JavaScript can’t be fixed

Posted in Uncategorized by Mike Stay on 2019 January 18

In 2015, Andreas Rossberg of the V8 team proposed making JavaScript a little saner by making classes, objects, and functions unalterable after their construction so that a sound type system becomes possible.

A year later, his team reported on their progress and said that there’s basically no way to do that without breaking compatibility with everything that came before.

Any sound type system that’s compatible with old code will have to admit that JavaScript “functions” really aren’t and will have to model a lot more of JavaScript’s insanity.



A whale-eating whale

Posted in Uncategorized by Mike Stay on 2019 January 16

The Basilosaurus is not, as its name suggests, a lizard, but a prehistoric cetacean, a mammal.  It grew up to 60 feet long (18 m).  They could bite with a force of 3,600 pounds per square inch (25,000 kPa), chewing and then swallowing their food.

The Gnostic Brotherhood of Mathematicians

Posted in Uncategorized by Mike Stay on 2019 January 15

3D printed cam system for drawing pictures with a laser

Posted in Uncategorized by Mike Stay on 2019 January 14

Bob Pepper’s Dragonmaster Cards

Posted in Uncategorized by Mike Stay on 2019 January 6

I got these cards for my birthday one year in the early 80s. We didn’t play the game very much—it was too hard for me as a little kid, and I was the oldest of my siblings—but I was always impressed by the art work and would pull them out just to look at them

New alternative to platinum in catalytic converters uses gelatin and molybdenum, tungsten or cobalt

Posted in Uncategorized by Mike Stay on 2018 December 26

These fossilized clams turned into opals

Posted in Uncategorized by Mike Stay on 2018 December 18

New Zealand batflies

Posted in Uncategorized by Mike Stay on 2018 December 18

South American batflies are parasitic on their bats, but the ones in New Zealand are vegetarian coprophages, eating the pre-digested fruit in the bats’ guano.

New cervical cancer test has 100% accuracy

Posted in Uncategorized by Mike Stay on 2018 December 17

“A new test for cervical cancer was found to detect all of the cancers in a randomised clinical screening trial of 15,744 women, outperforming both the current Pap smear and human papillomavirus (HPV) test at a reduced cost, according to a study led by Queen Mary University of London.”

Happy Quantum Day!

Posted in Uncategorized by Mike Stay on 2018 December 14

On this day in 1900, Max Planck presented his theory of blackbody radiation to the German Physical Society (Deutsche Physikalische Gesellschaft, DPG).

Sometime during December of 1925, Schrödinger came up with the wave equation of a nonrelativistic electron; he wrote about it in a letter to Wien on 1925-12-27. So we could say that Quantum day has a half-life on the order of 10 days!

No fees plus 3% interest on checking & savings balances

Posted in Uncategorized by Mike Stay on 2018 December 13

A recording of Mark Twain (sort of)

Posted in Uncategorized by Mike Stay on 2018 December 5

Not a recording of Twain, but a recording of a professional imitator who lived next to Twain for years.

The Spooky instrument

Posted in Uncategorized by Mike Stay on 2018 November 30

Scarab beetles and circularly polarized light

Posted in Uncategorized by Mike Stay on 2018 November 27

Time lapse of rocket launch as seen from ISS

Posted in Uncategorized by Mike Stay on 2018 November 26

Using microfluidics to direct T-cells to cancer cells, learn what antigens are on their surface, and create tumor-specific treatments

Posted in Uncategorized by Mike Stay on 2018 November 7

Hackberries apparently taste like peanut M&Ms

Posted in Uncategorized by Mike Stay on 2018 November 7

Lovecraftian tentacled horrors bursting from their leathery ova

Posted in Uncategorized by Mike Stay on 2018 November 7


Posted in Uncategorized by Mike Stay on 2018 October 29

I started parsing DFS as “discrete finite something-or-other” then realized I had to backtrack and try a different sibling of “discrete”.

Boys’ Life Tripods trilogy

Posted in Uncategorized by Mike Stay on 2018 October 26

This comic adaptation of John Christopher’s Tripods trilogy was the best part of Boys’ Life magazine for me. It started coming out the month after I became a boy scout.

Piano Genie

Posted in Uncategorized by Mike Stay on 2018 October 19

This project trained a neural net to predict, essentially, high res piano from low res input. You get to choose whether the note is higher or lower than the last one, but not the note itself; that’s up to the neural net to decide. There’s a version you can try out on the web!

Grail coin by romanbooteen

Posted in Uncategorized by Mike Stay on 2018 October 19

Custom coin carving with a knight, a removable sword, and a hidden grail!

Making of:

Meteor shower flows in 3d

Posted in Uncategorized by Mike Stay on 2018 October 19

A 3d interactive rendering of the solar system illustrating orbits of the meteors for each of the major showers.

G+ and website gone

Posted in Uncategorized by Mike Stay on 2018 October 10

I got rid of my Facebook account a month ago.  I used Google+ until it was announced that they’re shutting it down.  I also had a website on, but that server recently kicked the bucket, and it’s unclear whether the content is coming back.  The wayback machine has a copy from earlier this year, so not much would be lost if it doesn’t. But given those losses, it looks like I’ll be posting here more often. 

So to celebrate, here’s a fantastic video of a zebrafish embryo growing a nervous system.


Capability Myths Demolished

Posted in Uncategorized by Mike Stay on 2016 February 16

A nice summary of erights’ Capability Myths Demolished paper.

the morning paper

Capability Myths Demolished – Miller et. al 2003

Pretty much everyone is familiar with an ACL-based approach to security. Despite having been around for a very long time, the capabilities approach to security is less well-known. Today’s paper choice provides an excellent introduction to the capabilities model and how it compares to ACLs. Along the way we’ll learn about 7 fundamental properties of security systems, and which combinations of those are required to offer certain higher-level guarantees. Capabilities are central to the type system of the Pony language which we’ll be looking at tomorrow.

Let’s start out by looking at one of the fundamental differences between ACLs and capabilities, the direction of the relationship between subject and resource. Consider a classic access matrix such as the one below. Each row is a subject, and each column a resource. The entry in a given cell describes the permissions the subject has…

View original post 949 more words

Serializing Javascript Closures

Posted in Uncategorized by Mike Stay on 2016 January 21

In Javascript, the eval function lets you dynamically look up variables in scope, so you can do a terrible hack like this to sort-of serialize a closure if you’re in the same scope as the variables being closed over:

Given StringMap.js and atLeastFreeVarNames.js from SES, one can define the following:

var s = function(f) {
  // Find free vars in f. (This depends, of course, on
  // Function.prototype.toString being unchanged.)
  var code = f.toString();
  var free = ses.atLeastFreeVarNames(code);
  // Construct code that evaluates to an environment object.
  var env = ["({"];
  for (var i = 0, len = free.length; i < len; ++i) {
    env.push('":(function(){try{return eval("(');
    env.push(')")}catch(_){return {}}})()');
  return "({code:" + JSON.stringify(code) + ",env:" + env.join("") + "})";

// See or
// for versions of stringify that handle cycles.
var t = function(x) { return '(' + JSON.stringify(x) + ')'; }

Then you can use these definitions to serialize inline definitions that only close over “stringifiable” objects or objects behind cut points (see deserialization below):

var baz = {x: 1};
var serializedClosure = t(eval(s(
  function bar(foo) { console.log('hi'); return ++(baz.x)+foo /*fiddle*/; }
// serializedClosure === '({"code":"function bar(foo) { console.log('hi'); return ++(baz.x)+foo /*fiddle*/; }","env":{"function":{},"bar":{},"foo":{},"console":{},"log":{},"hi":{},"return":{},"baz":{"x":1},"x":{},"fiddle":{}}})'

The string serializedClosure can then be stashed somewhere.  When it’s time to deserialize, do the following:

var d = function(closure, localBindings) {
  localBindings = localBindings || {};
  return function() {
    with(closure.env) {
      with (localBindings) {
        return eval("(" + closure.code + ")").apply(this, arguments);
var closure = eval(serializedClosure);
// Hook up local values if you want them.
var deserializedFn = d(closure, {console: console});
// Prints "2foo" to the console.
// Prints "3bar" to the console.

If you want to store the updated state, just re-stringify the closure:

var newSerializedClosure = t(closure);
// newSerializedClosure === '({"code":"function bar(foo) { console.log('hi'); return ++(baz.x)+foo /*fiddle*/; }","env":{"function":{},"bar":{},"foo":{},"console":{},"log":{},"hi":{},"return":{},"baz":{"x":3},"x":{},"fiddle":{}}})'
// Note that baz.x is now 3.

As I said, a very ugly hack, but still might be useful somewhere.

HDRA part 1a

Posted in Uncategorized by Mike Stay on 2015 May 30

Again, comments are appreciated.

2-categories and lambda calculus

This is a follow up post, but not the promised sequel, from A 2-Categorical Approach to the Pi Calculus where I’ll try to correct various errors I made and try to clarify some things.

First, lambda calculus was invented by Church to solve Hilbert’s decision problem, also known as the Entscheidungsproblem. The decision problem was third on a list of problems he presented at a conference in 1928, but was not, as I wrote last time, “Hilbert’s third problem”, which was third on a list of problems he laid out in 1900.

Second, the problem Greg Meredith and I were trying to address was how to prevent reduction under a “receive” prefix in the pi calculus, which is intimately connected with preventing reduction under a lambda in lambda calculus. All programming languages that I know of, whether eager or lazy, do not reduce under a lambda.

There are two approaches in literature to the semantics of lambda calculus. The first is denotational semantics, which was originally concerned with what function a term computes. This is where computability and type theory live. Denotational semantics treats alpha-beta-eta equivalence classes of lambda terms as morphisms in a category. The objects in the category are called “domains”, and are usually “\omegaCPOs“, a special kind of poset. Lambek and Scott used this approach to show that alpha-beta-eta equivalence classes of lambda terms with one free variable form a cartesian closed category where composition is given by substitution.

The type of a term describes the structure of its normal form. Values are terms that have no beta reduction; they’re either constants or lambda abstractions. The types in lambda calculus are either base types or lambda abstractions, respectively. (While Lambek and Scott introduced new term constructors for products, they’re not strictly necessary, because the lambda term \lambda z.ztu can be used for the pair \langle t,u\rangle with projections \lambda p.p(\lambda xy.x) and \lambda p.p(\lambda xy.y).)

The second is operational semantics, which is more concerned with how a function is computed than what it computes. All of computational complexity theory and algorithmic analysis lives here, since we have to count the number of steps it takes to complete a computation. Operational semantics treats lambda terms as objects in a category and rewrite rules as morphisms. It is a very syntactical approach; the set of terms is an algebra of some endofunctor, usually presented in Backus-Naur Form. This set of terms is then equipped with some equivalence relations and reduction relations. For lambda calculus, we mod out terms by alpha, but not beta. The reduction relations often involve specifying a reduction context, which means that the rewrites can’t occur just anywhere in a term.

In pi calculus, terms don’t have a normal form, so we can’t define an equivalence on pi calculus terms by comparing their normal forms. Instead, we say two terms are equivalent if they behave the same in all contexts, i.e. they’re bisimilar. Typing becomes rather more complicated; pi calculus types describe the structure of the current term and rewrites should do something weaker than preserve types on the nose; it suggests using a double category, but that’s for next time.

Seely suggested modeling rewrites with 2-morphisms in a 2-category and showed how beta and eta were lax adjoints.  We’re suggesting a different, more operational way of modeling rewrites with 2-morphisms.  Define a 2-category L with

  • V and T as generating objects,
  • -, -(=), and \lambda as generating 1-morphisms,
  • alpha equivalence as an identity 2-morphism, and
  • beta reduction as a generating 2-morphism.

A closed term is one with no free variables, and the hom category L(I, T) consisting of closed lambda terms and beta reductions between them is a typical category you’d get from looking for the operational semantics of lambda calculus.

The 2-category L is a fine semantics for the variant of lambda calculus where beta can apply anywhere within a term, but there are too many morphisms if you want to model the lambda calculus without reduction under a prefix. Given closed terms t, t'\colon I \to T such that t beta reduces to t', we can whisker beta by \lambda x to get

\displaystyle \lambda x.\beta\colon \lambda x.t \Rightarrow \lambda x.t'.

To cut down the extra 2-morphisms, we need to model reduction contexts. The difference between the precontexts above and the contexts we need is the notion of the “topmost” context, where we can see enough of the surrounding term to determine that reduction is possible. To model a reduction context, we make a new 2-category L' by introducing a new generating 1-morphism

\displaystyle [-]\colon T \to T

and say that a 1-morphism with signature T^{\otimes n} \to T that factors as a precontext followed by [-] is an n-hole term context. We also interpret beta with a 2-morphism between reduction contexts rather than between precontexts. The hom category L'(I, T) is the operational semantics of lambda calculus without reduction under lambda.

In the next post, I’ll relate Seely’s 2-categorical approach and our 2-categorical by extending to double categories and using Melliès and Zeilberger’s notion of type refinement.


Posted in Uncategorized by Mike Stay on 2015 May 18

Greg Meredith and I have a short paper that’s been accepted to 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.


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 D such that the continuous functions from D 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 t and a variable x to produce the term \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 powerset 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 Melliès 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 T of terms is defined recursively, parametric in a countably infinite set V of variables. The base terms are the variables: if x is an element of V, then x is a term in T. Next, given any two terms t, t' \in T, we can apply one to the other to get t(t'). We say that t is in the head position of the application and 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 x and a term t, we get a term \lambda x.t.

The term constructors define an algebra, a functor LC from Set to Set that takes any set of variables V to the set of terms T = LC(V). The term constructors themselves become functions:

\displaystyle   \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^2 to be identical to the function f(y) = y^2. The third relation, eta, says that there’s no difference between a function f and a “middle-man” function that gets an input x and applies the function f to it: \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.

\displaystyle   \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 \lambda x.t applied to a term t', then we replace every free occurrence of x in t by t':

\displaystyle  (\lambda x.t)(t') \downarrow_\beta t\{t' / x\},

where we read the right hand side as “t with t' replacing x.” We see a similar replacement of y in action when we compose the following functions:

\displaystyle   \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) 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 \omega, takes an input and applies it to itself:

\displaystyle \omega = \lambda x.x(x)

If we apply \omega to itself, then beta reduction produces the same term, customarily called \Omega:

\displaystyle \Omega = \omega(\omega)

\displaystyle \Omega \downarrow_\beta \Omega.

It’s an infinite loop! Now consider this lambda term that has \Omega as a subterm:

\displaystyle (\lambda x.\lambda y.x)(\lambda x.x)(\Omega)

It says, “Return the first element of the pair (identity function, \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 V and T 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 t reduces to a term t', then by functoriality, \lambda x.t must reduce to \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 [-]\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 in an application; the second is beta reduction restricted to a reduction context.

\displaystyle  [t(t')]\, \downarrow_{ctx}\, [[t](t')]

\displaystyle  [[\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:

\displaystyle   \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 \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 \downarrow_{ctx} increases the number of occurrences of [-] in a term while \downarrow_\beta decreases the number. If we replace \downarrow_{ctx} by the rule

\displaystyle   [t(t')]\, \downarrow_{ctx'}\, [t](t')

then the number of occurences of [-] can never increase. By forming the term [[\cdots[t]\cdots]], we can bound the number of beta reductions that can occur in the computation of t.

If we have a nullary constructor c\colon 1 \to T, then we can define [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.


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 V and T as objects and Lambek and Scott’s approach that uses types as objects that uses Melliès and Zeilberger’s ideas about type refinement. Next time, I’ll talk about the pi calculus and types.

Q, Jokers, and Clowns

Posted in Uncategorized by Mike Stay on 2014 August 5

Conor McBride has a beautiful functional pearl, “Clowns to the Left of me, Jokers to the Right”, in which he discusses the idea of suspending the computation of a catamorphism. I just wanted to point out the relation of his work to the q-derivative. Since Star Trek’s character Q is a trickster god like Loki, Anansi, or Coyote, q-derivatives also fit nicely with McBride’s theme.

The usual definition of the derivative is

\displaystyle\frac{df(x)}{dx} = \lim_{h \to 0} \frac{f(x + h) - f(x)}{(x + h) - x}.

Instead of translating x by an infinitesimal amount h, we can scale x by an infinitesimal amount q; these two definitions coincide in the limit:

\displaystyle\frac{df(x)}{dx} = \lim_{q \to 1} \frac{f(qx) - f(x)}{qx - x}.

However, when people talk about the q-derivative, they usually mean the operator we get when we don’t take the limit and q \ne 1. It should probably be called the “q-difference”, but we’ll see that the form of the difference is so special that it deserves the exceptional name.

The q-derivative, as one would hope, behaves linearly:

\begin{array}{rl} \displaystyle \frac{d_qf(x)}{d_qx} & \displaystyle = \frac{f(qx) - f(x)}{qx - x} \\ \\  \displaystyle \frac{d_q(f(x) + g(x))}{d_qx} & \displaystyle = \frac{(f(qx) + g(qx)) - (f(x) + g(x))}{qx - x} \\  & \displaystyle = \frac{(f(qx) - f(x)) + (g(qx)- g(x))}{qx - x} \\  & \displaystyle = \frac{d_qf(x)}{d_qx} + \frac{d_qg(x)}{d_qx}\end{array}

Even better, the q-derivative of a power of x is separable into a coefficient that depends only on q and a single smaller power of x:

\begin{array}{rl}\displaystyle \frac{d_q x^n}{d_qx} & \displaystyle = \frac{(qx)^n - x^n}{qx - x} \\  &\displaystyle = \frac{q^n - 1}{q - 1}\frac{x^n}{x} \\  &\displaystyle = [n]_q x^{n-1},\end{array}


\displaystyle [n]_q = \frac{q^n - 1}{q - 1} = 1 + q + \cdots + q^{n-1}.

Clearly, as q \to 1, [n]_q \to n. Whereas n counts the number of ways to insert a point into an ordered list of n-1 items, [n]_q counts the number of ways to insert a linearly independent ray passing through the origin into an (n-1)-dimensional vector space over a field with q elements with a given ordered basis.

The q-derivative even works when q is an operator rather than a number. Polynomials work in any rig, so if x is, say, a function instead of a number, q could be the Fourier transform.

Let’s lift this to types. The derivative of a datatype is the type of its one-holed contexts, so we expect the q-derivative to have a similar interpretation. When we take Q and X to be types, the Q-derivative of a tuple X^n is

\displaystyle [n]_Q X^{n-1} = X^{n-1} + (QX)X^{n-2} + (QX)^2X^{n-3} + \cdots + (QX)^{n-1}.

Each option factors into two parts: the ‘clowns’ are a power of QX to the left of the hole, followed by the ‘jokers’, a power of X after the hole. This type is the one we expect for the intermediate state when mapping a function f\colon X \to Q\times X over the tuple; any function g\colon X \to Q can be lifted to such a function f(x) = (g(x), x).

Similarly, the Q-derivative of the list datatype 1/(1-X) is 1/(1-QX)(1-X); that is, the ‘clowns’ form the list of the outputs of type Q \times X for the elements that have already been processed and the ‘jokers’ form the list of the elements yet to process.

When we abstract from thinking of q as a number to thinking of it as an operator on ring elements, the corresponding action on types is to think of Q as a functor. In this case, QX is not a pair type, but rather a parametric type Q[X]. One might, for example, consider mapping the function f(x) = 1/x over a list of real numbers. The resulting outputs will be real except when the input is zero, so we’ll want to adjoin an undefined value. Taking Q = Option, the Q-derivative of List[Int] is List[Option[Int]]\times List[Int], consisting of the list of values that have already been processed, followed by the list of values yet to process.

Symmetric monoidal closed objects

Posted in Uncategorized by Mike Stay on 2012 April 12

I conjecture that there’s a compact closed bicategory Th(SMCC) such that the 2-category hom(Th(SMCC), Prof) of

  • sylleptic monoidal functors (of bicategories),
  • braided monoidal transformations and
  • monoidal modifications

is equivalent as a 2-category to the 2-category SMCC of

  • symmetric monoidal closed categories,
  • braided monoidal closed functors, and
  • monoidal closed natural isomorphisms.

If we model Th(SMCC) in the compact closed bicategory 3Cob2, then we get

where I didn’t draw

  • the right unitor
  • the pentagon equation for the associator
  • the triangle equations for the unitors
  • the hexagon equations for the braiding
  • the yanking for internal hom
  • a,b,c,l,r composed with their formal inverses equals the identity

but I think they’re pretty obvious given this other stuff.


Posted in Uncategorized by Mike Stay on 2011 October 31

I used this Halloween as motivation to start working out a few months ago, because I wanted to be a passable Wolverine; I made a lot of progress, but not so much that I feel comfortable posting pictures of muscles. Fortunately, my mom is a seamstress and made me a great jacket. Here’s the result:

Monster manual

Posted in Uncategorized by Mike Stay on 2011 April 13

What magical creature lives deep in the earth’s crust and has a horrible lisp? A BATHOLITHK.

Better than anything 2

Posted in Uncategorized by Mike Stay on 2011 April 1

Better than This Week from Baez,
Better than Lou Kauffman’s knots,
Better than all of the string theory
Witten’s forgot,

Better than Feynman’s graffiti,
Better than Calabi-Yau,
Better than Pauli’s neutrino,
Better than mu and than tau,

Better than Curie, Poincaré,
And Niels and Albert at Solvay
Better than anything except being in love.

More views of the metapixel

Posted in Uncategorized by Mike Stay on 2011 March 10

Double spiral:

Single spiral the other way:

Double spiral the other way:

Hooped-up 24-cell

Posted in Uncategorized by Mike Stay on 2011 March 1

The 24-cell is really cool. It’s the only self-dual regular polytope that’s not a simplex or a polygon. Its vertices form the multiplicative group of units in the Hurwitz quaternion ring. It tiles Euclidean 4-space. Spheres inscribed in the octahedra give the closest packing of spheres in 4 dimensions. And, well, there are 24 of them.

We ran out of tape while making the inner cuboctahedron–one of the triangular sides is missing. But we came darn close! Compare:

Yudkowski interview

Posted in Uncategorized by Mike Stay on 2011 February 16

Queensland flooding

Posted in Uncategorized by Mike Stay on 2011 February 9

Polarization for breakfast

Posted in Uncategorized by Mike Stay on 2011 February 5

Legos this morning

Posted in Uncategorized by Mike Stay on 2011 February 5

This one ended up looking rather shark-like:

William chose–who else?–Yoda to pilot the green ship.

Willie’s fugue

Posted in Uncategorized by Mike Stay on 2011 January 23

Toccata in D minor

Posted in Uncategorized by Mike Stay on 2011 January 22

This is what about ten minutes every other day or so for a year gets you.


Posted in Uncategorized by Mike Stay on 2011 January 18

Apollo. adj 1. brave, without fear (from L. ab- not + pollo chicken).

Silicon carbide

Posted in Uncategorized by Mike Stay on 2011 January 17

I took this photo of my chunk of silicon carbide.

With some time I could probably find a crystal pure enough to emit light.

Doodling like Vi

Posted in Uncategorized by Mike Stay on 2011 January 15


Say you’re me and you just watched Vi Hart’s video on infinity elephants and you totally missed the joke about Mr. Tusks even though you read Dinosaur Comics all the time but you liked the bit about Apollonian gaskets, which don’t blow out in Battle Mountain like the one in your car did but rather on the way to the L1 point and then you need Richard Feynman to tell you why. You thought she was going to draw the tiniest camels going through the eye of a needle, but you suppose that would ruin the hyperbole in the parable, so the ellipsis was justified. Anyway, you decide to avoid circular reasoning and doodle rectangles instead, filling them up with squares.

Eventually you wonder which ones you can fill up with finitely many squares and which ones you need infinitely many for, and so you start with squares and build up some rectangles. One square can only make one rectangle, itself. Two squares can only make one rectangle, but it can be lying down or standing up so you decide to say they’re different. Three squares make four rectangles and four squares make eight rectangles, and then you start thinking about Vi Hart’s video on binary trees. So you put the numbers into a tree, but it looks kind of stern, so you add some fareys to cheer it up. Then you see that the height of a rectangle is just the sum of its neighbors’ heights, and similarly for the width. You see lots of nice patterns in the dimensions involving flipping things over or running them backwards, kind of like the Blues Brothers’ police car when it was being chased by the Neo Nazis or that V6 racecar that Johann sent to Frederick.

Now instead of breadth, you decide to go for depth. Making the rectangles very long or very tall is too boring, so you add one square each time, alternately making it longer and taller. 1 1 2 3 5 8 13 21… You get the Fibonacci numbers; the limiting ratio is the golden ratio, [1+sqrt(5)]/2 to 1. This rectangle is the worst at being approximated by repeated squares so it shows up in systems where repetition is bad, like the angle at which plant leaves grow so they overlap the least and gather the most sunlight or how sunflowers pack the most seeds into a flowerhead and Roger Penrose thinks the Fibonacci spirals in the microtubules in the neurons in your brain are doing quantum error correction.

You decide to look at other irrational numbers to see if they have any nice patterns. e does. pi doesn’t. square roots of positive integers give repeating palindromes! You wonder whether all palindromes occur and if not which of the lyrics to Weird Al’s song Bob are special that way. And maybe then you make up a palindrome with vi hart’s name in it and turn it into a square root.


Maybe you decide that you want to doodle some circles after all, so you start with this gasket and figure out where the circles touch the line. The numbers look very familiar. You wonder what the areas of the circles are and how the gasket relates to the modular group and Poincare’s half-plane model of the hyperbolic plane and wish you had time to just sit in math class and doodle…

Math doodles

Posted in Uncategorized by Mike Stay on 2010 December 30

Vi Hart‘s got some great stuff.


Posted in Uncategorized by Mike Stay on 2010 July 19

Scimitry: the property that something remains the same after part has been cut off.  The Serpinski triangle is scimitric, since you can cut off the bottom two triangles and be left with something isomorphic to the whole.

A first attempt at re-winding Escher’s “Ascending and Descending”

Posted in Uncategorized by Mike Stay on 2010 May 19

And he dreamed, and behold a ladder set up on the earth, and the top of it reached to heaven: and behold the angels of God ascending and descending on it.

Edit (May 20):

Even though it’s not a conformal transformation, this version looks better in a lot of ways.

Rather than cramming the whole picture into a single window frame, it presumes there’s a concentric set of these castles, each half as small as the previous, and built within its open internal patio. Doing it really well would involve extending the walls out to the edge of the outer wall that obscures them.

I’m not this guy

Posted in Uncategorized by Mike Stay on 2010 May 18


Posted in Uncategorized by Mike Stay on 2010 May 13

Blind faith is the best antisceptic.

Tag clouds

Posted in Uncategorized by Mike Stay on 2010 May 7

Tag clouds are cumulonymous.