Nosimpler

Braverman & Yampolsky have shown that there exist noncomputable Julia sets, i.e., there exist $c \in \mathbb{C}$ such that the Julia set of $f(z) = c + z^2$ is not computable. "A set is computable, if, roughly speaking, its image can be generated by a computer with an arbitrary precision."

Braverman, Mark, and Michael Yampolsky. "Non-computable Julia sets." Journal of the American Mathematical Society (2006): 551-578. (PDF download.)

My questions are:

Q. Is an explicit such $c$ known? A computable $c$?

It seems likely these questions are answered, perhaps in the cited paper. If anyone is familiar enough with this line of work to answer, I'd appreciate it.

Braverman, Mark, and Michael Yampolsky. "Computability of Julia sets." arXiv link. 2007.

They prove there exist computable $c \in \mathbb{C}$ such that the Julia set of $c + z^2$ is not algorithmically computable, and provide an algorithm for computing such a $c$. Under the assumption of a complex dynamics conjecture (due to Buff & Chéritat), they obtain a polynomial-time algorithm for computing such a $c$, i.e., $n$ bits of $c$ can be computed in time polynomial in $n$.

No explicit $c$ is known, as far as I can tell. (Their algorithms would not be easy to implement.)

A study of the chestnut-crowned babbler bird from Australia revealed a method of communicating that has never before been observed in animals.

The bird combines sounds in different combinations to convey meaning.

The findings could help in the understanding of how language evolved in humans, researchers report in the online journal PLOS Biology.

Co-researcher Dr Andy Russell from the University of Exeter said: “It is the first evidence outside of a human that an animal can use the same meaningless sounds in different arrangements to generate new meaning.

“It’s a very basic form of word generation – I’d be amazed if other animals can’t do this too.”

There is more here.  You will find further coverage here.

Author(s): Jean-Baptiste Caussin and Denis Bartolo

Individual birds flying in a flock do not fly in straight lines, but weave in and out of each other. Topological invariant braiding statistics shows that this weaving has a coherent rotation: the birds create a braid.

[Phys. Rev. Lett. 114, 258101] Published Tue Jun 23, 2015

Acetaminophen, an effective and popular over-the-counter pain reliever (e.g., the active ingredient in Tylenol), has recently been shown to blunt individuals’ reactivity to a range of negative stimuli in addition to physical pain. Because accumulating research has shown that individuals’ reactivity to both negative and positive stimuli can be influenced by a single factor (an idea known as differential susceptibility), we conducted two experiments testing whether acetaminophen blunted individuals’ evaluations of and emotional reactions to both negative and positive images from the International Affective Picture System. Participants who took acetaminophen evaluated unpleasant stimuli less negatively and pleasant stimuli less positively, compared with participants who took a placebo. Participants in the acetaminophen condition also rated both negative and positive stimuli as less emotionally arousing than did participants in the placebo condition (Studies 1 and 2), whereas nonevaluative ratings (extent of color saturation in each image; Study 2) were not affected by drug condition. These findings suggest that acetaminophen has a general blunting effect on individuals’ evaluative and emotional processing, irrespective of negative or positive valence.

The New England Conference of United Methodist Churches, a group of 600 churches, has issued a resolution calling for an end to the war on drugs. The resolution draws on ethical principles and also a remarkably astute reading of economics and social science:

Whereas: The public policy of prohibition of certain narcotics and psychoactive substances, sometimes called the “War on Drugs,” has failed to achieve the goal of eliminating, or even reducing, substance abuse and;

Whereas: There have been a large number of unintentional negative consequences as a result of this failed public policy and;

Whereas: One of those consequences is a huge and violent criminal enterprise that has sprung up surrounding the Underground Market dealing in these prohibited substances and;

Whereas: Many lives have been lost as a result of the violence surrounding this criminal enterprise, including innocent citizens and police officers and;

Whereas: Many more lives have been lost to overdose because there is no regulation of potency, purity or adulteration in the production of illicit drugs and;

Whereas: Our court system has been severely degraded due to the overload caused by prohibition cases and;

Whereas: Our prisons are overcrowded with persons, many of whom are non-violent, convicted of violation of the prohibition laws and;

Whereas: Many of our citizens now suffer from serious diseases, contracted through the use of unsanitary needles, which now threaten our population at large and;

Whereas: To people of color, the “War on Drugs” has arguably been the single most devastating, dysfunctional social policy since slavery and;

Whereas: Huge sums of our national treasury are wasted on this failed public policy and;

Whereas: Other countries, such as Portugal and Switzerland, have dramatically reduced the incidence of death, disease, crime, and addiction by utilizing means other than prohibition to address the problem of substance abuse and;

Whereas: The primary mission of our criminal justice system is to prevent violence to our citizens and their property, and to ensure their safety, therefore;

Be it Resolved: That the New England Annual Conference supports seeking means other than prohibition to address the problem of substance abuse; and is further resolved to support the mission of the international educational organization Law Enforcement Against Prohibition (LEAP) to reduce the multitude of unintended harmful consequences resulting from fighting the war on drugs and to lessen the incidence of death, disease, crime, and addiction by ending drug prohibition.

Significance

Younger, better performing adults typically show greater brain signal variability than older, poorer performers, but the mechanisms underlying this observation remain elusive. We attempt to restore deficient functional-MRI–based blood oxygen level-dependent (BOLD) signal variability (SDBOLD) levels in older adults by boosting dopamine via d-amphetamine (AMPH). Notably, older adults met or exceeded young adult SDBOLD levels under AMPH. AMPH-driven changes in SDSDBOLD also predicted AMPH-driven changes in reaction time speed and variability on a working memory task, but depended greatly on age and drug administration order. These findings (i) suggest that dopamine may account for adult age differences in brain signal variability and (ii) highlight the importance of considering practice effects and state dependencies when evaluating the neurochemical basis of age- and cognition-related brain dynamics. 

Abstract

Better-performing younger adults typically express greater brain signal variability relative to older, poorer performers. Mechanisms for age and performance-graded differences in brain dynamics have, however, not yet been uncovered. Given the age-related decline of the dopamine (DA) system in normal cognitive aging, DA neuromodulation is one plausible mechanism. Hence, agents that boost systemic DA [such as d-amphetamine (AMPH)] may help to restore deficient signal variability levels. Furthermore, despite the standard practice of counterbalancing drug session order (AMPH first vs. placebo first), it remains understudied how AMPH may interact with practice effects, possibly influencing whether DA up-regulation is functional. We examined the effects of AMPH on functional-MRI–based blood oxygen level-dependent (BOLD) signal variability (SDBOLD) in younger and older adults during a working memory task (letter n-back). Older adults expressed lower brain signal variability at placebo, but met or exceeded young adult SDBOLD levels in the presence of AMPH. Drug session order greatly moderated change–change relations between AMPH-driven SDBOLD and reaction time means (RTmean) and SDs (RTSD). Older adults who received AMPH in the first session tended to improve in RTmean and RTSD when SDBOLD was boosted on AMPH, whereas younger and older adults who received AMPH in the second session showed either a performance improvement when SDBOLD decreased (for RTmean) or no effect at all (for RTSD). The present findings support the hypothesis that age differences in brain signal variability reflect aging-induced changes in dopaminergic neuromodulation. The observed interactions among AMPH, age, and session order highlight the state- and practice-dependent neurochemical basis of human brain dynamics.

Figure. Increased BOLD variability and improved cognitive performance under AMPH. Multivariate partial least-squares model of relation between SDBOLD, Age Group, AMPH, and Task Condition. Higher brain scores reflect higher BOLD signal variability. Error bars represent bootstrapped 95% confidence intervals (1,000× with replacement). Brain images are plotted in neurological orientation (left is Left). AMPH, amphetamine; BSR, bootstrap ratio.

Computers have never been good at answering the type of verbal reasoning questions found in IQ tests. Now a deep learning machine unveiled in China is changing that.

"In the USSR work is a duty and a matter of honor for every able-bodied citizen, in accordance with the principle: “He who does not work, neither shall he eat.”

 1936 Soviet Constitution

For the past two years, Luanda—not Tokyo, Moscow, or Hong Kong—has been named, (...) as the world’s most expensive city for expatriates.(...) The country now produces 1.8 million barrels of oil a day(...)The boom has transformed a failed state into one of the world’s fastest-growing economies.(...) Almost nothing is made in Angola, so nearly every car, computer, crate of oranges, tin of caviar, jar of peanut butter, pair of bluejeans, and bottle of wine arrives by boat. Every day, a trail of container ships backs up from the port through the Bay of Luanda and out into the sea.(...) Grotesque inequality long ago became a principal characteristic of the world’s biggest and most crowded cities. But there is no place quite like Luanda, where a bottle of Coke can sell for ten dollars(...). Per-capita income in Angola has nearly tripled in the past dozen years, and the country’s assets grew from three billion dollars to sixty-two billion dollars. Nonetheless, by nearly every accepted measure, Angola remains one of the world’s least-developed nations. Half of Angolans live on less than two dollars a day, infant mortality rates are among the highest in the world, and the average life expectancy—fifty-two—is among the lowest. (...) Nearly half the population is undernourished, rural sanitation facilities are rare, malaria accounts for more than a quarter of all childhood deaths(...). One businessman famously distributed Rolexes to guests as party favors at a wedding. Each member of parliament recently received a new hundred-thousand-dollar Lexus. Isabel dos Santos, the President’s forty-two-year-old daughter, is typically described as the richest woman in Africa; Forbes puts her net worth at more than three billion dollars. (...) In 2011, as president of the Red Cross, dos Santos paid Mariah Carey a million dollars to perform for two hours at the organization’s annual gala. (...)Hotels, luxury apartment buildings, shopping arcades, and modern office complexes compete for space in the city center with shantytowns made from corrugated tin and heavy cardboard and with tens of thousands of people who live on mounds of dirt, in the scrapped remains of rusted and abandoned vehicles, or out in the open, next to fetid, unused water tanks.  Extreme City - The New Yorker  

Author(s): Yohai Bar-Sinai and Eran Bouchbinder

According to the equipartition theorem, in a classical system at equilibrium, thermal energy is equally distributed among the degrees of freedom appearing as quadratic forms in the Hamiltonian. Asking the question what is the spatial distribution of the thermal energy, the authors find a general upper bound and show that the details depend on dimensionality, interactions, and disorder.

[Phys. Rev. E 91, 060103(R)] Published Tue Jun 09, 2015

As everyone knows, nobody reads their user agreements when they sign up for apps or services. Even if they did, it wouldn’t matter, because most of them stipulate that they can change at any moment. That moment has come.

You might not be concerned, but I’d like to point out that there’s a reason you’re not. Namely, you haven’t actually seen what this enormous loss of privacy translates into yet.

You see, there’s also a built in lag where we’ve given up our data, and are happily using the corresponding services, but we haven’t yet seen evidence that our data was actually worth something. The lag represents the time it takes for the market in personal data to mature. It also represents the patience that Silicon Valley venture capitalists have or do not have between the time of user acquisition and profit. The less patience they have, the sooner they want to exploit the user data.

The latest news (hat tip Gary Marcus) gives us reason to think that V.C. patience is running dry, and the corresponding market in personal data is maturing. Turns out that EBay and PayPal recently changed their user agreements so that, if you’re a user of either of those services, you will receive marketing calls using any phone number you’ve provided them or that they have “have otherwise obtained.” There is no possibility to opt out, except perhaps to abandon the services. Oh, and they might also call you for surveys or debt collections. Oh, and they claim their intention is to “benefit our relationship.”

Presumably this means they might have bought your phone number from a data warehouse giant like Acxiom, if you didn’t feel like sharing it. Presumably this also means that they will use your shopping history to target the phone calls to be maximally “tailored” for you.

I’m mentally tacking this new fact on the same board as I already have the Verizon/AOL merger, which is all about AOL targeting people with ads based on Verizon’s GPS data, and the recent broohaha over RadioShack’s attempt to sell its user data at auction in order to pay off creditors. That didn’t go through, but it’s still a sign that the personal data market is ripening, and in particular that such datasets are becoming assets as important as land or warehouses.

Given how much venture capitalists like to brag about their return, I think we have reason to worry about the coming wave of “innovative” uses of our personal data. Telemarketing is the tip of the iceberg.

Robots that can adapt like animals

Nature 521, 7553 (2015). doi:10.1038/nature14422

Authors: Antoine Cully, Jeff Clune, Danesh Tarapore & Jean-Baptiste Mouret

Robots have transformed many industries, most notably manufacturing, and have the power to deliver tremendous benefits to society, such as in search and rescue, disaster response, health care and transportation. They are also invaluable tools for scientific exploration in environments inaccessible to humans, from distant planets to deep oceans. A major obstacle to their widespread adoption in more complex environments outside factories is their fragility. Whereas animals can quickly adapt to injuries, current robots cannot ‘think outside the box’ to find a compensatory behaviour when they are damaged: they are limited to their pre-specified self-sensing abilities, can diagnose only anticipated failure modes, and require a pre-programmed contingency plan for every type of potential damage, an impracticality for complex robots. A promising approach to reducing robot fragility involves having robots learn appropriate behaviours in response to damage, but current techniques are slow even with small, constrained search spaces. Here we introduce an intelligent trial-and-error algorithm that allows robots to adapt to damage in less than two minutes in large search spaces without requiring self-diagnosis or pre-specified contingency plans. Before the robot is deployed, it uses a novel technique to create a detailed map of the space of high-performing behaviours. This map represents the robot’s prior knowledge about what behaviours it can perform and their value. When the robot is damaged, it uses this prior knowledge to guide a trial-and-error learning algorithm that conducts intelligent experiments to rapidly discover a behaviour that compensates for the damage. Experiments reveal successful adaptations for a legged robot injured in five different ways, including damaged, broken, and missing legs, and for a robotic arm with joints broken in 14 different ways. This new algorithm will enable more robust, effective, autonomous robots, and may shed light on the principles that animals use to adapt to injury.

guest post by Mike Stay

Greg Meredith and I have a short paper that’s been accepted for Higher-Dimensional Rewriting and Applications (HDRA) 2015 on modeling the asynchronous polyadic pi calculus with 2-categories. We avoid domain theory entirely and model the operational semantics directly; full abstraction is almost trivial. As a nice side-effect, we get a new tool for reasoning about consumption of resources during a computation.

It’s a small piece of a much larger project, which I’d like to describe here in a series of posts. This post will talk about lambda calculus for a few reasons. First, lambda calculus is simpler, but complex enough to illustrate one of our fundamental insights. Lambda calculus is to serial computation what pi calculus is to concurrent computation; lambda calculus talks about a single machine doing a computation, while pi calculus talks about a network of machines communicating over a network with potentially random delays. There is at most one possible outcome for a computation in the lambda calculus, while there are many possible outcomes in a computation in the pi calculus. Both the lazy lambda calculus and the pi calculus, however, have as an integral part of their semantics the notion of waiting for a sub-computation to complete before moving onto another one. Second, the denotational semantics of lambda calculus in Set is well understood, as is its generalization to cartesian closed categories; this semantics is far simpler than the denotational semantics of pi calculus and serves as a good introduction. The operational semantics of lambda calculus is also simpler than that of pi calculus and there is previous work on modeling it using higher categories.

History

Alonzo Church invented the lambda calculus as part of his attack on Hilbert’s third problem, also known as the Entscheidungsproblem, which asked for an algorithm to solve any mathematical problem. Church published his proof that no such algorithm exists in 1936. Turing invented his eponymous machines, also to solve the Entscheidungsproblem, and published his independent proof a few months after Church. When he discovered that Church had beaten him to it, Turing proved in 1937 that the two approaches were equivalent in power. Since Turing machines were much more “mechanical” than the lambda calculus, the development of computing machines relied far more on Turing’s approach, and it was only decades later that people started writing compilers for more friendly programming languages. I’ve heard it quipped that “the history of programming languages is the piecemeal rediscovery of the lambda calculus by computer scientists.”

The lambda calculus consists of a set of “terms” together with some relations on the terms that tell how to “run the program”. Terms are built up out of “term constructors”; in the lambda calculus there are three: one for variables, one for defining functions (Church denoted this operation with the Greek letter lambda, hence the name of the calculus), and one for applying those functions to inputs. I’ll talk about these constructors and the relations more below.

Church introduced the notion of “types” to avoid programs that never stop. Modern programming languages also use types to avoid programmer mistakes and encode properties about the program, like proving that secret data is inaccessible outside certain parts of the program. The “simply-typed” lambda calculus starts with a set of base types and takes the closure under the binary operation →\to to get a set of types. Each term is assigned a type; from this one can deduce the types of the variables used in the term. An assignment of types to variables is called a typing context.

The search for a semantics for variants of the lambda calculus has typically been concerned with finding sets or “domains” such that the interpretation of each lambda term is a function between domains. Scott worked out a domain DD such that the continuous functions from DD to itself are precisely the computable ones. Lambek and Scott generalized the category where we look for semantics from Set to arbitrary cartesian closed categories (CCCs).

Lambek and Scott constructed a CCC out of lambda terms; we call this category the syntactical category. Then a structure-preserving functor from the syntactical category to Set or some other CCC would provide the semantics. The syntactical category has types as objects and equivalence classes of certain terms as morphisms. A morphism in the syntactical category goes from a typing context to the type of the term.

John Baez has a set of lecture notes from Fall 2006 through Spring 2007 describing Lambek and Scott’s approach to the category theory of lambda calculus and generalizing it from cartesian closed categories to symmetric monoidal closed categories so it can apply to quantum computation as well: rather than taking a functor from the syntactical category into Set, we can take a functor into Hilb instead. He and I also have a “Rosetta stone” paper summarizing the ideas and connecting them with the corresponding generalization of the Curry-Howard isomorphism.

The Curry-Howard isomorphism says that types are to propositions as programs are to proofs. In practice, types are used in two different ways: one as propositions about data and the other as propositions about code. Programming languages like C, Java, Haskell, and even dynamically typed languages like JavaScript and Python use types to talk about propositions that data satisfies: is it a date or a name? In these languages, equivalence classes of programs constitute constructive proofs. Concurrent calculi are far more concerned about propositions that the code satisfies: can it reach a deadlocked state? In these languages, it is the rewrite rules taking one term to another that behave like proofs. Melliès and Zeilberger’s excellent paper “Functors are Type Refinement Systems” relates these two approaches to typing to each other.

Note that Lambek and Scott’s approach does not have the sets of terms or variables as objects! The algebra that defines the set of terms plays only a minor role in the category; there’s no morphism in the CCC, for instance, that takes a term tt and a variable xx to produce the term λx.t\lambda x.t. This failure to capture the structure of the term in the morphism wasn’t a big deal for lambda calculus because of “confluence” (see below), but it turns out to matter a lot more in calculi like Milner’s pi calculus that describe communicating over a network, where messages can be delayed and arrival times matter for the end result (consider, for instance, two people trying to buy online the last ticket to a concert).

The last few decades have seen domains becoming more and more complicated in order to try to “unerase” the information about the structure of terms that gets lost in the domain theory approach and recover the operational semantics. Fiore, Moggi, and Sangiorgi, Stark and Cattani, Stark, and Winskel all present domain models of the pi calculus that recursively involve the power set in order to talk about all the possible futures for a term. Industry has never cared much about denotational semantics: the Java Virtual Machine is an operational semantics for the Java language.

What we did

Greg Meredith and I set out to model the operational semantics of the pi calculus directly in a higher category rather than using domain theory. An obvious first question is, “What about types?” I was particularly worried about how to relate this approach to the kind of thing Scott and Lambek did. Though it didn’t make it into the HDRA paper and the details won’t make it into this post, we found that we’re able to use the “type-refinement-as-a-functor” idea of Melliés and Zeilberger to show how the algebraic term-constructor functions relate to the morphisms in the syntactical category.

We’re hoping that this categorical approach to modeling process calculi will help with reasoning about practical situations where we want to compose calculi; for instance, we’d like to put a hundred pi calculus engines around the edges of a chip and some ambient calculus engines, which have nice features for managing the location of data, in the middle to distribute work among them.

Lambda calculus

The lambda calculus consists of a set of “terms” together with some relations on the terms. The set TT of terms is defined recursively, parametric in a countably infinite set VV of variables. The base terms are the variables: if xx is an element of VV, then xx is a term in TT. Next, given any two terms t,t′∈Tt, t' \in T, we can apply one to the other to get t(t′)t(t'). We say that tt is in the head position of the application and t′t' in the tail position. (When the associativity of application is unclear, we’ll also use parentheses around subterms.) Finally, we can abstract out a variable from a term: given a variable xx and a term t,t, we get a term λx.t\lambda x.t.

The term constructors define an algebra, a functor LCLC from Set to Set that takes any set of variables VV to the set of terms T=LC(V)T = LC(V). The term constructors themselves become functions: −: V→T mboxvariable −(−): T×T→T mboxapplication λ: V×T→T mboxabstraction \begin{array}{rll} -\colon & V \to T &\mbox{variable}\\ -(-)\colon & T \times T \to T &\mbox{application}\\ \lambda\colon & V \times T \to T &\mbox{abstraction} \end{array}

Church described three relations on terms. The first relation, alpha, relates any two lambda abstractions that differ only in the variable name. This is exactly the same as when we consider the function f(x)=x 2f(x) = x^2 to be identical to the function f(y)=y 2f(y) = y^2. The third relation, eta, says that there’s no difference between a function ff and a “middle-man” function that gets an input xx and applies the function ff to it: λx.f(x)=f\lambda x.f(x) = f. Both alpha and eta are equivalences.

The really important relation is the second one, “beta reduction”. In order to define beta reduction, we have to define the free variables of a term: a variable occurring by itself is free; the set of free variables in an application is the union of the free variables in its subterms; and the free variables in a lambda abstraction are the free variables of the subterm except for the abstracted variable. FV(x)= {x} FV(t(t′))= FV(t)∪FV(t′) FV(λx.t)= FV(t)/{x} \begin{array}{rl} \mathrm{FV}(x) = & \{x\} \\ \mathrm{FV}(t(t')) = & \mathrm{FV}(t) \cup \mathrm{FV}(t') \\ \mathrm{FV}(\lambda x.t) = & \mathrm{FV}(t) / \{x\} \\ \end{array}

Beta reduction says that when we have a lambda abstraction λx.t\lambda x.t applied to a term t′t', then we replace every free occurrence of xx in tt by t′t': (λx.t)(t′)↓ βt{t′/x}, (\lambda x.t)(t') \downarrow_\beta t\{t' / x\}, where we read the right hand side as “tt with t′t' replacing xx.” We see a similar replacement of yy in action when we compose the following functions: f(x)= x+1 g(y)= y 2 g(f(x))= (x+1) 2 \begin{array}{rl} f(x) = & x + 1 \\ g(y) = & y^2 \\ g(f(x)) = & (x + 1)^2 \\ \end{array}

We say a term has a normal form if there’s some sequence of beta reductions that leads to a term where no beta reduction is possible. When the beta rule applies in more than one place in a term, it doesn’t matter which one you choose to do first: any sequence of betas that leads to a normal form will lead to the same normal form. This property of beta reduction is called confluence. Confluence means that the order of performing various subcomputations doesn’t matter so long as they all finish: in the expression (2+5)*(3+6)(2 + 5) * (3 + 6) it doesn’t matter which addition you do first or whether you distribute the expressions over each other; the answer is the same.

“Running” a program in the lambda calculus is the process of computing the normal form by repeated application of beta reduction, and the normal form itself is the result of the computation. Confluence, however, does not mean that when there is more than one place we could apply beta reduction, we can choose any beta reduction and be guaranteed to reach a normal form. The following lambda term, customarily denoted ω\omega, takes an input and applies it to itself: ω=λx.x(x)\omega = \lambda x.x(x) If we apply ω\omega to itself, then beta reduction produces the same term, customarily called Ω\Omega: Ω=ω(ω)\Omega = \omega(\omega) Ω↓ βΩ.\Omega \downarrow_\beta \Omega. It’s an infinite loop! Now consider this lambda term that has Ω\Omega as a subterm: (λx.λy.x)(λx.x)(Ω)(\lambda x.\lambda y.x)(\lambda x.x)(\Omega) It says, “Return the first element of the pair (identity function, Ω\Omega)”. If it has an answer at all, the answer should be “the identity function”. The question of whether it has an answer becomes, “Do we try to calculate the elements of the pair before applying the projection to it?”

Lazy lambda calculus

Many programming languages, like Java, C, JavaScript, Perl, Python, and Lisp are “eager”: they calculate the normal form of inputs to a function before calculating the result of the function on the inputs; the expression above, implemented in any of these languages, would be an infinite loop. Other languages, like Miranda, Lispkit, Lazy ML, and Haskell and its predecessor Orwell are “lazy” and only apply beta reduction to inputs when they are needed to complete the computation; in these languages, the result is the identity function. Abramsky wrote a 48-page paper about constructing a domain that captures the operational semantics of lazy lambda calculus.

The idea of representing operational semantics directly with higher categories originated with R. A. G. Seely, who suggested that beta reduction should be a 2-morphism; Barney Hilken and Tom Hirschowitz have also contributed to looking at lambda calculus from this perspective. In the “Rosetta stone” paper that John Baez and I wrote, we made an analogy between programs and Feynman diagrams. The analogy is precise as far as it goes, but it’s unsatisfactory in the sense that Feynman diagrams describe processes happening over time, while Lambek and Scott mod out by the process of computation that occurs over time. If we use 2-categories that explicitly model rewrites between terms, we get something that could potentially be interpreted with concepts from physics: types would become analogous to strings, terms would become analogous to space, and rewrites would happen over time. The idea from the “algebra of terms” perspective is that we have objects VV and TT for variables and terms, term constructors as 1-morphisms, and the nontrivial 2-morphisms generated by beta reduction. Seely showed that this approach works fine when you’re unconcerned with the context in which reduction can occur.

This approach, however, doesn’t work for lazy lambda calculus! Horizontal composition in a 2-category is a functor, so if a term tt reduces to a term t′t', then by functoriality, λx.t\lambda x.t must reduce to λx.t′\lambda x.t'—but this is forbidden in the lazy lambda calculus! Functoriality of horizontal composition is a “relativity principle” in the sense that reductions in one context are the same as reductions in any other context. In lazy programming languages, on the other hand, the “head” context is privileged: reductions only happen here. It’s somewhat like believing that measuring differences in temperature is like measuring differences in space, that only the difference is meaningful—and then discovering absolute zero. When beta reduction can happen anywhere in a term, there are too many 2-morphisms to model lazy lambda calculus.

In order to model this special context, we reify it: we add a special unary term constructor [−]:T→T[-]\colon T \to T that marks contexts where reduction is allowed, then redefine beta reduction so that the term constructor [−][-] behaves like a catalyst that enables the beta reduction to occur. This lets us cut down the set of 2-morphisms to exactly those that are allowed in the lazy lambda calculus; Greg and I did essentially the same thing in the pi calculus.

More concretely, we have two generating rewrite rules. The first propagates the reduction context to the head position of the term; the second is beta reduction restricted to a reduction context. [t(t′)]↓ ctx[[t](t′)] [t(t')]\, \downarrow_{ctx}\, [[t](t')] [[λx.t](t′)]↓ β[t]{t′/x} [[\lambda x.t](t')]\, \downarrow_\beta\, [t]\, \{t'/x\} When we surround the example term from the previous section with a reduction context marker, we get the following sequence of reductions: [(λx.λy.x)(λx.x)(Ω)] ↓ ctx [[(λx.λy.x)(λx.x)](Ω)] ↓ ctx [[[λx.λy.x](λx.x)](Ω)] ↓ β [[λy.(λx.x)](Ω)] ↓ β [λx.x] \begin{array}{rl} & [(\lambda x.\lambda y.x)(\lambda x.x)(\Omega)] \\ \downarrow_{ctx}& [[(\lambda x.\lambda y.x)(\lambda x.x)](\Omega)] \\ \downarrow_{ctx}& [[[\lambda x.\lambda y.x](\lambda x.x)](\Omega)] \\ \downarrow_{\beta}& [[\lambda y.(\lambda x.x)](\Omega)]\\ \downarrow_{\beta}& [\lambda x.x] \\ \end{array} At the start, none of the subterms were of the right shape for beta reduction to apply. The first two reductions propagated the reduction context down to the projection in head position. At that point, the only reduction that could occur was at the application of the projection to the first element of the pair, and after that to the second element. At no point was Ω\Omega ever in a reduction context.

Compute resources

In order to run a program that does anything practical, you need a processor, time, memory, and perhaps disk space or a network connection or a display. All of these resources have a cost, and it would be nice to keep track of them. One side-effect of reifying the context is that we can use it as a resource.

The rewrite rule ↓ ctx\downarrow_{ctx} increases the number of occurrences of [−][-] in a term while ↓ β\downarrow_\beta decreases the number. If we replace ↓ ctx\downarrow_{ctx} by the rule [t(t′)]↓ ctx′[t](t′) [t(t')]\, \downarrow_{ctx'}\, [t](t') then the number of occurences of [−][-] can never increase. By forming the term [[⋯[t]⋯]][[\cdots[t]\cdots]], we can bound the number of beta reductions that can occur in the computation of tt.

If we have a nullary constructor c:1→Tc\colon 1 \to T, then we can define [t]=c(t)[t] = c(t) and let the program dynamically decide whether to evaluate an expression eagerly or lazily.

In the pi calculus, we have the ability to run multiple processes at the same time; each [−][-] in that situation represents a core in a processor or computer in a network.

These are just the first things that come to mind; we’re experimenting with variations.

Conclusion

We figured out how to model the operational semantics of a term calculus directly in a 2-category by requiring a catalyst to carry out a rewrite, which gave us full abstraction without needing a domain based on representing all the possible futures of a term. As a side-effect, it also gave us a new tool for modeling resource consumption in the process of computation. Though I haven’t explained how yet, there’s a nice connection between the “algebra-of-terms” approach that uses VV and TT as objects and Lambek and Scott’s approach that uses types as objects, based on Melliès and Zeilberger’s ideas about type refinement. Next time, I’ll talk about the pi calculus and types.

Take a look at this article (hat tip Felix Salmon), which has me absolutely raging this morning, about new legislation in Kansas that prevents poor people on welfare from taking out more than $25 per day using their state-issued debit cards.

To be clear, you have to round up to the nearest $20 if you want to take out money from an ATM, so that’s really the limit.

And to be clear, there’s a $1 fee to take out money, and then typically an extra $2.50 fee if you don’t have a bank account, which many of the affected people do not.

So altogether, they’re giving $3.50 for every $20 of their welfare benefits, which I’d characterize as a bank tax of 17.5%. Because poor people don’t need that money, never mind the convenience of paying their actual bills.

For fuck’s sake, Kansas.

Author(s): Tim Wogan

Rotating bacterial cells suck one another into a 2D crystal structure, an unprecedented pattern for living organisms.

[Physics 8, 35] Published Fri Apr 17, 2015

Like hallucinogenic drugs, full-field flickering visual stimulation produces regular, geometric hallucinations such as radial or spiral patterns. Computational and theoretical models have revealed that the geometry of these hallucinations can be related to functional neuro-anatomy. However, while experimental evidence links both visual flicker and hallucinogenic drugs to upward and downward modulations of brain oscillatory activity, the exact relation between brain oscillations and geometric hallucinations remains a mystery. Here we demonstrate that, in human observers, this link is bidirectional. The same flicker frequencies that preferentially induced radial (<10 Hz) or spiral (10–20 Hz) hallucinations in a behavioral experiment involving full-field uniform flicker without any actual shape displayed, also showed selective oscillatory EEG enhancement when observers viewed a genuine static image of a radial or spiral pattern without any flicker. This bidirectional property constrains the possible neuronal events at the origin of visual hallucinations, and further suggests that brain oscillations, which are strictly temporal in nature, could nonetheless act as preferential channels for spatial information.

Dr. Greene, working with a student, has also found that “squirrels understand ‘bird-ese,’ and birds understand ‘squirrel-ese.’ ” When red squirrels hear a call announcing a dangerous raptor in the air, or they see such a raptor, they will give calls that are acoustically “almost identical” to the birds, Dr. Greene said. (Researchers have found that eastern chipmunks are attuned to mobbing calls by the eastern tufted titmouse, a cousin of the chickadee.)

The titmice are in on it too.  The article has numerous further points of interest.

It's a familiar scene to most Americans. The poignant moment when a soldier is honored for his or her service before a cheering crowd during halftime of an NFL game.  It turns out, however, that at least some of these patriotic displays are not what they seem.

A New Jersey-based website, NJ.com, has a detailed report that reveals the Department of Defense is paying millions of dollars to many NFL teams in what are essentially paid promotions to honor America's heroes...

This does not mean, of course, that all halftime events featuring troops or veterans are paid promotions. However, the fact that many are could undermine such efforts and "leaves a bad taste in your mouth" one lawmaker said.

"Those of us go to sporting events and see them honoring the heroes," said Arizona Sen. Jeff Flake in an interview with NJ.com. "You get a good feeling in your heart. Then to find out they're doing it because they're compensated for it, it leaves you underwhelmed. It seems a little unseemly."

It's hardly a secret that the NFL is one of the leading recruitment vehicles for the U.S. military. The problem, Flake implies, is that these events are portrayed as genuine moments of gratitude expressed to America's servicemen, not advertisements.

Comparing mathematics with poetry is an infinitely rich game. For every opinion you express, there is an equally valid counter-opinion. Contrasted to Hilbert’s dismissal of a student who had left mathematics for poetry, “I always thought he didn’t have enough imagination for mathematics”, someone said to me recently that the early death of Schubert was a greater tragedy than that of Galois, since what Galois could have achieved would sooner or later be done by someone else, whereas Schubert’s potential was lost forever.

So it isn’t so surprising that a book by Ron Aharoni, newly translated into English, doesn’t come to a definite conclusion one way or the other. The best we can do in a book entitled Mathematics, Poetry and Beauty is to give many examples of beautiful mathematics and beautiful poetry and discuss what the similarities and differences are.

Ron Aharoni is a mathematician whose field is combinatorics. He has collaboration distance 2 from me (we are both co-authors of Paul Erdős). I hadn’t heard from him for a while. In the book he explains that he made a deliberate move from university to elementary school.

Many, though not all, of his examples of poetry are taken from Israeli poets. I don’t know whether Hebrew is a particularly good language for poetry, but some of these poets pack many layers of meaning into a few words. But other poets appear, including Johann Wolfgang von Goethe, John Donne, Emily Dickinson, Federico Garcia Lorca, Constantine Cafavy, William Carlos Williams, and Matsuo Basho.

Here is an example of the argument. Displacement is a mechanism which, according to Freud, occurs in almost every area of human thought. By focussing on a subsidiary idea, the main message slips through almost unnoticed, although it may be too painful to face directly. Aharoni suggests that this is a technique used by poets for diving inside themselves, and for mathematicians stuck on a problem who look at a seemingly irrelevant detail in the hope of a breakthrough. He gives several examples, both poetic and mathematical.

One of his telling comparisons, expanded over four chapters, is that both a poem and a mathematical proof constitute a game of ping-pong between the abstract and the concrete. A poem can have several such switches in a few lines, as in this example “Written in pencil in the sealed railway-car” by Dan Pagis (translated by Stephen Mitchell):

Here in this carload
I am Eve
with my son Abel
if you see my older boy
Cain son of Adam
tell him that I …

In mathematics, both finding a proof and (if you are kind to your readers) presenting it involve frequent shifts of focus between logical argument and examples. But Aharoni’s conclusion is

… the heart of the poem is given to the concrete, and it is in this direction that the poem goes. This is the diametric opposite of the ping-pong of mathematics, in which the last shot is always towards the abstract.

It is also true that, as he remarks, in many published proofs (most notoriously, those of Gauss, the “fox who effaces his tracks in the sand with his tail”, according to Abel), all traces of the concrete are covered and only the shots to the abstract remain.

This points to another important difference. A finished poem can convey its beauty to any open-minded reader; knowledge of the poet’s biography often gets in the way. But the beauty in mathematics lies in the experience of the discovery of the proof; this can be reproduced to some extent in a reader who follows the argument carefully, but does not reside in the published proof, and still less in the statement of the theorem (in most cases).

And on the same theme, Aharoni invites us to watch the mathematician and the poet at work. The striking secret he reveals is that the mathematician spends most of his time staring into space.

Both poets (as many thinkers have observed) and mathematicians are concerned, not with ever more florid invention, but with the truth. It seems like a different kind of truth. Poets remind us of things we already know. But almost every mathematician, no matter which side they take on the “discovered or invented” question, are in their ordinary work Platonists, and act as if that their mental constructions are “out there”. However, the means they have for diving inside themselves, described in detail by Hadamard in his book, and divided into four stages (preparation, incubation, illumination, and verification) go well beyond the subjective ways that poets operate.

There is much more thought-provoking material in the book, many more dimensions on which mathematics and poetry can be compared. The chapter titles give some indication, including “The miracle of order”, “The power of the oblique”, “Reality or imagination”, “Unexpected combinations”, “Symmetry”, “Content and husk”, and “Change”.

Ug99 is a strain of wheat stem rust that was first identified in Uganda in 1998 (and named in 1999). By 2001, Ug99 began appearing in fields in Kenya; in Ethiopia by 2003; Sudan and Yemen by 2006; and Iran a year later. It now plagues wheat plants in nine African and Middle Eastern countries. Should the pathogen establish a global presence, 90 percent of wheat varieties could succumb, with whole crops flopping over and rotting within weeks or months of infection. The annual global harvest of some 700 million tons of wheat would be decimated...

University of Minnesota’s Anderson says GM corn and soybeans may have received more acceptance because they’re often processed or fed to animals, whereas most wheat enters the human food stream. Some consumers fear that tinkering with crop genomes could reduce the nutritive value of food, introduce toxins, increase the use of pesticides, or propel our already heavily-processed diets further away from what nature provided...

Nevertheless, GM wheat may still make its debut. In 2010, Monsanto announced that it was re-entering the field of biotech wheat and would work to genetically engineer crops that are higher yielding, stress tolerant, or herbicide resistant... Wheat growers are also starting to come around to the idea of cultivating GM plants. Several years ago, the National Association of Wheat Growers and US Wheat Associates expressed their support of biotech wheat. “We just have to prepare ourselves for a future where GM crops are more accepted,” Wulff says.

There is some excitement in the net about some news of Harold White’s experiment at NASA. I have uncovered it by chance at a forum. This is a well-frequented site with people at NASA posting on it and regularly updating about the work that they are carrying out. You can also have noticed some activity in the Wikipedia’s pages about it (see here at the section on EmDrive and here). Wikipedia’s section on EmDrive explains in a few lines what is going on. Running a laser inside the RF cavity of the device they observed an unusual effect. They do not know yet if this could be better explained by more mundane reasons like air heating inside the cavity itself. They will repeat the measurements in a vacuum chamber to exclude such a possibility. I present here some of the slides used by White to recount about this This is the current take by Dr. White as reported by one of his colleagues too prone to leak on nasaspaceflight forum:

 …to be more careful in declaring we’ve observed the first lab based space-time warp signal and rather say we have observed another non-negative results in regards to the current still in-air WFI tests, even though they are the best signals we’ve seen to date. It appears that whenever we talk about warp-drives in our work in a positive way, the general populace and the press reads way too much into our technical disclosures and progress.

I would like to remember that White is not using exotic matter at all. Rather, he is working with strong RF fields to try to develop a warp bubble. This was stated here even if implicitly. Finally, an EmDrive device has been properly described here. Using strong external fields to modify locally a space-time has been described here. If this will be confirmed in the next few months, it will represent a major breakthrough in experimental general relativity since Eddington  confirmed the bending of light near the sun. Applications would follow if this idea will appear scalable but it will be a shocking result anyway. We look forward to hear from White very soon.

Marco Frasca (2005). Strong coupling expansion for general relativity Int.J.Mod.Phys.D15:1373-1386,2006 arXiv: hep-th/0508246v3

Physicists have mapped out how to create quantum music, an experience that will be profoundly different for every member of the audience, they say.

One of the features of 20^th century art is its increasing level of abstraction from cubism and surrealism in the early years to abstract expressionism and mathematical photography later. So an interesting question is what further abstractions can we look forward to in the 21^th century?

I’m helping run a workshop on Information and Entropy in Biological Systems at NIMBioS, the National Institute of Mathematical and Biological Synthesis, which is in Knoxville Tennessee.

I think you’ll be able to watch live streaming video of this workshop while it’s taking place from Wednesday April 8th to Friday April 10th. Later, videos will be made available in a permanent location.

To watch the workshop live, go here. Go down to where it says

Investigative Workshop: Information and Entropy in Biological Systems

Then click where it says live link. There’s nothing there now, but I’m hoping there will be when the show starts!

Below you can see the schedule of talks and a list of participants. The hours are in Eastern Daylight Time: add 4 hours to get Greenwich Mean Time. The talks start at 10 am EDT, which is 2 pm GMT.

Schedule

There will be 1½ hours of talks in the morning and 1½ hours in the afternoon for each of the 3 days, Wednesday April 8th to Friday April 10th. The rest of the time will be for discussions on different topics. We’ll break up into groups, based on what people want to discuss.

Each invited speaker will give a 30-minute talk summarizing the key ideas in some area, not their latest research so much as what everyone should know to start interesting conversations. After that, 15 minutes for questions and/or coffee.

Here’s the schedule. You can already see slides or other material for the talks with links!

Wednesday April 8

• 9:45-10:00 — the usual introductory fussing around.

• 10:00-10:30 — John Baez, Information and entropy in biological systems.

• 10:30-11:00 — questions, coffee.

• 11:00-11:30 — Chris Lee, Empirical information, potential information and disinformation.

• 11:30-11:45 — questions.

• 11:45-1:30 — lunch, conversations.

• 1:30-2:00 — John Harte, Maximum entropy as a foundation for theory building in ecology.

• 2:00-2:15 — questions, coffee.

• 2:15-2:45 — Annette Ostling, The neutral theory of biodiversity and other competitors to the principle of maximum entropy.

• 2:45-3:00 — questions, coffee.

• 3:00-5:30 — break up into groups for discussions.

• 5:30 — reception.

Thursday April 9

• 10:00-10:30 — David Wolpert, The Landauer limit and thermodynamics of biological organisms.

• 10:30-11:00 — questions, coffee.

• 11:00-11:30 — Susanne Still, Efficient computation and data modeling.

• 11:30-11:45 — questions.

• 11:45-1:30 — lunch, conversations.

• 1:30-2:00 — Matina Donaldson-Matasci, The fitness value of information in an uncertain environment.

• 2:00-2:15 — questions, coffee.

• 2:15-2:45 — Roderick Dewar, Maximum entropy and maximum entropy production in biological systems: survival of the likeliest?

• 2:45-3:00 — questions, coffee.

• 3:00-6:00 — break up into groups for discussions.

Friday April 10

• 10:00-10:30 — Marc Harper, Information transport and evolutionary dynamics.

• 10:30-11:00 — questions, coffee.

• 11:00-11:30 — Tobias Fritz, Characterizations of Shannon and Rényi entropy.

• 11:30-11:45 — questions.

• 11:45-1:30 — lunch, conversations.

• 1:30-2:00 — Christina Cobbold, Biodiversity measures and the role of species similarity.

• 2:00-2:15 — questions, coffee.

• 2:15-2:45 — Tom Leinster, Maximizing biological diversity.

• 2:45-3:00 — questions, coffee.

• 3:00-6:00 — break up into groups for discussions.

Participants

Here are the confirmed participants, just so you can get a sense of who is involved:

• John Baez - mathematical physicist.

• Romain Brasselet - postdoc in cognitive neuroscience knowledgeable about information-theoretic methods and methods of estimating entropy from samples of probability distributions.

• Katharina Brinck - grad student at Centre for Complexity Science at Imperial College; did masters at John Harte’s lab, where she extended his Maximum Entropy Theory of Ecology (METE) to trophic food webs, to study how entropy maximization on the macro scale together with MEP on the scale of individuals drive the structural development of model ecosystems.

• Christina Cobbold - mathematical biologist, has studied the role of species similarity in measuring biodiversity.

• Troy Day - mathematical biologist, works with population dynamics, host-parasite dynamics, etc.; influential and could help move population dynamics to a more information-theoretic foundation.

• Roderick Dewar - physicist who studies the principle of maximal entropy production.

• Barrett Deris - MIT postdoc studying the studying the factors that influence evolvability of drug resistance in bacteria.

• Charlotte de Vries - a biology master’s student who studied particle physics to the master’s level at Oxford and the Perimeter Institute. Interested in information theory.

• Matina Donaldson-Matasci - a biologist who studies information, uncertainty and collective behavior.

• Chris Ellison - a postdoc who worked with James Crutchfield on “information-theoretic measures of structure and memory in stationary, stochastic systems - primarily, finite state hidden Markov models”. He coauthored Intersection information based on common randomness. The idea: “The introduction of the partial information decomposition generated a flurry of proposals for defining an intersection information that quantifies how much of “the same information” two or more random variables specify about a target random variable. As of yet, none is wholly satisfactory.” Works on mutual information between organisms and environment (along with David Krakauer and Jessica Flack), and also entropy rates.

• Cameron Freer - MIT postdoc in Brain and Cognitive Sciences working on maximum entropy production principles, algorithmic entropy etc.

• Tobias Fritz - a physicist who has worked on “resource theories” and haracterizations of Shannon and Rényi entropy and on resource theories.

• Dashiell Fryer - works with Marc Harper on information geometry and evolutionary game theory.

• Michael Gilchrist - an evolutionary biologist studying how errors and costs of protein translation affect the codon usage observed within a genome. Works at NIMBioS.

• Manoj Gopalkrishnan - an expert on chemical reaction networks who understands entropy-like Lyapunov functions for these systems.

• Marc Harper - works on evolutionary game theory using ideas from information theory, information geometry, etc.

• John Harte - an ecologist who uses the maximum entropy method to predict the structure of ecosystems.

• Ellen Hines - studies habitat modeling and mapping for marine endangered species and ecosystems, sea level change scenarios, documenting of human use and values. Her lab has used MaxEnt methods.

• Elizabeth Hobson - behavior ecology postdoc developing methods to quantify social complexity in animals. Works at NIMBioS.

• John Jungk - works on graph theory and biology.

• Chris Lee - in bioinformatics and genomics; applies information theory to experiment design and evolutionary biology.

• Maria Leites - works on dynamics, bifurcations and applications of coupled systems of non-linear ordinary differential equations with applications to ecology, epidemiology, and transcriptional regulatory networks. Interested in information theory.

• Tom Leinster - a mathematician who applies category theory to study various concepts of ‘magnitude’, including biodiversity and entropy.

• Timothy Lezon - a systems biologist in the Drug Discovery Institute at Pitt, who has used entropy to characterize phenotypic heterogeneity in populations of cultured cells.

• Maria Ortiz Mancera - statistician working at CONABIO, the National Commission for Knowledge and Use of Biodiversity, in Mexico.

• Yajun Mei - statistician who uses Kullback-Leibler divergence and how to efficiently compute entropy for the two-state hidden Markov models.

• Robert Molzon - mathematical economist who has studied deterministic approximation of stochastic evolutionary dynamics.

• David Murrugarra - works on discrete models in mathematical biology; interested in learning about information theory.

• Annette Ostling - studies community ecology, focusing on the influence of interspecific competition on community structure, and what insights patterns of community structure might provide about the mechanisms by which competing species coexist.

• Connie Phong - grad student at Chicago’s Institute of Genomics and System biology, working on how “certain biochemical network motifs are more attuned than others at maintaining strong input to output relationships under fluctuating conditions.”

• Petr Plechak - works on information-theoretic tools for estimating and minimizing errors in coarse-graining stochastic systems. Wrote “Information-theoretic tools for parametrized coarse-graining of non-equilibrium extended systems”.

• Blake Polllard - physics grad student working with John Baez on various generalizations of Shannon and Renyi entropy, and how these entropies change with time in Markov processes and open Markov processes.

• Timothee Poisot - works on species interaction networks; developed a “new suite of tools for probabilistic interaction networks”.

• Richard Reeve - works on biodiversity studies and the spread of antibiotic resistance. Ran a program on entropy-based biodiversity measures at a mathematics institute in Barcelona.

• Rob Shaw - works on entropy and information in biotic and pre-biotic systems.

• Matteo Smerlak - postdoc working on nonequilibrium thermodynamics and its applications to biology, especially population biology and cell replication.

• Susanne Still - a computer scientist who studies the role of thermodynamics and information theory in prediction.

• Alexander Wissner-Gross - Institute Fellow at the Harvard University Institute for Applied Computational Science and Research Affiliate at the MIT Media Laboratory, interested in lots of things.

• David Wolpert - works at the Santa Fe Institute on i) information theory and game theory, ii) the second law of thermodynamics and dynamics of complexity, iii) multi-information source optimization, iv) the mathematical underpinnings of reality, v) evolution of organizations.

• Matthew Zefferman - works on evolutionary game theory, institutional economics and models of gene-culture co-evolution. No work on information, but a postdoc at NIMBioS.

Zoology: Here be dragons

Nature 520, 7545 (2015). doi:10.1038/520042a

Authors: Andrew J. Hamilton, Robert M. May & Edward K. Waters

Emerging evidence indicates that dragons can no longer be dismissed as creatures of legend and fantasy, and that anthropogenic effects on the world's climate may inadvertently be paving the way for the resurgence of these beasts.

That means when you search for “cold sores,” for instance, and click the highly ranked “Cold Sores Topic Overview WebMD” link, the website is passing your request for information about the disease along to one or more (and often many, many more) other corporations...

Thus, Libert has discovered that the vast majority of health sites, from the for-profit WebMD.com to the government-run CDC.gov, are loaded with tracking elements that are sending records of your health inquiries to the likes of web giants like Google, Facebook, and Pinterest, and data brokers like Experian and Acxiom.

From there, it becomes relatively easy for the companies receiving the requests, many of which are collecting other kinds of data (in cookies, say) about your browsing as well, to identify you and your illness...

WebMD, for instance, is the 106th most-visited site in the US, according to Alexa, and figures prominently in search results for most commonly searched diseases. It sends third party requests to a whopping 34 separate domains, including the data brokers Experian and Acxiom.“WebMD is basically calling up everybody in town and telling them that’s what you’re looking at..."

With nonprofit sites like the CDC and the Mayo Clinic, again, it’s not due to any insidious intent; it’s simply because developers are installing “free” tools like Google Analytics and social media “share” buttons on their sites, and most users have no idea that means information about their searches is being shared with third parties. “The problem is that using these 'free' third-party tools is really easy for web developers. What developers don’t consider is, why are these tools free?”