities the simulation can be done in polynomial time on average. For Hamiltonian systems we relate this to the volume of "almost singularities" in phase space and give some general criteria to show that a Hamiltonian system can be simulated efficiently on average. As an application we show that the planar circular-restricted three-body problem is average-case polynomial-time computable.

The last years have seen an increasing interest in classifying (existence claims in) classical mathematical theorems according to their strength. We pursue this goal from the refined perspective of computational complexity. Specifically, we establish that rigorously solving the Dirichlet Problem for Poisson's Equation is in a precise sense ‘complete’ for the complexity class ${\#\mathcal{P}}$ and thus as hard or easy as parametric Riemann integration (Friedman 1984; Ko 1991.Complexity Theory of Real Functions).

We promote the theory of computational complexity on metric spaces: as natural common generalization of (i) the classical discrete setting of integers, binary strings, graphs etc. as well as of (ii) the bit-complexity theory on real numbers and functions according to Friedman, Ko (1982ff), Cook, Braverman et al.; as (iii) resource-bounded refinement of the theories of computability on, and representations of, continuous universes by Pour-El&Richards (1989) and Weihrauch (1993ff); and as (iv) computational perspective on quantitative concepts from classical Analysis: Our main results relate (i.e. upper and lower bound) Kolmogorov's entropy of a compact metric space X polynomially to the uniform relativized complexity of approximating various families of continuous functions on X. The upper bounds are attained by carefully crafted oracles and bit-cost analyses of algorithms perusing them. They all employ the same representation (i.e. encoding, as infinite binary sequences, of the elements) of such spaces, which thus may be of own interest. The lower bounds adapt adversary arguments from unit-cost Information-Based Complexity to the bit model. They extend to, and indicate perhaps surprising limitations even of, encodings via binary string functions (rather than sequences) as introduced by Kawamura&Cook (SToC'2010, §3.4). These insights offer some guidance towards suitable notions of complexity for higher types.

We connect both discrete and algebraic complexity theory with the satisfiability problem in certain non-Boolean lattices. Specifically, quantum logic was introduced in 1936 by Garrett Birkhoff and John von Neumann as a framework for capturing the logical peculiarities of quantum observables: in the 1D case it coincides with Boolean propositional logic but, starting with dimension two, violates the distributive law. We introduce the weak and strong satisfiability problem for quantum logic propositional formulae. It turns out that in dimension two, both are also NP --complete. For higher-dimensional spaces ℝ d and ℂ d with d ≥ 3 fixed, on the other hand, we show both problems to be complete for the nondeterministic Blum-Shub-Smale (BSS) model of real computation. This provides a unified view on both Turing and real BSS complexity theory, and extends the (still relatively scarce) list of problems established NP ℝ --complete with one, perhaps, closest in spirit to the classical Cook-Levin Theorem. More precisely, strong satisfiability of ∧ ∨ ∧ --terms is complete, while that of ∧ ∨--terms (i.e., those in conjunctive form) can be decided in polynomial time in dimensions d ≥ 2. The decidability of the infinite-dimensional case being still open, we proceed to investigate the case of indefinite finite dimensions. Here, weak satisfiability still belongs to NP R and strong satisfiability is still hard; the latter, in fact, turns out as polynomial-time equivalent to the feasibility of noncommutative integer polynomial equations over matrix rings.

The synthesis of (discrete) Complexity Theory with Recursive Analysis provides a quantitative algorithmic foundation to calculations over real numbers, sequences, and functions by approximation up to prescribable absolute error 1/2n (roughly corresponding to n binary digits after the radix point). In this sense Friedman and Ko have shown the seemingly simple operators of maximization and integration 'complete' for the standard complexity classes NP and #P — even when restricted to smooth (=C∞) arguments. Analytic polynomial-time computable functions on the other hand are known to get mapped to polynomial-time computable functions: non-uniformly, that is, disregarding dependences other than on the output precision n. The present work investigates the uniform parameterized complexity of natural operators Λ on subclasses of smooth functions: evaluation, pointwise addition and multiplication, (iterated) differentiation, integration, and maximization. We identify natural integer parameters k=k(f) which, when given as enrichment to approximations to the function argument f, permit to computably produce approximations to Λ(f); and we explore the asymptotic worst-case running time sufficient and necessary for such computations in terms of the output precision n and said k. It turns out that Maurice Gevrey's 1918 classical hierarchy climbing from analytic to (just below) smooth functions provides for a quantitative gauge of the uniform computational complexity of maximization and integration that, non-uniformly, exhibits the phase transition from tractable (i.e. polynomial-time) to intractable (in the sense of NP-'hardness'). Our proof methods involve Hard Analysis, Approximation Theory, and an adaptation of Information-Based Complexity to the bit model.

Quantum logic generalizes, and in dimension one coincides with, Boolean propositional logic. We introduce the weak and strong satisfiability problem for quantum logic formulas, and show both NP-complete in dimension two as well. For higher-dimensional spaces Rdand Cdwith d≥3 fixed, on the other hand, we show the problem to be complete for the nondeterministic Blum-Shub-Smale model of real computation. This provides a unified view on both Turing and real BSS complexity theory, and adds (a perhaps more natural and combinatorially flavoured) one to the still sparse list of NPR-complete problems, mostly pertaining to real algebraic geometry. Our proofs rely on (a careful examination of) works by John von Neumannas well as contributions by Hagge et. al (2005,2007,2009). We finally investigate the problem over Indefinite finite dimensions and relate it to NON-commutative semi algebraic geometry.

We study and compare in two degree-theoretic ways (iterated Halting oracles analogous to Kleene's arithmetical hierarchy and the Borel hierarchy of descriptive set theory) the capabilities and limitations of three models of analytic computation: BSS machines (aka real-RAM) and strongly/weakly analytic machines as introduced by Hotz et. al. (1995).