Logical Statements as Tasks

In the next step of our exploration of type theory, we’re going to take a step away from the set-based stuff. There are set-based intepretations of every statement in set theory. But what we really want to focus on is the interpretation of statements in computational terms.

What that means is that we’re going to take logical statements, and view them as computation tasks – that is, as formal logical specifications of a computation that we’d like to do. Under that interpretation, a proof of a statement S is an implementation of the task specified by S.

This interpretation is much, much easier for computer science types like me than the set-based interpretations. We can walk through the interpretations of all of the statements of our intuitionistic logic in just a few minutes.

We’ll start with the simple statements.

A \land B is a specification for a program that produces a pair (a, b) where a is a solution for A, and b is a solution for B.
A \lor B is a specification for a program that produces either a solution to A or a solution to B, along with a way of identifying which of A and B it solved. We do that using a version of the classical projection functions: A \lor B produce either \text{inl}(A) (that is, the left projection), or \text{inr}(B) (the right projection).
A \supset B is a specification for a program that produces a solution to B given a solution to A; in lambda calculus terms, it’s a form like \lambda x: b(x).

Now, we can move on to quantified statements. They get a bit more complicated, but if you read the quantifier right, it’s not bad.

(\forall x \in A) B(x) is a program which, when executed, yields a program of the form \lambda x.b(x), where b(x) is an implementation of B, and x is an implementation of A. In other words, a universal statement is a program factory, which produces a program that turns one program into another program.

To me, the easiest way to understand this is to expand the quantifier. A quantified statement \forall x \in A: B(x) can be read as \forall x: x \in A \Rightarrow B(x). If you read it that way, and just follow the computational interpretation of implication, you get precisely the definition above.

Existential quantification is easy. An existential statement \exists x \in A: B(x) is a two part problem: it needs a value for a (that is, a value of x for which a proof exists that x \in A), and a proof that for that specific value of x, x \in B. A solution, then, has two parts: it’s a pair (a, b), where a is a value in A, and b is a program that computes the problem B(a).

This is the perspective from which most of Martin-Loff’s type theory pursues things. There’s a reason why ML’s type theory is so well-loved by computer scientists: because what we’re really doing here is taking a foundational theory of mathematics, and turning it into a specification language for describing computing systems.

That’s the fundamental beauty of what Martin-Loff did: he found a way of formulating all of constructive mathematics so that it’s one and the same thing as the theory of computation.

And that’s why this kind of type theory is so useful as a component of programming languages: because it’s allowing you to describe the semantics of your program in terms that are completely natural to the program. The type system is a description of the problem; and the program is the proof.

With full-blown Martin-Loff type system, the types really are a full specification of the computation described by the program. We don’t actually use the full expressiveness of type theory in real languages – among other things, it’s not checkable! But we do use a constrained, limited logic with Martin-Loff’s semantics. That’s what types really are in programming languages: they’re logical statements! As we get deeper into type theory, well see exactly how that works.

Propositions as Proofsets: Unwinding the confusion

My type theory post about the different interpretations of a proposition caused a furor in the comments. Understand what’s going on that caused all of the confusion is going to be important as we continue to move forward into type theory.

The root problem is really interesting, once you see what’s going on. We’re taking a statement that, on the face of it, isn’t about sets. Then we’re appyling a set-based interpretation of it, and looking at the subset relation. That’s all good. The problem is that when we start looking at a set-based interpretation, we’re doing what we would do in classical set theory – but that’s a different thing from what we’re doing here. In effect, we’re changing the statement.

For almost all of us, math is something that we learned from the perspective of axiomatic set theory and first order predicate logic. So that’s the default interpretation that we put on anything mathematical. When you talk about a a proposition as a set, we’re programmed to think of it in that classical way: for any set S, there’s a logical predicate P_s such that by definition, \forall x: x \in S \Leftrightarrow P_s(x). When you see P \Rightarrow Q in a set-theory context, what you think is something like \forall x: x \in P \Rightarrow x \in Q. Under that intepretation, the idea that P \supset Q is equivalent to P \rightarrow Q is absolutely ridiculous. If you follow the logic, implication must be the reverse of the subset relation!

The catch, though, is that we’re not talking about set theory, and the statement P \Rightarrow Q that we’re looking at is emphatically not \forall x : P(x) \Rightarrow Q(x). And that, right there, is the root of the problem.

P \rightarrow Q always means P \rightarrow Q – it doesn’t matter whether we’re doing set theory or type theory or whatever else. But in set theory, when we talk about the intepretation of P as a set, right now, in the world of type theory, we’re talking about a different set.

Super set doesn’t suddenly mean subset. Implication doesn’t start working backwards! and yet, I’m still trying to tell you that i really meant it when i said that superset meant implication! how can that possibly make sense?

In type theory, we┬┤re trying to take a very different look at math. In particular, we’re building everything up on a constructive/computational framework. So we’re necessarily going to look at some different interpretations of things – we’re going to look at things in ways that just don’t make sense in the world of classical set theory/FOPL. We’re not going to contradict set theory – but we’re going to look at things very differently.

For example, the kind of statement we’re talking here about is a complete, closed, logical proposition, not a predicate, nor a set. The proposition P is a statement like “‘hello’ has five letters”.

When we look at a logical proposition P, one of the type theoretic interpretations of it is as a set of facts: P can be viewed as the set of all facts that can be proven true using P. In type theory land, this makes perfect sense: if I’ve got a proof of P, then I’ve got a proof of everything that P can prove. P isn’t a statement about the items in Ps proof-set. P is a logical statement about something, and the elements of the proof-set of P are the things that the statement P can prove.

With that in mind, what does P \Rightarrow Q mean in type theory? It means that everything provable using Q is provable using nothing but P.

(It’s really important to note here that there are no quantifiers in that statement. Again, we are not saying \forall p: P(x) \Rightarrow Q(x). P and Q are atomic propositions – not open quantified statements.)

If you are following the interpretation that says that P is the set of facts that are provable using the proposition P, then if P \Rightarrow Q, that means that everything that’s in Q must also be in P. In fact, it means pretty much exactly the same thing as classical superset. Q is a set of facts provable by the statement Q. The statement Q is provable using the statement P – which means that everything in the provable set of Q must, by definition! be in the provable set of P.

The converse doesn’t hold. There can be things provable by P (and thus in the proof-set of P) which are not provable using Q. So taken as sets of facts provable by logical propositions, P \supset Q!

Again, that seems like it’s the opposite of what we’d expect. But the trick is to recognize the meaning of the statements we’re working with, and that despite a surface resemblance, they’re not the same thing that we’re used to. Type theory isn’t saying that the set theoretic statements are wrong; nor is set theory saying that type theory is wrong.

The catch is simple: we’re trying to inject a kind of quantification into the statement P \Rightarrow Q which isn’t there; and then we’re using our interpretation of that quantified statement to say something different.

But there’s an interpretation of statements in type theory which is entirely valid, but which trips over our intuition: our training has taught us to take it, and expand it into an entirely different statement. We create blanks that aren’t there, fill them in, and by doing so, convert it into something that it isn’t, and confuse ourselves.

Lenovo and the Superfish Scam

I’m a bit late to the scene here, but I think it’s still worth making a stab at this. Also, this is being written on an airplane, where I am travelling home from San Francisco to NY with a case of bronchitis. So I am not exactly at my best.

Lenovo, one of the largest makers of windows-based laptops, sold out its customers as part of one of the worst deliberate violations of computer security I’ve ever seen, by shipping a piece of software called Superfish pre-installed on its computers. Superfish is, with absolutely no exaggeration, one of the most serious, unethical, despicable things I’ve seen in quite a lot time. It’s appalling.

So what is it, and what’s the big deal?

We need to start with some background, and talk a bit about how secure connections work on the internet.

Every time that you visit a website with a secure connection (a URL that starts with https), you’re using a protocol called TLS (formerly SSL). TLS is designed to do two things:

  1. Ensure that you are talking to who you think you’re talking to.
  2. Ensure that no one but you and the person you wanted to talk to can actually see what you’re saying.

The way that it does both of those is based on encryption. Every time you create a secure connect to a website, you’re exchanging credentials with the site to ensure that they’re who they say they are, and then based on those credentials, you establish an encryption key for the rest of your communication.

That connection-establishment process is the critical bit. You need to get some information that allows you to trust them to be who they claim to be. The security and integrity of everything that happens over the connection depends on the truth and integrity of that initial piece of identity verification.

The identity verification piece of TLS is built using public key cryptography, as part of a standard infrastructure for key maintenance and verification called X.509.

I’ve written about public key crypto before – see here. The basic idea behind public key crypto is that you’ve got two keys, called the public and private keys. Anything which is encrypted with the public key can only be decrypted with the private key; anything which is encrypted with the private key can only be decrypted using the public key. Your public key is available to anyone who wants it; no one but you has access to your private key.

If you receive a message from me, and you can decrypt it with my public key, then you know, without a doubt, that you can be sure that I’m the one who encrypted it. Only my private key could have encrypted a message that could then be decrypted with my public key.

For that to work, though, there’s one thing that you need to be sure of: you need to be absolutely sure that the public key that you’ve got is really my public key. If some clever person managed to somehow give you a different key, and convince you that it’s mine, then they can send you messages, and they’ll look exactly as if they came from me. If I handed you my public key on a USB thumbdrive, then you’re sure that the key came from me – but if you received it online, haw can you be sure that it was really me that gave it to you? Maybe someone switched it along the way?

In X.509, we use an idea of delegated trust. That is, we have some small collection of fundamental trusted authorities. Those authorities can issue public/private key pairs, so when someone need a public key, they can go to them and ask for it, and they’ll create one. The authority gives them a certificate, which is a copy of the new public key encrypted by the authority using their private key.

Now, when someone connects to a website, the target site can state who they are by sending a copy of the certificate. The client site recieves the certificate, decrypts it using the authorities public key, and then starts using that public key to encrypt their communications.

If the two sides can keep talking, then the client knows who it’s talking to. It got a public key, and it’s using that public key to talk to the server; so the server couldn’t decrypt the communication unless it had the public key; and it trusts that that it got the right public key, because it was encrypted with the private key of the certificate authority.

This is great as far as it goes, but it leaves us with a single certificate authority (or, at best, a small group). With billions of human users, and possibly trillions of networked devices, having a single authority isn’t manageable. They simple can’t produce enough keys for everyone. So we need to use our trust in the certificate authority to expand the pool of trust. We can do that by saying that if the certificate authority can declare that a particular entity is trustworthy, then we can use that entity itself as a verifier. So now we’ve taken a single trusted authority, and expanded that trust to a collection of places. Each of those newly trusted entities can now also issue new keys, and can certify their validity, by showing their certificate, plus the new encrypted public key. In general, anyone can issue a public key – and we can check its validity by looking at the chain of authorities that verified it, up to the root authority.

There’s a catch to this though: the base certificate providers. If you can trust them, then everything works: if you’ve got a certificate chain, you can use it to verify the identity of the party you’re talking to. But if there’s any question about the validity of the root certificate provider, if there’s any question whether or not you have the correct, valid public key for that provider, then you’re completely hosed. Ultimately, there’s some piece of seed information which you have to start off with. You need to accept the validity of an initial certificate authority based on some other mechanism. The people who sold you your computer, or the people who built your web browser, generally install a root certificate – basically the public key for a trusted certificate authority.

If that root certificate isn’t trustworthy, then nothing that results from it can be trusted. The untrustworthy root certificate can be used by an unscrupulous person to create new certificates allowing them to masquerade as anything that they want.

In particular, an untrustworthy root certificate it makes it easy to perform a man-in-the-middle attack. Suppose you want to talk to your bank, and somehow Joe Schmoe has planted a bad root certificate on your computer:

  1. You try to connect to Bank.
  2. Joe intercepts the request. He accepts your connection request, pretending to be the bank. Using his fake root certificate, he sends you a certificate claiming to be the bank.
  3. Now, you’re connected to Joe, believing that you’re connected to the bank. You try to log in to your account, sending your username and password.
  4. Joe receives your request, connects to the bank, passing on your request.
  5. Joe logs in to the bank on your behalf. The bank returns its successful login screen, showing your account numbers and balances.
  6. Joe copies the successful login screen from his connection to the bank to you.
  7. Every time that you send information to “the bank”, Joe decrypts it using his private key, and then sends it on to the bank masquerading as you. Similarly, when the bank sends something to you, he decrypts it using the banks public key, and then encrypts it with his his private key to send it to you.
  8. You now believe that you are connected to the bank over a secure connection. Everything that you do looks exactly as if you’re connected to the bank over a security encrypted link.
  9. Joe now has your login information, and control of your bank accounts.

In the Lenovo fiasco, Lenovo installed a system called Superfish, which deliberately installs a bad root certificate, and then uses that root certificate to create man-in-the-middle attacks on every secure connection ever made with the computer.

Why does it do that? Purportedly for ad retargeting: it uses its man-in-the-middle to decrypt supposedly secure connections so that it can replace ads in the pages that you view. That way, Lenovo and Superfish get advertising money for the pages you view, instead of the page-owners.

It’s spectacularly despicable. It’s fundamentally compromising everything you do with your computer. It’s doing it in a way that you can’t detect. And it’s doing it all in a sleazy attempt to steal advertising money.

Based on this, I’ve got two pieces of advice:

  1. Don’t ever buy a Lenovo computer. Never, ever, ever. A company that would even consider installing something like this isn’t a company that you can ever trust.
  2. If you have a Lenovo computer already, get rid of it. You could reformat your disks and install a fresh untainted copy of your operating system – but if your hardware manufacturer is a sleezy as Lenovo has demonstrated themselves to be, then there’s no reason to believe that that’s going to be sufficient.

This is, by far, the worst thing that I’ve ever seen a computer manufacturer do. They deserve to be run out of business for this.

Simpler Consensus with Raft

A few weeks ago, I wrote about Paxos, which is (at least in my experience), the most widely used algorithm for consensus in distributed systems. I’m a huge fan of Paxos – I think that it’s a remarkably elegant system.

But Paxos does have its problem.

  1. Paxos has a lot of roles: client, proposer, learner, acceptor, leader, follower. When you want to implement Paxos, you need to figure out all of those roles, and how you’re going to implement them. In general, you end up merging roles – but there are lots of ways of doing that merge. Each particular way of setting up the roles has its own properties, and thus its own tradeoffs that you need to understand.
  2. Paxos, as we normally talk about it, is really a single-consensus protocol – that is, the basic protocol is designed to get a group of agents to come to consensus just once. If you want to be able to repeatedly seek new consensus values, you’re actually going to be using an extension to the basic paxos protocol. There are a ton of Paxos extensions that work to add repeated consensus. Paxos itself is simple and elegant, with well-defined formal properties that we care about – the moment we start modifying it, we can no longer count on those properties unless we can also prove them in our extension!
  3. Paxos was originally described in a truly awful paper. Leslie Lamport was trying to write a paper that would be less dull than the typical bone-dry technical snoozer – but the way that he wrote it actually makes it much harder to understand.

In short: Paxos has more complexity than it needs, and despite that, it needs to be tweaked to be really useful, and getting those tweaks right is hard. There are, sadly, a lot of incorrect Paxos implementations – and their incorrectness has all-too-often come as a surprise to the people who rely on them.

To avoid those problems, there are other consensus algorithms out there. In this post, we’re going to look at one of the Paxos competitors – a consensus algorithm/protocol called raft.

Raft does away with the role complexity of Paxos. In Raft, you have a collection of cooperating agents. There are no distinct proposers, acceptors, or learners: there are just servers. Communication between the servers in raft is done entirely with synchronous remote procedure calls.

In Raft, the target of consensus is a log containing a sequence of events. The log is the history of the distributed system. The goal of raft is that the log be maintained in a consistent state throughout the raft network. Just like in Paxos, if we have 2n+1 servers, up to n can fail without the network losing its consistency.

Raft is designed in terms of remote procedure calls between the elements of the network. In Raft, we never talk about single messages – every communication between servers is a pair of messages: a request from caller to callee; and a response from callee to caller. When a message gets lost, we’ll just talk about it as a failed remote procedure call.

Within a Raft network, at any given time, each server has a state. It can be a follower, a leader, or a candidate. Within the network, there is at most one leader. When there is a leader, all of the other servers are in the follower state. The followers are almost entirely passive. Followers don’t talk to clients at all – they just wait for RPCs from the leader. The leader is the only participant that’s allowed to talk to clients: any client request must go through the current leader. The leader is also the only server that’s allowed to add new entries to the consensus log.

Raft divides time into a sequence of terms. In each term, the servers in the raft network need to select a leader using a process called an election. Raft is a strong leader protocol – no interactions with a client can take place except through a leader. If there’s no leader, then we can’t process client requests without a leader.

So, to understand Raft, there’s three processes that we need to

  1. Leader election
  2. Transitions between terms
  3. Appending an entry to a log.

In those processes, the servers have a collection of variables
that they use for the Raft protocol:

the current term for the server.
the serverID that this server voted for in the current term, or “none”.
the list of entries in the log.
The index of the highest log entry known to be committed by the server.
The index of the highest log entry that’s been added to the log – but not necessarily committed. (It doesn’t become committed until a majority of servers accept it.)

Leader Election

In each term, the Raft cluster needs to have a leader. The way that a leader is selected called election.

Elections are triggered by a term transition. When a server in the cluster decided that it needs to start a new term, it increments its term number, puts itself into the candidate state, and sends a RequestVote(term, candidateId) RPC to each of the other servers in the cluster. This request asks the other servers in the cluster to select it as the leader. If it receives enough “yes” votes, it will become the leader.

When a server receives a RequestVote RPC, it checks the term. If it’s smaller than the server’s current term, then it replies “No” – meaning that it cannot support the requestor as leader.

If the term in the request is greater that the receiver’s term, then the receiver cannot have voted in the new term. So it updates to the term from the request, and then it replies “Yes”.

If the term in the request equals the receiver’s term, then the receiver has already updated its term. If it’s already voted for someone else as leader, then it can’t support the requestor, so it replies “No”. If it hasn’t voted for a leader in the term, then it votes for the requestor, and replies “Yes”.

If the requester receives “Yes” votes from more than 1/2 of the cluster (counting itself), then it becomes the leader, and starts both processing requests from clients, and sending heartbeats to the other servers in the cluster.

If it doesn’t receive enough votes, then it waits to see if anyone else becomes the leader and starts sending heartbeats. If it doesn’t get a heartbeat in time, then it starts over: it would increment its term again, and try to start a fresh election.

Term Transitions

For a given server, term transitions happen in three ways:

  1. Timeout: the leading server needs to periodically communicate with each of the followers. This process is called heartbeat: even if the leader has no updates for its followers, it sends RPC calls to the followers just to say “I’m still here”. If a client goes too long without receiving a heartbeat, it decides that the leader was lost, and it will increment the term number, and trigger a new election.
  2. Leader resignation: the current leader can, at any time, decide to stop being the leader. (This is typically done by an implementation as part of a system that says that there’s a maximum period between leader elections. For example, in the Aurora scheduler, we had leader elections at least once per day. In a raft consensus, the leader would trigger this by deciding it was time for it to stop being a leader, and triggering an election by starting a new term.)
  3. External term change: every RPC received by a server includes a term number. If any RPC to a server ever includes a term number greater than the current term for that server, the server will update its term to the new number. As a special case of this, when a leader server decides to resign, it does that by sending an RPC to the other servers with an incremented term number.

Appending to the log

We just spent a fair bit of time talking about leaders and elections. That’s almost beside the point. What we really want to do is just maintain a consistent log across the cluster of servers. Everything except creating log entries is just the book-keeping that’s necessary to make the consistent log work. The log itself is maintained using the AppendEntries RPC call.

When a client request does something that alters the state of the cluster, the leader describes that change by adding an entry to the log. It builds a proposed log entry, and sends it to the other members of the cluster using an RPC. If it gets enough “Yes” votes from other cluster members, then the log entry becomes committed, and the leader updates its commitIndex to the index of the new log entry to reflect that.

The RPC request takes a bunch of parameters:

  1. term: the leader’s term.
  2. leaderId: the id of the leader.
  3. prevLogIndex: the index number of the last log entry in the consensus log preceeding this new entry.
  4. prevLogTerm: the number of the term where the last log entry was committed.
  5. entries: a set of new log entries to be appended to the log.
  6. leaderCommit: the index of the commitlog on the leader after this set of entries has been committed.

When an AppendEntries call is received by a follower, what it does is:

  • If the receiver’s term is greater than the request term, then the receiver rejects the request by replying “No”.
  • If the the receivers commit index is larger than the commit index of the request, then it rejects the request by replying “No”.
  • If the receiver’s log doesn’t contain an entry at prevLogIndex, or that entry’s term doesn’t match the request term, then it rejects the request by replying “No”.
  • If there’s an entry in the log with the same index as the new log entries, and the term in the request matches the receiver’s term, then the receiver removes all entries after prevLogIndex from its log.
  • The receiver then appends the new entries from the request to its log.
  • If the leaderCommit is greater than the commitIndex on the receiver, then the receiver updates its commitIndex.
  • Finally, the receiver replies “Yes”.

When a majority of the cluster members have accepted an AppendEntries call, then the log entry gets committed.

The one part of this that’s confusing is how the logs get managed. The leader creates a new log entry, and sends it to the other servers. The complexity comes from dealing with cases where something doesn’t reach consensus.

For example, the leader sends entries 5, 6, and 7 to server S. S adds the entries to its copy of log – it now contains [1, 2, 3, 4, 5, 6, 7]. Meanwhile, the leader also sends those entries to server T, but the RPC to T fails due to a network fault. Another client request happens, and now the leader sends [5, 6, 7, 8] to S. S sees that it’s got entry 5 already: so it discards everything after 5, and then re-appends.

So the trailing segment of the log can change! How do we handle consensus?

The next time that the leader sends an AppendEntries to a follower, it contains the leader’s commitIndex. The follower updates its commit index to that value. Once it’s done that, any request from a leader that tries to modify anything that comes before that commit index will be rejected.

The consensus commit thus doesn’t really occur until the next heartbeat call after a log update.

Raft versus Paxos

That’s the basics of Raft.

In comparison to Paxos, there’s a couple of things to notice:

  1. There’s a lot less confusion around roles. Paxos had a ton of different roles, and rules for interactions between the different roles. Raft doesn’t have any of that: it’s just servers, with one of the servers designated as the leader.
  2. Raft explicitly manages a log, and it adds complexity around log management. In Paxos, you’re just managing a single consensus value; in Raft, you’ve got a sequence of log entries.
  3. Paxos is defined in terms of messages; Raft is designed in terms of remote procedure calls.

So is Raft really simpler than Paxos? I think that’s up for discussion. Personally, I prefer Paxos. There’s a lot of complexity hidden under the covers of the RPC system. It looks simple on the surface, but all of the complexity of message passing, lost messages, message duplication – it’s still there. It’s just been swept under the carpet, as if that really makes it easier.

The way that the logs get maintained is confusing. That’s inevitable: getting distributed knowledge is never easy. Raft at least makes that part of things explicit, whereas it’s a common part of Paxos implementations, but it’s not really specified in the protocol.

Quick Vaccine Math

A friend of mine asked me to write about the math of vaccines. A lot of people have been talking about it lately, so I’m not really sure if I’ve got anything new to add, but I can at least give my usual mathy spin to it.

Vaccines have been getting a lot of attention lately, for good reason. There are a lot of people in America who’ve bought in to a bogus line about the supposed danger of vaccines, and the supposed benign-ness of the diseases that they can prevent. That’s led to many children not getting vaccinated as they should, which has culminated in a recent outbreak of measles caused by a contagious but not yet symptomatic child at DisneyLand.

When it comes to vaccines, there’s two things that a lot of people don’t understand. One is herd immunity; the other is probability of infection.

Herd immunity is a really important concept. In an ideal world, if you got vaccinated against measles, there’d be no chance that you’d ever catch it. But it doesn’t work that way. What a vaccine does is reduce the probability that you’ll catch the disease. Due to lots of random factors involving the way that a given individual’s immune system works, the vaccine can’t be perfect. Beyond that, there are also many people who either can’t be immunized, or whose immune system is not functioning correctly. For example, people who are getting chemotherapy for cancer have severely depressed immune systems, and even if they’ve been immunized, their immune systems aren’t capable of preventing the disease.

So just relying on the fact that you’ve been immunized isn’t really enough. To prevent outbreaks of the disease, we rely on an emergent property of a vaccinated population. If enough people are immune to the disease, then even if one person somehow gets infected with it, they won’t be able to cause it to spread.

Let’s walk through a simple example. Suppose we’ve got a disease where the vaccine is 95% effective – that is, 95 out of every hundred people who received it are completely immune to infection by it. Let’s also suppose that this is a highly infectious disease: out of every 100 non-immune people who are exposed to it, 95 will become ill. If everyone is immunized, how many people need to be exposed to a sick person in order for the disease to spread?

Infections turn into outbreaks when the number of infected people grows – if each sick person infects more than one other person, then the infection will start to grow exponentially. The severity of the outbreak will depend on how many people get infected by each sick person.

Suppose that the first sick person has contact with 20 people while they’re contagious. 95% of them are immune – which means that only one out of that twenty is succeptible. There’s a 95% chance that that person will get infected. This isn’t good, but if it’s kept to that rate, we won’t have an outbreak: each sick person will probably infect one other person on average – and not always even that. So the infection will die out without exploding into a significant outbreak.

What happens in 5% of the population doesn’t get vaccinated? Then the pool of infected people grows to 10%. And in our contrived example, we now have a 90% probability of the sick person making two other people ill. That’s more than enough to cause a major outbreak! On average, each sick person will cause 1.8 other people to become sick!

When the population’s immunity rate (either through vaccine, or through prior infection) gets to be high enough that an infection can no longer spread, the population is said to have herd immunity: even individuals who can’t be immunized no longer need to worry about catching it, because the population doesn’t have the capacity to spread it around in a major outbreak.

(In reality, the effectiveness of the measless vaccine really is in the 95 percent range – actually slightly higher than that; various sources estimate it somewhere between 95 and 97 percent effective! And the infectivity of most diseases is lower than the example above. Measles (which is a highly, highly contagious disease, far more contagious than most!) is estimated to infect between 80 and 90 percent of exposed non-immune people. So if enough people are immunized, herd immunity will take hold even with more that 20 people be exposed by every sick person.)

Moving on: there’s a paradox that some antivaccine people use in their arguments. If you look at an outbreak of an illness that we vaccinate for, you’ll frequently find that more vaccinated people become ill than unvaccinated. Therefore, they say, it’s not the fault of the unvaccinated. We’ll look at the math to see the problem with that.

Let’s use the same numbers as above: 95% vaccine effectiveness, 95% contagion. In addition, let’s say that 2% of people choose to go unvaccinated.

That means thats that 98% of the population has been immunized, and 95% of them are immune. So now 92% of the population has immunity.

If each infected person has contact with 20 other people, then we can expect expect 8% of those 20 to be infectable – or 1.6; and of those, 95% will become ill – or 1.52. So on average, each sick person will infect 1 1/2 other people. That’s enough to cause a significant outbreak. Without the non-immunized people, the infection rate is less than 1 – not enough to cause an outbreak. The non-immunized population reduced the herd immunity enough to cause an outbreak.

Now, within the population, how many immunized versus non-immunized people will get sick?

Out of every 100 people, there are 5 who got vaccinated, but aren’t immune. Out of that same 100 people, there are 2% that didn’t get vaccinated, or 2. So we’d expect that in 100 cases of the disease, about 70 of them to be vaccinated, and 30 unvaccinated.

The vaccinated population is much, much larger – 50 times larger! – than the unvaccinated, so we’d expect more vaccinated people to become ill, even though it’s the smaller unvaccinated group that broke the herd immunity! The easiest way to see that is to take those numbers, and normalize them.

In the vaccinated community, those 70 sick people are – in the worst possible case, where every single non-immune vaccinated person became ill! – the 5% non-immune from a population of 1400 people. So the worst possible infection rate in the vaccinated population is just 5% – and in reality, it’s more like 4.75%. But those 30 sick people from the unvaccinated pool are 30 out of about 32 non-immunized people who were exposed. The unvaccinated people were more than 20 times more likely to be infected.

The reality of vaccines is pretty simple.

  1. Vaccines are highly effective.
  2. The diseases that vaccines prevent are not benign.
  3. Vaccines are really, really safe. None of the horror stories told by anti-vaccine people have any basis in fact. Vaccines don’t damage your immune system, they don’t cause autism, and they don’t cause cancer.
  4. Not vaccinating your children (or yourself!) doesn’t just put you at risk for illness; it dramatically increases the chances of other people becoming ill. Even when more vaccinated people than unvaccinated become ill, that’s largely caused by the unvaccinated population.

In short: everyone who is healthy enough to be vaccinated should get vaccinated. If you don’t, you’re a despicable free-riding asshole who’s deliberately choosing to put not just yourself but other people at risk.

The Program is the Proof: Propositions in Type Theory

As usual, I’m going in several different directions. I’m going to continue doing data structure posts, but at the same time I also want to get back to the type theory stuff that I was writing about before I put the blog on hiatus.

So let’s get back to a bit of Martin-Loff type theory! (The stuff I’m writing about today corresponds, roughly, to chapter 2 of the Nordstrom/Petersson/Smith text.)

One of the key ideas of Martin-Loff’s type theory is that a logical statement is exactly the same thing as a specification of a computation. When you define a predicate like “Even”, the definition specifies both the set of numbers that satisfy the predicate, and the computation that tests a number for membership in the set of even numbers. If you haven’t provided enough information to fully specify the computation, then in Martin-Loff type theory, you haven’t defined a predicate or a set.

When you say “2 is even”, what you’re really saying in terms of the type theory is that “The computation for ‘even(2)’ will succeed”. The computation and the logical statement are the same thing.

In functional programming, we like to say that the program is the proof. Martin-Loff type theory is where that came from – and today we’re going to take a first look in detail at exactly what it means. In the world of type theory, the program is the proof, and the proof doesn’t exist without the program.

This creates an interesting set of equivalent interpretations. When you see a statement like “x : T” (or x \in T), that could be interpreted in the following ways, all of which are really equivalent in type theory.

  1. Set theoretic: x is a member of the set T.
  2. Intuitionistic: x is a proof object for the proposition T.
  3. Computational: x is a program that satisfies the specification T.
  4. Abstract: x is a solution for the problem T.

In the rest of this post, I’m going to focus on those four interpretations, and explain how each of them makes sense in this version of type theory.

The set theoretic interpretation is obvious – as the name suggests, it’s nothing but what we all learned from basic set theory. An object is a member of a set – which means, from set theory, that the object satisfies some predicate in first order predicate logic – because that’s what it means to be a member of a set.

The intuitionistic interpretation is almost the same as the set theoretic, but rewritten for intuitionistic logic. In intuitionistic logic, the predicate over the set is written as a proposition T, and if we know that x is a member of the set T, then that means that we have a proof that x demonstrates that T is true.

The computational interpretation takes the intuitionistic one, and rephrases it in computational terms. A logical proposition, rendered into a computational setting, is just a specification of a program; and a proof of the proposition is a program that satisfies the specification.

Finally, the abstract interpretation just rephrases the computational one into terms that aren’t tied to a computing device. A predicate is a problem that needs to be solved; anything that provides a solution to the problem is demonstrating a member of the set.

The key takeaway though is the basic idea here of what a type is. What we’re talking about as a type here is something that goes far beyond any programming language’s idea of what a type is. In intuitionistic type theory, a type is a specification of a computation. If we had this in a real language, that would mean that any program that typechecked would be guaranteed to work: asserting that x has type T means, precisely, that x is a computation that matches the specification!

(Of course, that’s not the panacea that you might think the first time you hear that. The catch is simple: the type is a specification of the computation. That means that just writing a type is a form of programming! And that means that your type descriptions are going to have bugs. But we’ll come back to that in a much later post.)

What type theory is doing is taking something like set theory, and re-rendering it entirely in a computational world. It still has a mapping from the computations to the abstract concepts that we use when we’re thinking, but if we can talk about those abstract concepts in type theory, we’ll always do it by mapping them into some kind of computation.

In type theory, we’re not dealing in a world of pure mathematical objects that exist if we can describe them; instead, we’re building a world where everything is at least theoretically computable. That might seem constraining, but every proof already corresponds to a computation of some sort; the only additional constraint here is that we can’t play tricks like the axiom of choice, where we can “prove” the existence of some unattainable, intangible, nonsensical object.

To make that work, we’re going to take all of the construct that we’re used to seeing in intuitionistic logic, and give them a meaning in terms of computations.

For example, in set theory, we can have a statement A \supset B – meaning that A is a superset of B, that every element of B is also necessarity an element of A. In type theory, since A and B are specifications of computations, that means that a member (or proof) of A \supset B is a computation that given a proof of A, generates a proof of B – in short, that A implies B.

Now, suppose that we want to prove A \supset A. How could we do that? We need a program that given a proof of A generates a proof of A. That is, we need an implementation of the identity function: \lambda a . a.

In fact, using the computation interpretation of things, we can interpret A \supset B as being the type of a function that takes an instance of a, and generates an instance of b – that is, that if f : (A \supset B), then f is a function from an instance of A to an instance of B!

The only trick to that is understanding that in type theory, saying that a is an element of A means that a is a proof of A. Using the same interpretation, that means that f: A \supset B means that f is a proof of A \supset B – which means the same thing as saying that given an example of an element of A (a proof of A), f will produce an element of B (a proof of B). The statement A \supset B is exactly the same thing as the logical implication A \rightarrow B, which is exactly the same thing as the type of a function from A to B.

Notes on sources: I’m working from two main references in this series of posts, both of which are available in PDF form online.

  1. “Programming in Martin-Lof’s Type Theory”, by Nordstrom, Petersson, and Smith, which you can download here.
  2. “Type Theory and Functional Programming” by Simon Thompson (available here).

In addition, I first learned a lot of this from reading some papers by Phil Wadler and Simon Peyton Jones. The exactly references but the exact references to those are long-lost in the shadows of my memory. But any of their papers are well-worth reading, so just read them all!)

Mathematical Data Structures Part 1: Binary Heaps

Being a PhD in computer science, it’s only natural that I love data structures. In particular, I’m fascinated by the way that the math factors in to the way we structure data. Data structures fit into a beautiful intersection between information theory and algorithms: the way that a good data structure is built is a reflection of what information it really needs to maintain. The best data structure encodes exactly the information it needs in order to do it’s job – no more, and no less. The mathematical impacts of that are beautiful, and sometimes surprising. To try to demonstrate that, I’m going to take a couple of posts, and work my way through one of my favorite examples of a surprising outcome in a structure called a fibonacci heap.

A heap is a structure designed to solve a common problem. You’ve got a collection of objects, each of which has an associated numeric value. You want, at any time, to be able to find and remove the largest value in the collection, and to be able to add new elements to it. Those two operations are the core of the heap. Some variations also allow you to increase the value of objects inside the heap, or to remove values other than the maximum.

There are a lot of different ways to implement a heap. One obvious one is to just maintain a sorted sequence of objects. The problem with that is performance: some of the common operations are painfully slow!

Using the sorted sequence approach, removing the largest value is easy: you just remove the last element of the sequence. That’s very fast: it’s constant time. But you also need to be able to add values to the heap, and that’s not so good.

There’s two basic ways of doing a sequence: an array, or a linked list. In both cases, the performance isn’t acceptable. If we used an array,then in order to add a new object to the collection, we’d need to:

  1. Find the correct position for it in the array. We can do that by doing a binary search, which takes time O(lg n) where n is the length of the array. This step isn’t bad – in general, we’re pretty happy with O(lg n) operations.
  2. Insert the value into the array – which means shifting all of the elements that come after it one place to the right. That’s O(n) time, which is pretty crappy.

In the linked list approach, inserting the value isn’t a problem – it’s a constant time operation. But finding the position where it should be inserted is linear time. So we’re still talking about linear time.

Similarly, we could use a linked list, where inserting the element is constant time, but then finding its position is O(n) – again, unacceptable.

The problem with the sorted sequence approach isn’t really related to the kind of structure we use to maintain the sorted list; the problem is that we’re maintaining more information that we need. At any time, we want to be able to find the largest element of the heap quickly – we don’t care about the relative positions of any pair of values that don’t include the largest element of the collection! But if we keep a sorted list, every time we insert an element, we’re spending a lot of time comparing things whose comparison we don’t really care about!

To be able to make it faster, we need to build a data structure that doesn’t waste time and effort computing and maintaining information that we don’t want.

So our goal is to find ways of building structures that always let us both find the largest element quickly, and add new elements quickly, without maintaining more information that is really necessary. We’ll start off with a simple but good version, and then work our way through to better ones.

The most basic implementation of a heap is called a binary heap. A binary heap is a binary tree with two key properties:

  1. Every node in the tree is larger than its children.
  2. The tree is left-full: every level of the tree is full except for the last; and the last level is filled in from left to right.

The left-full property might seem a bit strange, but it turns out to be pretty straightforward. A binary heap can be implemented using an array. The root node is stored in the first position of the array; its children are in positions 2 and 3; the children of node 2 are stored in positions 4 and 5; the childen of position 3 are stored in positions 6 and 7. Using one-based indices, for any node N, it’s children are stored in positions 2N and 2N+1. Adding a new leaf to the tree can always be done by just appending one value to the array. The left-full property just means that you always extend the array by adding an element onto the left.

Implementing a heap this way is simple:

  1. To get the maximum value, you just look at the first element of the array – O(1).
  2. To remove the largest element from the array, you get the value from the first element of the array, and save it. Then you remove the last element from the array, and bubble it down – swapping it with one of its children if they’re bigger than it. We’ll look at this in more detail, but the bubble down process is O(lg n) in the worst case.
  3. Inserting a new element is done by adding it to the end of the array, and then bubbling up, by comparing it to its parent, and swapping if it’s bigger than its parent. Again, it’s O(lg n).

I’m going to show code for this. For fun, I wrote the code in a language called xtend. Xtend is a Java extension that cleans up the syntax, gets rid of semicolons, improves the type system, adds lambdas, and does a few other really neat things.

The whole beast is just a wrapper around an array:

class BinHeap> {
  val ArrayList _contents

  new() {
    _contents = new ArrayList()


If you know Java, this is mostly clear. In xtend, you write constructors using the name “new” instead of the name of the class being constructed.

Then we’ll set up some utilities to make other stuff easier to write.

  def leftChildPosition(int pos) {
    2 * (pos + 1) - 1

  def rightChildPosition(int pos) {
    2 * (pos + 1)

  def int parentPosition(int pos) {
    if (pos == 0) {
      throw new MaxHeapException()
    } else {
      (pos + 1)/ 2 - 1

  def void swap(int one, int two) {
    val T first = _contents.get(one)
    _contents.set(one, _contents.get(two))
    _contents.set(two, first)

Again, these should be straightforward. The only tricky thing is that the JVM uses zero-based arrays – so the left child of the node in position N is (2*(N+1) - 1): we need to add one to the node number to shift to one-based position; and then subtract one from the result to switch back to zero-based position. We do a similar thing for each of the other position computations.

Now we can get to the interesting bits. How do we get values into the heap?

def insert(T v) {
  val idx = _contents.size()

Insert is exactly what I described in prose above: append the new value onto the end of the array, and then bubble it up. Bubbling is the interesting part:

  private def void bubbleUp(int pos) {
    if (pos > 0) {
      val parentPos = parentPosition(pos)
      if (_contents.get(pos) > _contents.get(parentPos)) {
        swap(pos, parentPos)

Bubbling up from a position P compares P to its parent. If it’s bigger than its parent, it swaps positions with the parent, and then tries to continue bubbling up from its new position.

For example, imagine we had a tree like:


Now, suppose we wanted to add the value “10” to this. We’d add 10 to the end of the array, which would make it a child of 6. That would give us:


So, we’d compare 10 to its parent – it’s bigger, so we’d swap:


Then we’d compare 10 to its new parent, 8. It’s bigger, so we swap:


And finally, we’d compare 10 to its new parent, 9. It’s bigger so we swap, and then we’re done.


Appending to the end of the array is constant time, so the dominant time cost is the bubbling. The maximum possible number of swaps in the doubling process is depth of the tree minus 1 – and the depth of a full binary tree with N members is \lceil ln N \rceil. So it’s O(lg n) swaps, and the overall cost of inserts is O(lg n).

Getting the largest value is trivial:

def getMax() {

Removing the largest value is a lot like adding a value: we really play with the last element of the array, and then do a bubbling process – only this time we’ll bubble in the opposite direction:

  def removeMax() {
    if (_contents.size == 0) {
      throw new MaxHeapException()
    } else {
      val result = getMax()
      val last = _contents.remove(_contents.size() - 1)
      if (_contents.size() > 0) {
        _contents.set(0, last)

Bubbling down is similar to bubbling up, but it’s a bit more complicated, because we need to look at both children.

  private def void bubbleDown(int pos) {
    val rightChildPos = rightChildPosition(pos)
      val leftChildPos = leftChildPosition(pos)
      if (leftChildPos >= _contents.size) {
      // Try to bubble left if there is no right child, or if the lift child is
      // bigger than the right.
      if (rightChildPos >= _contents.size || _contents.get(leftChildPos) > _contents.get(rightChildPos)) {
        if (_contents.get(pos) < _contents.get(leftChildPos)) {
          swap(pos, leftChildPos)

      } else {
        // Try to bubble right
        if (_contents.get(pos) < _contents.get(rightChildPos)) {
          swap(pos, rightChildPos)

The process is almost the same as bubbling up, but moving in the opposite direction. We're starting with a parent node, and comparing it to its children. If it's bigger than either of its children, then we swap it with the largest child, and then continue bubbling down.

For example, let's look at the same heap we looked at for insert:


If we want to remove 9, we set the value 9 aside, and then remove 3 from the end of the array, and put it at the root of the tree:


Then we'd compare 3 against its two children, 8 and 7. Since 8 is the larger child, we swap 8 for 3:


Now we compare 3 with its new children, 5 and 6. 6 is bigger, so we swap 6 with 3:


3 has no children, so we're done: it's bubbled down as far as it can go.

Note: I messed up this example in the original version of the post. Thanks to John Armstrong for pointing it out.

The cost here is the same as insert, for the same reason. The dominant cost is the bubbling, and the bubbling is bounded by the depth of the tree. So removing the maximum is also O(lg n).

It's worth noting that heaps can be used to build a very reasonable sorting algorithm. To sort a collection, just insert all of the elements of the collection, and then remove them one by one. It's O(n lg n), and it's conceptually quite simple. It's not widely used, because the old classic quicksort is faster - not in big(0) notation, but it ends up with a smaller constant. (In big-O notation, something that takes 3(lg n) steps and something that takes 6(lg n) steps are both O(lg n), but the one whose constant is 3 is still twice as fast as the one whose constant is 6.)

The beauty of math; the humor of stupidity.

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