There are multiple phenomena referred to by this name in domains such as sampling, combinatorics, machine learning and data mining. The common theme of these problems is that when the dimensionality increases, the volume of the space increases so fast that the available data becomes sparse. This sparsity is problematic for any method that requires statistical significance.

Sending invoices and business documents in Japan is deeply ingrained with various cultural nuances, and is ruled by a series of business manners and traditions. We had no idea if businesses would actually use an online system to send their invoices, since you cannot get less ‘traditional’ than using an online system to send invoices.

The underlying issue is that statically safe finalization is inherently not a monad and requires a parametrized monad, but I want to be able to decompose the problem so that users can decide for themselves whether they want a monad or static guarantees. However, the decomposition still eludes me.

beautiful folding

There are multiple reasons why pieces of GCC cannot be reused as libraries, including rampant use of global variables, weakly enforced invariants, poorly designed data structures, sprawling code base, and the use of macros that prevent the codebase from being compiled to support more than one front-end/target pair at a time

> Wait, are you saying that if you apply ContT to any monad that has the
> "left recursion on >>= takes quadratic time" problem, and represent
> all primitive operations via lift (never using >>= within "lift"),
> that you will get a new monad that doesn't have that problem?

Since all the finalization primitives in Control.Exception are implemented on top of exception handling primitives like catch and mask, I initially believed that finalization would follow automatically from exception handling capabilities in pipes.

Unfortunately, there is a fundamental operational difference between IO and Pipe, which makes exception handling alone insufficient to guarantee finalization of resources.

Selenium-WebDriver makes direct calls to the browser using each browser’s native support for automation. How these direct calls are made, and the features they support depends on the browser you are using. Information on each ‘browser driver’ is provided later in this chapter.

Reading mathematics too quickly results in frustration. A half hour of concentration in a novel might net the average reader 20-60 pages with full comprehension, depending on the novel and the experience of the reader. The same half hour in a math article buys you 0-10 lines depending on the article and how experienced you are at reading mathematics. There is no substitute for work and time. You can speed up your math reading skill by practicing, but be careful. Like any skill, trying too much too fast can set you back and kill your motivation.

Axiomatic proofs are harder to construct than sequent proofs
because we cannot use conditional proofs or reductio ad absurdum.

I will discuss rewrite rules later in a post about optimizing pipes and frames, but one of my big motivations for strictly enforcing all the class laws as strong as possible is that it then permits rewrites so powerful that all the pipe code completely disappears, leaving behind the hand-written loop code (think: stream fusion on steroids).

Intuitionistic systems have inspired a variety of interpretations, including Beth's tableaus, Rasiowa and Sikorski's topological models, formulas-as-types, Kleene's recursive realizabilities, and the Kleene and Aczel slashes. Kripke's [1965] possible-world semantics, with respect to which intuitionistic predicate logic is complete and consistent, most resembles classical model theory.

A specific real number such as 2 is also a set, but not in such a familiar way—it contains a very complex construction of sets, a kind of machine inside of it that causes it to behave according to the laws for real numbers.

One of the things that we always say is that we can recreate all of mathematics using set theory as a basis. What does that mean? Basically, it means that given some other branch of math, which works with some class of objects O using some set of axioms A, we can define a set-based construction of the objects of S(O), and them prove the axioms A about S(O) using the axioms of ZFC.

A specific real number such as 2 is also a set, but not in such a familiar way—it contains a very complex construction of sets, a kind of machine inside of it that causes it to behave according to the laws for real numbers.

You are correct that substitution in Metamath is the simple replacement of a variable with a (well-formed) expression and thus isn't hygienic, in Scheme's terminology. In logic, the term used for “hygienic substitution” is “proper substitution,” which Metamath does not implement directly.

Metamath can be used with species of logic as different as Hilbert-style logics or sequents-based logics or even with lambda calculus. In contrast, it is largely incompatible with logical systems which uses other things than formulas and inference rules. The original natural deduction system (due to Gerhard Gentzen), which uses an extra stack, is an example of a system that cannot be implemented with Metamath. In the case of natural deduction however it is possible to append the stack to the formulas (transforming the natural deduction formulas into a sort of sequent) so that Metamath's requirements are met.

Most variants of Hilbert systems take a characteristic tack in the way they balance a trade-off between logical axioms and rules of inference.[1] Hilbert systems can be characterised by the choice of a large number of schemes of logical axioms and a small set of rules of inference. Systems of natural deduction take the opposite tack, including many deduction rules but very few or no axiom schemes.

In mathematical logic, the deduction theorem is a metatheorem of first-order logic.[1] It is a formalization of the common proof technique in which an implication A → B is proved by assuming A and then proving B from this assumption. The deduction theorem explains why proofs of conditional sentences in mathematics are logically correct. Though it has seemed "obvious" to mathematicians literally for centuries that proving B from A conjoined with a set of theorems is sufficient to proving the implication A → B based on those theorems alone, it was left to Herbrand and Tarski to show (independently) this was logically correct in the general case—another instance, perhaps, of modern logic "cleaning up" mathematical practice.

Computer scientist Stuart Staniford said that as the robot population surpasses humans and takes most of our jobs,"the least disruptive approach to managing this is for the underclass to disappear into technologically mediated secondary universes."

Persistent and Ephemeral Truth. It has long been argued by philosophers
that truth is not universal, but depends on the world in which we consider a
proposition. This could depend on time (It is raining now, but it did not rain
yesterday."), place (it is hot in the Sahara but cold at the North Pole.") or
simply the state of a circumscribed system (The white king is on square e1.").
Studying logical inference in these situations is the domain of temporal, spatial,
or linear logic, respectively.

You can certainly send even a binary payload with an HTTP request, or in an HTTP response. Just write the bytes of the protocol buffer directly into the request/response, and make sure to set the content type to "application/octet-stream". The client, and server, should be able to take care of the rest easily. I don't think you need anything more special than that on either end.

These are RPC implementations that work with Protocol Buffers. Some of these actually work with Protocol Buffers service definitions (defined using the service keyword in .proto files) while others just use Protocol Buffers message objects.

Typically, a rule of inference preserves truth, a semantic property. In many-valued logic, it preserves a general designation. But a rule of inference's action is purely syntactic, and does not need to preserve any semantic property: any function from sets of formulae to formulae counts as a rule of inference.

The most important judgments in logic are of the form "A is true". The letter A stands for any expression representing a proposition; the truth judgments thus require a more primitive judgment: "A is a proposition". Many other judgments have been studied; for example, "A is false" (see classical logic), "A is true at time t" (see temporal logic), "A is necessarily true" or "A is possibly true" (see modal logic), "the program M has type τ" (see programming languages and type theory), "A is achievable from the available resources" (see linear logic), and many others. To start with, we shall concern ourselves with the simplest two judgments "A is a proposition" and "A is true", abbreviated as "A prop" and "A true" respectively.