Reuben Cohn-Gordon

Functional Reactive Probabilistic Programming

2026-09-22T18:40:00Z

Interactive demos: Rhine-Bayes on the web

The Unfolding of Language

2026-09-21T21:00:00Z

How languages come to exist, treated as a dynamical systems problem.

The One with the Common Ground

2020-01-26T16:07:24Z

The One with the Common Ground

26 January 2020

“They think they are so slick messing with us. But they don’t know that we know that they know”

Somewhere in my PhD between feeling anxious about research and feeling frustrated by it, there was this halcyon time period (on reflection, no actual continuous stretch of time but more like a badly pieced-together action montage) where I paced around poorly lit rooms and thought really hard about the way common ground works in language.

The common ground is this shimmering mirage of a concept. It’s the information everyone in a conversation not only knows but acts as if everyone else knows (and knows they know, and so on), and it sort of only exists on the basis of a shared pretense perpetuated by all the people taking part. Really there’s only one cultural touchstone which can make this tangible, and I’m going to surprise you by saying that no, it’s not Friends.

Okay, I lied, it’s Friends. What happens is that Monica and Chandler (party 1) are covertly sleeping together - Joey knows but that’s it. Joey for the purposes of this example is not relevant. For reasons that will become apparent, it will be convenient to call this fact, that Chandler and Monica are, both literally and figuratively in bed together: \(\phi_0\). That is, \(\phi_0\) is the proposition: Monica and Chandler are sleeping together.

Then Rachel and Phoebe (party 2) find out. Let’s denote by \(\phi_1\) the proposition: Party 2 knows \(\phi_0\). So \(\phi_1\) and furthermore, party 1 do not know that \(\phi_1\). In fact, maybe that deserves a handy shorthand too; let’s say that \(\psi_1\) is the proposition: Party 1 do not know \(\phi_1\).

In a commendably sly turn, party 2 realize that \(\psi_1\) makes the situation ripe for trickery (“we could not tell them we know, and have a little fun of our own”). Party 2’s plan is that Phoebe will maintain the pretense that \(\phi_0\) is false, so as to preserve \(\psi_1\), but do so in such a way as to mess with Chandler and Monica. So the upshot is that Phoebe flirts with Chandler, who is unaware that this is a pretense, and is taken aback.

So far, so good. But this is where the epistemic game theorists must have infiltrated the script writers’ meeting, because then party 1 infer that party 2’s weird behavior can only be explained by \(\phi_1\) being true. Immediately, \(\psi_1\) ceases to be true, but \(\phi_2\) and \(\psi_2\) become true instead.

For the sake of clarity, and I mean that in the broadest possible sense, I’ll introduce the following definitions. Here, \(X_n\) is 2 when \(n\) is odd, and \(1\) when \(n\) is even (for \(n>0\)):

\(\phi_n\): Party \(X_n\) know that \(\phi_{n-1}\).
\(\psi_n\): Party \(X_{n+1}\) do not know that \(\phi_{n}\).

“The messers become the messees” and Chandler flirts back, alarming Phoebe who then is led to believe that his commitment to Monica is somewhat less than exemplary. This eventually makes her realize that of course! they know we know (\(\phi_2\)), but - aha!! - now we know that (\(\phi_3\)), and they don’t know we know they know we know (\(\psi_3\)). One again armed with the upper hand, Phoebe accepts Chandler invitation to “have all the sex”, beckoning in the final act of the episode.

To paraphrase Lemony Snicket, I don’t need to drag you any further through the predictable details of this narrative, other than to say that eventually Chandler abandons the pretense, confesses his love for Monica, and, as Joey exclaims in self-aware theatricality, like the chorus at the end of Oedipus Rex, finally everyone knows! (Except Ross, cue the after-credit scene).

I am writing this right after turning 26, on the eve of moving to a new city and starting a new job, so I rewatched The One Where Everybody Finds Out with a closer-than-normal eye for its tropes and a strange sense of whatever the reverse of foreboding is (watching the future solidify from a fine mist to a concrete slab, but without a sense of dread?)

But inner turmoil aside, let me pause here to take stock (I hate to, but may I?). Granted, the convoluted plot of the episode is noteworthy for showcasing the potential nuance of social interaction (obviously the show has made a comical point out of this nuance, a point which, yes, I am knowingly grinding into a fine dust), but how does it shed light on the idea of the common ground?

The idea is this: by the end of the episode, \(\phi_0\), the fact that Monica and Chandler are dating, is in the common ground (among Rachel, Phoebe, Joey, Monica and Chandler). But at every stage before this, even though everyone knows \(\phi_0\), and then knows that everyone knows, and so on, it isn’t in the common ground, as shown by the fact that no-one acts like it’s true. In a sense, the episode could almost have been designed as a thought experiment to show that any degree of higher order knowledge of a fact (we know that they know that…) is qualitatively different to that fact being in the common ground

But why spend time thinking about such a niche idea as common ground anyway? The answer, maybe predictably, is that it’s not niche at all. The notion of common ground doesn’t just arise in the contrived scenario above, but rather it’s a basic element of communication generally.

Everything from the rule that we drive on the right (or left) to the fact that certain words refer to certain objects is not just something everyone knows, but something that everyone knows everyone knows, and so on.

For example, I know what “My hair is on fire - please help” means, and you know what “My hair is on fire - please help” means, but in order for say it to any effect, I must also assume that you know, and in hearing my frenzied request, you must assume that I know you know, and have issued this series of strangely but deliberately choreographed noises with the intent of it being understood to mean that my hair is on fire (please help). To put it much more succinctly, the convention that words mean particular things is true only because everyone acts according to that convention. It’s not enough that we all know English - our knowledge of English has to also be in the common ground.

Actually, that leads to a working definition what the common ground is, namely: The common ground between some set of people is set of all facts that they all know and that everyone assumes are in the common ground. (Note how this definition is recursive)

One of the consequences of this definition is that the common ground \(C\) is such that some fact \(x\) being in \(C\) and everyone acting as if \(x\in C\) amount to the same thing. There is something profoundly abnormal about that: imagine if everyone acting like it never rains in England actually makes it true. Or imagine playing a version of Guess the Weight of the Fruit Cake¹ in which the cake’s weight depends on what everyone believes the weight to be.

In that vein, I like to think of the common ground from a sort of magical realist point of view, as if it’s this mutable object which metamorphoses into what everyone acts as if it is, like in the book American Gods, where the old-world gods disappear when people stop believing in them.

There are many questions you have, ranging from: how do you build a precise (let alone testable) model of communication that encodes the idea of common ground in the right way? to: if he can get this much mileage out of the social nuances of Friends, would he ever shut up about Seinfeld?

And then there’s one question best put by Baldrick, to Blackadder:

Baldrick: the way I see it, these days there’s a war on, right, and ages ago there wasn’t, right, so there must have been a moment when there not being a war on went away and there being a war on came along, so, what I want to know is, how did we get from the one case of affairs to the other case of affairs?

Blackadder: Do you mean, how did the war start?

Baldrick, in the tradition of philosophers the world over, has phrased his simple question with so much care that it seems complicated. He’s asking about how the 1st world war started, but in this case, this analogous question might be:

How did we go from the one case of affairs, where \(\phi_0\) wasn’t in the common ground, to another state, where it was?

How, in fact, does anything enter the common ground?

Let me make the following broad claim: things can enter the common ground in (at least) two ways. The first is by some collectively witnessed event. For example, if Joey (now counterfactually relevant) had said in the presence of both parties: “I assert that \(\phi_0\) is true” (this is the sort of offhand remark that Joey often makes, to reliable comic effect), then \(\phi_0\) would be in the common ground. This is the power of public announcements, like group emails, or shouting “I’m so lonely” in the middle of a large crowd². Chandler’s concession to Phoebe at the end of the episode has the same effect re. \(\phi_0\).

As an aside, this first way of things entering the common ground requires that the triggering event be sufficiently “strong” evidence. For example, certain events which provide evidence towards \(\phi_0\), like party 2 walking in on party 1 standing unusually close, would be sufficiently compatible with \(\phi_0\) being false that both parties could, in a strangely collaborative act, tacitly agree to ignore it. This, by the way, seems to be what happens when something very awkward happens at a social gathering: all parties simply agree that it in fact has not happened. (Think: someone forgetting your name at a cocktail party and you acting as in they have not forgotten and taking pains not to put them in a position in which this lack of knowledge will be clearly exposed). Some events, on the other hand, like party 2 walking in on party 1 in flagrante delictu - I’m assuming that if unfamiliar, the meaning of this phrase is recoverable from context - would be just too compelling evidence for either party to ignore. That is, Rachel and Monica couldn’t walk in on Chandler and Monica having sex, make eye contact, leave the room, and then pretend that nothing had happened. Actually, as I write that, I realize that pretending you have not walked in on someone having sex is an extremely plausible scenario, so I’ll revise my point. What I really mean is just that based on my model of how the cast of Friends would behave in a counterfactual version of this episode, I think this scenario would trigger an end to the pretense that \(\phi_0\) is false³.

So things can enter the common ground through collectively witnessed events. What about the second way? The second way is stranger. In a nutshell it’s that everyone in a conversation can simply pretend that something was already in the common ground, and lo and behold, it starts to be there, by virtue of always having been there.

I’ll need another example. In the interest of continuity, let’s say it’s also from Friends, but as this one will be strictly contrived, so let’s say it’s from the as yet unreleased, much feted (but sadly delayed) Friends reunion show.

We start watching with baited breath as the screen pans from the New York skyline, much as it ever was, to Phoebe and Joey holding umbrellas and standing in a fountain. Finally, the significance of the opening credits scene is going to be revealed.

This is their first time seeing each other in years, so as the dialog begins, it’s unclear what is and isn’t in the common ground.

Phoebe might not be sure, for example, if Joey remembers her song Smelly Cat. Perhaps, at one point, their conversation goes like this:

Phoebe: I played Smelly Cat the other day

Joey: no way!

This exchange shores things up: the existence of Smelly Cat is back in the common ground, on account of Phoebe acting as if it already was in the common ground, and Joey acting the same way too.

What’s interesting about this is that you can assume information was already in the common ground even if it couldn’t have possibly been there before.

For example, suppose Joey wants it in the common ground that he is married. He could announce this fact - that was method one for entering something into the common ground - but he could also just say:

The other day, my partner said the funniest thing to me…

And Phoebe might reply:

Oh, what was that?

In this exchange, Joey has simply acted as if it was already in common ground that he was married, and Phoebe has not challenged him. His marriage is a presupposition of his utterance. The conversation continues, but the fact of Joey’s marriage has slid into the DMs of their common ground.

To see that this way of entering a fact into the common ground is different from a direct announcement, note that there are some things which, when asserted directly are just surprising, but when presupposed, make you sound unhinged.

For example: if say to a new acquaintance: “I have a pet alpaca”, then sure, they may be surprised or even doubtful, but if I say, without first mentioning the previous fact: “Last week, while I was taking my pet alpaca for a walk, I had an interesting idea”, you would be confused, because while it’s implausible that I have a pet alpaca, it is much much more implausible that the fact of my having one was already in the common ground. (This is context dependent: if this conversation took place at an alpaca conference, the presupposition of alpaca ownership might be downright plausible.)

I’ll wind up with a second example, of a great party game (to my knowledge nameless) which works like this. Person 1 starts with a sufficiently vague statement about an event, like:

Gosh, wasn’t that thing we went to last week fantastic?

To be clear, there is no actual event - the point of the game is that both people just make it up as they go along. As such, Person 2 responds in assent, accommodating the presupposition that they shared this collective experience, and maybe adding some details:

Oh man, that was so good - I love the way they were dressed!

The conversation continues in this vein until, after a while, both participants feel they have established to some degree of clarity what musical/flood/funeral/baking show they collaboratively conjured out of the ether, and compare notes. (I really love this game, but admittedly, it is also the kind of thing I imagine people who describe themselves as thespians doing in large groups).

If there’s any punchline, it’s that language is exactly like this game. No-one goes through the dictionary, carefully checking with the friends, neighbors and loved ones that they are all in agreement about the meanings of each word. But without doing this, we all agree that there is some incredibly complex and baroque set of rules, that we already are in agreement on those rules, and then just sort of talk to each other on this basis. In my soon-to-be-professional opinion, having thought about it for several years, I can confirm that language is pretty odd.

Game played at an English country fair, rules self-explanatory↩︎
In fact, this puzzle and the follow up discussion are a good, if obviously idealized, example of how public announcements of a proposition can have a very surprising effect even if that proposition was already known by everyone (but not in the common ground). If you read it, consider that in the analogy with Friends, Joey plays the role of the impartial Guru - ostensibly a flesh and blood human, but for all intents and purposes exempt from the complexity of the social machinations.↩︎
You might wonder (and indeed I think this is an empirically fascinating question which at some point in the future we will have the tools to investigate rigorously): what is the threshold where an event provides sufficiently strong evidence of \(\phi_0\) that it enters the common ground? In other words, how undeniable does the evidence have to be before all parties abandon any pretense? I think, if I had to really pinpoint why I’m fixating on this point, it is because strength of evidence is a continuous variable, but the effect of something entering the common ground is not continuous. That is, you can imagine increasingly strong signs of Monica and Chandler’s romance being ignored until some threshold, at which point \(\phi_0\) enters the common ground and everyone starts to behave entirely differently, sort of like a phase transition.↩︎

Dependent Types

2020-01-26T16:07:24Z

Dependent Types

26 January 2020

Dependent types are most easily understood by looking at a program that requires them. I’ll first give a contrived example, then a real one.

Contrived example

Suppose we want a function that takes an integer, and returns it if it is less than \(10\), and otherwise returns the unit value ().

Haskell functions must have a single return type, so you could either have Int -> Int or Int -> (), but a function cannot sometimes return an Int and sometimes return a (), depending on the input.

The standard solution is to return a sum type, i.e. Int -> Either () Int. This works, but it is then the job of the consumer of this function to handle the Either, and moreover, we don’t have a static guarantee that when the input is less than \(10\), the returned Either () Int is really a Left ().

Dependent types allow a degree of expressivity which solves this problem. In pseudocode, the desired type would be:

foreach (x :: Int) -> if x < 10 then x else ()

Two key things to note about this:

the types here are treated like values, to the extent that we can write an if-statement involving types
the foreach keyword associates a value with the type Int, so that this value can be used later on in the type.

This isn’t a valid Haskell type, and won’t be for a while.

Real example

Suppose I want to write an interpreter. The idea is that I’ll parse my input expression (say “4 + 3 = 8”) into a syntax tree like:

       /\
      /  \
     /    \
    /\     \  
   /\ \    /\  
  4 +  3  =  8

I then assign a value to each leaf, and recursively compute the nodes of the tree until I get to the top.

But what values should I assign to the leafs? The typical functional answer would be:

“4” –> 4 :: Int
“+” –> (+) :: (Int -> Int -> Int)
“3” –> 3 :: Int
“=” –> (=) :: (Int -> Int -> Bool)
“8” –> 8 :: Int

Think of this as a lexicon, which takes strings to their meanings.

But note that the values have different types. So if I want a function that implemented this lexicon, i.e. that takes the strings and returns their meaning, without dependent types I would need to make a sum type like:

data Expression =
    Number Int
    | Operator (Int -> Int -> Int)
    | Prop (Int -> Int -> Bool)

and then a lexicon of type:

type LexiconNaive = String -> Expression

This is a workable solution, but it causes pain points down the road. In particular, when it comes to combining nodes of the tree, one has the awkward task of pattern matching on all possible cases, and throwing an error if neither the lefthand expression can be applied to the right nor vice versa. This scales poorly.

A nicer solution is dependently typed, and uses the implementation of dependent types from the singletons library (which the reader need not understand in depth):

{-# LANGUAGE TypeOperators, PatternSynonyms, TypeFamilies, FlexibleContexts, TypeApplications, ScopedTypeVariables,
    GADTs, AllowAmbiguousTypes, DataKinds, TemplateHaskell, StandaloneKindSignatures, PolyKinds #-}

data Category =  P | N | BS Category Category | FS Category Category
genSingletons [''Category]

type family Semantic (s :: Category) where  
    Semantic N  = Int
    Semantic P = Bool
    Semantic (FS a b) = (Semantic b -> Semantic a)
    Semantic (BS a b) = (Semantic b -> Semantic a)

type Syntactic a = Sing a

Category is a new type, but also gets lifted, so that it’s a new kind. We then write a type family, Semantic, on types of this kind, that takes a type like N (standing for “number”) and gives the type that a number’s meaning should have, which for our purposes is Int.

Syntactic a is a convenient synonym for the singleton library’s Type-kinded Sing a.

With this, we can now give a dependent type for lexicons, which beautifully expresses the intent:

type Lexicon = forall (a :: Category). Syntactic a -> Text -> Semantic a

An example of a lexicon, as we might need for an interpreter, is:

example :: Maybe Integer
example = (fmap . fmap) sum Just [1, 2, 3]

exampleLexicon :: Lexicon
exampleLexicon = \case
    Number -> \case
        "1" -> 1
        "2" -> 2
        "3" -> 3
        "4" -> 4
        "5" -> 5
        "6" -> 6
        "7" -> 7
        "8" -> 8
        "9" -> 9

    Proposition -> \case
        "True" -> True
        "False" -> False


    Number `ForwardSlash` Number -> \case
        "-" -> negate

    (Proposition `BackSlash` Number ) `ForwardSlash` Number  -> \case
        "=" -> (==)

    (Number `ForwardSlash` Number ) `BackSlash` Number  -> \case
        "+" -> (+)

    _ -> error "not implemented"

data Category =  
    Proposition
    | Number
    | Branch Category Category

genSingletons [''Category]

type family Semantic e (s :: Category) where    Semantic e Number  = e
    Semantic _ Proposition = Bool
    Semantic e (Branch a b) = (Semantic e b -> Semantic e a)

For instance:

examples
exampleLexicon Proposition "True"
>>> True
exampleLexicon Number "3"
>>> 3
exampleLexicon (Number `To` Number) "-"
>>> No instance for (Show (Int -> Int))
(exampleLexicon (Number `To` Number) "-") 5
>>> (-5)

Social Reasoning in Arcadia

2018-06-26T16:07:24Z

Social Reasoning in Arcadia

26 June 2018

A tour of the Bayesian perspective on pragmatics

This is an introduction to the nested reasoning models (I think that you think that I think…) that I worked on in my Phd. I’ve tried to make this light on mathematical detail (barring the occasional technical digression) in favour of the big picture point, that Bayesian inference and nested reasoning are really great tools for thinking about language and meaning.

Picture the scene: it’s midday in Arcadia and Echo is waiting for Narcissus to finish his lengthy beauty routine:

Echo: Will you be done soon?

Narcissus: Don’t hold your breath.

We gather from Narcissus’ response that the answer is no, but how? After all, if Echo had asked “what is a useful piece of advice when deep sea diving”, Narcissus’ reply would take on a totally different character. So it would seem that the meaning we infer from Narcissus’ utterance depends not only on the utterance itself, but the context in which it is said.

Meaning, to understate the issue, is a bit of a head scratcher. We could venture to say that Narcissus’ statement has semantic content (it’s a recommendation to not hold your breath), and meanings in different contexts (“No I won’t be ready any time soon.”, “Holding your breath is a poor way to dive.”), which is inferred from the context. (Deciding what content belongs to the statement as opposed to the context is often tricky. For instance, “don’t hold your breath” is an idiom in English - the meaning “It will take a long time” is at least somewhat baked into its semantic content.)

This inference to obtain the “full” meaning from the semantic content and context, which we make so easily, is complicated to spell out, even if a vague, informal way:

if Narcissus had been able to answer truthfully that he would be done soon, he would have done, but he did not. Given that, and since “Don’t hold your breath” is relevant advice in situations where you’re going to have to wait a long time, it seems that Narcissus is trying to convey that he won’t be done soon.

The study of the semantic content belongs to the field of semantics, while the richer meaning derived from reasoning about the situation - the pragmatic content - is the focus of pragmatics (The distinction was described as follows by a student in a class I TA’ed: “Semantics is about what things mean, and pragmatics is about what they like actually mean.”. Relatedly, I’ve always wanted to write a philosophy paper introducing a like actually operator LA, such that LA(P) iff P is like actually true.). The hope is that by factoring the relation between language and the world into these two pieces, we can simplify and refine our understanding of how that relationship works.

I want to show how we can boil down the essence of the above reasoning, and get out a paradigm for formalizing pragmatics which involves nested inference: an inference about another agent’s inference. What I’ll describe is the Rational Speech Acts (RSA) paradigm, and comes from previous work on game theory going back to Lewis.

My goal here is to show why the language of Bayesian probability (and in particular, recursive inference models like RSA) are the Right Tool for the Job: they neatly incorporate and generalize a logical semantics, can be computationally modeled, experimentally tested, integrated with machine learning, and what for my money matters most, formalize the essence of social reasoning. In short, they are for pragmatics what classical logic is for semantics.

Starting Simple

OK, so in the interest of tractability, let’s exchange our complex example from above for something much simpler and more mundane. Our Classical couple are looking for two sheep that have wandered off from the herd (shepherding is really the only profession in Arcadia):

Echo: Did you find both sheep?

Narcissus: I found one of them.

This example, or something similar, is the drosophila of pragmatics: a scalar implicature. It’s like the previous example in that the semantic and pragmatic content differ; Narcissus doesn’t explicitly say that he didn’t find both of them. That’s a state of affairs compatible with his utterance. After all, if you’ve found both, it’s true that you’ve found one. A logician might represent the semantic content of his utterance as

\[ (1) \quad \exists s. found(N,s)\]

However, if it was true that he’d found both sheep, he would have said as much, so we can infer that he found one but not both. We could represent this pragmatic content as

\[(2) \quad \exists s. found(N,s) \wedge \neg\forall s. found(N,s)\]

Note that (1) does not logically imply (2). That is, knowing that (1) is true is not enough in itself to know that (2) is true. And yet, we do know, or at least strongly suspect (2) is the case on the basis of Narcissus’ utterance.

So if we can’t get from (1) to (2) by logical means, we’ll need something else, capable of representing the counterfactual reasoning: “since (2) is a more informative statement than (1) (on account of implying (1)), if Narcissus been in a position to say (2), he would have. But he didn’t, so he wasn’t.”.

Fortunately, this example is simple enough that we can formalize it - i.e. build a model (with nice interactive code) which captures everything we are presently interested in about it. So let’s.

Preliminaries

We’re going to refer to the set of all utterances as U. U represents all the things Narcissus could have said as a reply to Echo’s question. Because this is a simple model, we’ll replace the infinitude of possible utterances with a more modest number, 2. U = { I found one of the sheep, I found both of the sheep}.

There’s another set we need to consider, the set W of all possible states (i.e. things which could be the case in the world). Again, our present purposes allow us to keep this simple too

one: the state in which Narcissus has found exactly 1 sheep
two: the state in which Narcissus has found exactly 2 sheep

We’re now in a position to talk about literal meaning. In a state w \(\in\) W, an utterance is either true or false. For example, if Narcissus has found one sheep, so that the world state is one, then saying I found both of the sheep is untrue. He’d be deluded or deceitful to say it.

OK, so formally, that all means that the semantics is a relation, which is a function of type \(((U,W)\to\\{\mathit{True},\mathit{False}\\}\)) (we write \(\llbracket u \rrbracket(w)\) to mean that the thing u means is compatible with the world w). More visually, a world w and an utterance u are related if there’s a line between them, as in the diagram below:

To make things a bit more interactive, here’s some code to play with in a probabilistic programming language (this introduction uses PPLs to model cognition) which represents the semantics. Nothing probabilistic yet, but WebPPL will feature again below in a more sophisticated capacity.

var worlds = [{totalSheepFound:1},{totalSheepFound:2}]
var utterances = [
	"I found one of the sheep",
	"I found both of the sheep"]

var meaning = function(utterance, world){
  (utterance === "I found one of the sheep")
  && (world['totalSheepFound']>0)  ? true :
  (utterance === "I found both of the sheep")
  && (world['totalSheepFound']==2)  ? true :
  false}

meaning("I found both of the sheep",{totalSheepFound:1})

Overview of The Model

And now for the Bayesian part. We’ll start by modeling literal interpretation, via a model I’ll call \(L_0\), which is hardly anything more than the semantics we already have in a slightly different shape. We’ll use \(L_0\) to build a model of production (i.e. choice of utterance given world state) called \(S_1\), which in turn we’ll use to build our end goal, \(L_1\). \(L_1\) is a model of interpretation which accounts not just for semantic meaning, but for pragmatic meaning. We can think of \(L_1\) as a model which reasons about a speaker \(S_1\) which is itself reasoning about \(L_0\). Sorry if that’s a bit of a mouthful. The big picture idea is that by reasoning about your interlocutor reasoning about you, you can infer extra, pragmatic, meaning beyond the semantic content of what you hear.

This image graphically represents the overview above. On the left we have the space of utterances, and on the right, the space of worlds. Models of interpretation, often called “listeners”, are shown as red arrows (in a precise sense discussed below) from U to W, while models of production, sometimes called “speakers”, are depicted as blue arrows in the opposite direction. Finally, the vertical arrow between speaker and listener models are there to suggest that the \(L_1\) is build from the \(S_1\), and the \(S_1\) from the \(L_0\).

(Brief digression with technical hat on: for the computer scientists, there’s a more general recursive definition: \(S_n\) is defined in terms of \(L_n\\__1\), which is defined in terms of \(S_n\\__1\), and so on. \(L_0\) is the base case of the recursion, and the fix point \(L_m\) such that \(L_m\) = \(L_m\\__1\) represents the ideal listener, which is closely related to the notion of a game theoretic equilibrium.)

The Literal Listener \(L_0\)

First of all, what type of thing is \(L_0\)? It’s going to be a function which takes u \(\in\) U and returns a distribution over all w \(\in\) W. This is just a way of saying it’s a conditional distribution \(L_0\)(w|u), but I prefer the function perspective (*putting on the pointiest most arcane ivory tower shaped hat*, a conditional distribution is a morphism in a very special category - the Kleisli category of the distribution monad - which is precisely why it makes sense to view them as arrows, and implicitly is what we’re doing when we do probabilistic programming. If you do probabilistic programming in Haskell, then it’s also explicitly what you’re doing.).

Here’s the (simplest possible) definition of \(L_0\) (I’m ignoring things like cost, non-uniform priors on worlds and utterances, rationality parameters - all useful, but unnecessary for deriving scalar implicatures):

\[L_0(w|u) = \frac{\llbracket u \rrbracket(w)}{\sum_{w'} \llbracket u \rrbracket(w')}\]

If you’re like me, this equation might seem less than helpful. Here’s an explanation of what it actually amounts to: After hearing an utterance u, \(L_0\) thinks all worlds compatible with the utterance they just heard are equally likely. Here’s code that implements the \(L_0\):

var worlds = [{totalSheepFound:1},{totalSheepFound:2}]
var utterances = [
	"I found one of the sheep",
	"I found both of the sheep"]

var meaning = function(utterance, world){
  (utterance === "I found one of the sheep")
  && (world['totalSheepFound']>0)  ? true :
  (utterance === "I found both of the sheep")
  && (world['totalSheepFound']==2)  ? true :
  false}

var l0 = function(utterance){
  Infer({model: function(){
    var world = uniformDraw(worlds);
    condition(meaning(utterance, world))
    return world}})}

viz(l0("I found one of the sheep"))
viz(l0("I found both of the sheep"))

So \(L_0\) is a simple generalization of a logical semantics. Probabilistic programming is useful for defining this sort of model, particularly when things start getting complicated. Oh, and here’s a visualization of the \(L_0\) posterior conditional distributions:

The Informative Speaker \(S_1\)

There’s a sense in which production is the dual of interpretation. A production model is a conditional distribution p(u|w); given a state, it gives a distribution over utterances. The particular production model we’re interested in is \(S_1\), defined as:

\[S_1(u|w) = \frac{L_0(w|u)}{\sum_{u'} L_0(w|u')}\]

This production model’s goal is to maximize informativity; it has some state w it wants to convey, and it put the most weight on the utterance u which gets the literal listener \(L_0\) to place the most weight on w after hearing u. Again, code, to make that interactive:


var worlds = [{totalSheepFound:1},{totalSheepFound:2}]
var utterances = [
	"I found one of the sheep",
	"I found both of the sheep"]

var meaning = function(utterance, world){
  (utterance === "I found one of the sheep")
  && (world['totalSheepFound']>0)  ? true :
  (utterance === "I found both of the sheep")
  && (world['totalSheepFound']==2)  ? true :
  false}

var l0 = function(utterance){
  Infer({model: function(){
    var world = uniformDraw(worlds);
    condition(meaning(utterance, world))
    return world}})}

var s1 = function(world){
  Infer({model: function(){
    var utterance = uniformDraw(utterances)
    factor(l0(utterance).score(world))
    return utterance}})}

viz(s1({totalSheepFound:1}))
viz(s1({totalSheepFound:2}))

And a diagram of the conditional distributions:

The Pragmatic Listener \(L_1\)

OK, so we had a listener \(L_0\). And we had \(S_1\) thinking about \(L_0\). Now we’re going to have \(L_1\), which is a model of a listener who thinks about \(S_1\) thinking about \(L_0\):

\[L_1(w|u) = \frac{S_1(u|w)}{\sum_{w'} S_1(u|w')}\]

You can think of \(L_1\) hearing an utterance u and asking the following question: what world state must \(S_1\) have been in to have said u. See what happens when you run the code.


var worlds = [{totalSheepFound:1},{totalSheepFound:2}]
var utterances = [
	"I found one of the sheep",
	"I found both of the sheep"]

var meaning = function(utterance, world){
  (utterance === "I found one of the sheep")
  && (world['totalSheepFound']>0)  ? true :
  (utterance === "I found both of the sheep")
  && (world['totalSheepFound']==2)  ? true :
  false}

var l0 = function(utterance){
  Infer({model: function(){
    var world = uniformDraw(worlds);
    condition(meaning(utterance, world))
    return world}})}

var s1 = function(world){
  Infer({model: function(){
    var utterance = uniformDraw(utterances)
    factor(l0(utterance).score(world))
    return utterance}})}

// pragmatic listener
var l1 = function(utterance){
  Infer({model: function(){
    var world = uniformDraw(worlds)
    factor(s1(world).score(utterance))
    return world }})}

viz(l1("I found one of the sheep"))
viz(l1("I found both of the sheep"))

Or just see the figure below:

The takeaway is that \(L_1\) hears I found one of the sheep and infers that it’s more likely to be the case that only one sheep has been found. Tada, it’s a scalar implicature!

So to wrap up, we’ve seen how to model a simple type of pragmatic meaning using nested Bayesian models. This example was simple, but the core idea of pragmatic phenomena arising naturally from a semantics and nested reasoning is powerful. All sorts of pragmatic phenomena can be tackled with tools of this ilk, like vagueness, metaphor, hyperbole, focus, m-implicature, questions, generic language and politeness.

With a logical semantics, we had a way to get from utterances to compatible world states, but no way to handle pragmatic meaning formally. By making things probabilistic, we get to do semantics and pragmatics in a unified framework: pragmatic and semantic meanings exist in the same space. That’s good.

Moreover, in this paradigm, pragmatic meaning arises naturally from a recursive process of inter-agent reasoning where the base case is a semantics, i.e. a conventional relationship between states of the world and utterances.

Next time, we’ll see that by changing U and W to represent different spaces, similar models take on a different character and can be used to model sociolinguistic phenomena.

FAQ Addendum:

Q: Why start with a literal listener, not a literal speaker? A: No reason - the other way works too. In fact, we could also start with both and do a mutual recursion. I’m becoming increasingly convinced that this is the right thing to do.
Q: Can we add more layers above \(L_1\)? A: Yes! the more we add, the closer the model gets to making hard (i.e. non-probabilistic) decisions. See the above digression about fix points.
Q: Do we ever need more? A: Yes. But only for more complicated phenomena. For scalar implicature, this many layers does just fine.
Q: What is Bayesian probability adding here? A: there are many answers, but here’s my favourite: in classical logic, an implication \(p\to q\) allows information to flow from p to q. But if you know the value of q, you don’t know anything about p. The essence of Bayesian probability is precisely that if you have \(p\to q\) and you know about q, you know about p. Information flows backwards. That’s a pretty abstract answer, but can be made precise, albeit with more technical details added. That said, there are non-probabilistic approaches available too.

Informativity and Galois Connections

2015-06-26T16:07:24Z

Informativity and Galois Connections

26 June 2015

Here is an observation that I talked about at MIT’s category theory seminar at some point, relating Grice’s maxims of Quantity and Quality to the mathematical notion of a Galois connection.

Abstract Summary:

For a state space W, the left adjoint of a monotone map (i.e. a left Galois connection), \(L_0\), from a set of utterances U to the poset (ordered by inclusion) \(\mathcal{P}(W)\) is the monotone map \(S_1: \mathcal{P}(W)\to U\) which takes a set of states and returns the strongest true utterance with respect to \(L_0\). This happens to be a very natural way to encode Grice’s maxims of Quality (roughly: speak truthfully) and Quantity (roughly: be as informative as possible, relative to what is relevant) simultaneously.

Concrete Explanation:

Each \(w \in W\) is a state of the world, or, so that each element of \(\mathcal{P}(W)\) is a set of states.

As usual, the simplest possible example is a reference game, where “state” just means the intended referent. Concretely, say that W = \(\\{R_1, R_2, R_3\\}\) as pictured below, and U = {red dress, dress, hat, silence}. Obviously arbitrary choices, but just for illustration.

Say that the literal listener \(L_0\) maps an utterance u to the set of referents (i.e. states, i.e. worlds) compatible with u, mapping red dress to \(\\{R_1\\}\), dress to \(\\{R_1, R_2\\}\) , hat to \(\\{R_3\\}\) and silence to \(\\{R_1, R_2, R_3\\}\).

Note that we can make U a poset by defining the partial ordering on U where \(u \leq u' \leftrightarrow L_0(u) \leq L_0(u')\). For example, \(\mathit{dress} \leq \mathit{silence}\). Note that \(u \leq u'\) means that u is stronger than u’.

It then follows (by the definition of the ordering on U) that \(L_0\) is a monotone map (i.e. a function that preserves the poset ordering) from U to W.

Galois Connections

So far just definitions. Just one more: for monotone maps f and g, f is the left Galois connection of g iff:

\[f(s) \leq u \leftrightarrow s \leq g(u)\]

It takes a bit of thinking to make sense of this strange definition, but the intuition is this: there’s no obvious notion of an exact inverse of g, because g might well not be surjective (or injective). But for a monotone map, there’s a notion of the best approximation of such an inverse. That approximation is f, as defined above. (In fact there are two, the left and right Galois connections, and more broadly, the left and right adjoints of a functor. A monotone map is a very simple case of a functor between very simple categories, namely posets).

Maybe this direct corollary of the above definition will help: if I know g, then its left adjoint f is defined as:

\[f(s) = \bigwedge(\\{u : s \leq g(u)\\})\].

I write \(\bigwedge(X)\) for a poset X to mean the greatest lower bound of X.

OK, so now you can ask: what’s the left Galois connection of the literal listener \(L_0\)? Let’s call this left Galois connection \(S_1\), for reasons that will soon be clear. Again, note that \(L_0\) can’t just be inverted, because it’s in general not the case that for any subset s of W (i.e. element of \(\mathcal{P}(W)\)), there’s an expression which means exactly s under \(L_0\).

It’s illustrative to work through an example, to see what \(S_1\) looks like. Using our case from above, what’s \(S_1(\\{R_2\\})\)?

Well, \(S_1(\\{R_2\\}) = \bigwedge\\{u : \\{R_2\\} \leq L_0(u)\\})\) = \(\bigwedge(\\{dress, silence\\})\) = \(dress\).

First you find all the utterances that map to supersets of \(\\{R_2\\}\). These are all the true utterances (Quality). Then you take the greatest lower bound (Quantity).

So in other words, the definition of an left Galois connection gives you the following informative speaker \(S_1\): consider the set of all utterances compatible with your (possibly singleton) set of worlds, and choose the strongest of these. There’s something nice about how the maxims of Quality and Quantity fall out from this.

We also obtain similar results for pragmatic implicatures (that I won’t sketch out here for reasons of laziness) namely that a literal speaker, in the form of a monotone map \(S_0\) from w \(\in\) W to us in \(\mathcal{P}(U)\), admits a left Galois connection \(L_1\) which returns the exhaustification (linguistics term) of the literal meaning of us. So this would model, for example, the fact that the pragmatic interpretation of a (possibly singleton) set of utterances us should give the smallest set of possible worlds that could have produced every \(u \in \mathit{us}\).

The niceness of this correspondence between Galois connections and pragmatics suggests that something relatively deep is going on here, but I haven’t thought about it too much further. The sensible thing to do would be to consider the categorical generalization of Galois connections, namely adjoint functors, and to see if we get the same effect when we invert a more sophisticated functorial semantics.

I came up with this idea thanks to John Baez’s fantastic category theory course, based off of David Spivak and Brendan Fong’s Seven Sketches in Compositionality.

Summary:

*An informative speaker is a left Galois connection to a literal listener

*A pragmatic listener is a left Galois connection to a literal speaker

Eulogy

2015-05-26T07:01:22Z

Eulogy

26 May 2015

–

To my father (late)

I dream about you all the time. Sometimes we’re in a car, and I try to gently remind you that it is dangerous for people who have suffered a fatal heart attack to drive.

Living in memory doesn’t really suit you, with its proclivity for the grandiose. Epithets like “wise” and “gentle” roll off you slightly awkwardly, though true. They misleadingly suggest an air of purposeful humility. In my mother’s words, you were “the rare case of a man without an ego”.

Some assorted thoughts:

You loved to be early - to airports, especially. This, part of a perennial but mild neuroticism.
You cooked wonderful food every day - your homespun repertoire, now lost for good, was of vaguely hippyish vegetarian rice dishes, which you’d experiment on over time.
You loved to do chores (who loves to do chores?) and were sometimes comically meticulous. When I sent you an email asking if I’d left some item of clothing at home, I’d get 10 photos of potential culprits in return, with an offer to send any of them over immediately.
You didn’t get angry.
You drove us, as children, to playgrounds and amusement parks, and later sports events and parties, unfailingly, at all hours of the day, and without complaint.
When I lost my temper, you knew exactly how to defuse my anger with a joke that indicated you understood my side.
You had a subtle sense of humor - we would often send each other comedy skits - and I keep old emails of yours marked unread in my inbox that still make me laugh, like memories of your jocular expressions, slightly ludicrous but self aware.
On the other hand, you were subdued in the presence of strangers. Even old friends remarked on your quietness.
You’d clean my room against my will every time I left for a term at uni. You claimed, computer scientist that you were, that this process was “structure preserving” - you just mapped the books and clothes strewn over my floor to new, tidier positions. This was not true.
You owned at least 20 pairs of glasses, lost behind every sofa in the house, like some mad academic. In your later years you’d start to prefer ones with perfectly round, small lenses, that made you look like a 19th century German professor. “They’re quite good, aren’t they”, you’d say with a wry grin to my mother’s disparagement, that makes me smile.
You’d pick mushrooms in the fields, and point them out when we went on walks and talked about logic and my half baked thoughts about philosophy.
Your chair is empty at dinner now, like the chair left for Elijah at passover, who is always welcome but never shows up.

More thoughts:

I remember you, in an atemporal sort of way, sitting in your room reading a spy novel in your shirt and boxers, while watching the news. You once picked me up from the station wearing that. You once cut onions wearing a snorkeling mask to protect your eyes. I miss all your small absurdities.

I remember your inflexible hands and carpal-tunnel-ridden handwriting. A characteristically forthright hotel manager in the Midlands had once suggested I do your signature for you. You were also of a departing generation of computer scientists who typed slower than they wrote.

I remember once finding a book in the house and noticing with surprise that you had written it. It had just never occurred to you to mention that. You didn’t have a retirement party, because the whole idea was “embarrassing”. Even in your death. you were unassuming; you just dropped the pan you were holding, fell back, and that was pretty much that.

I remember how you booked me into an arachnophobia course without telling me (you had cured your own arachnophobia by capturing and keeping a spider as a pet, but then, that approach isn’t for everyone).

I remember how I once turned up for an interview, only to find that my interviewer was an old friend of yours. you had deliberately avoided mentioning this, feeling uncomfortable with any possibility of nepotism.

I remember how you would always check the lights were off at night (usually several times), and find me downstairs. There would always be some thought you had to share, or some mild concern. I always teased you about worrying too much. You’d exaggerate it in turn. In the afternoon, you’d drop in to my room to ask what I wanted for dinner. I also remember my excitement in early childhood to hear the sound of your keys in the door, coming home from work.

And also, and also.

You talked about your work so little that my own nascent interests in logic at university seemed like a coincidence when I found out about your limitless knowledge about that topic. Months before your death we’d stood on a beach in California, near where you are now buried, and worked out some logical definition with a stick in the sand.

I inherited your social awkwardness, sense of humour, academic interests and as I’ve been occasionally told, your appearance.

I always hoped, or rather assumed, that you’d be around to help me out my whole life, but as soon as you died, it became suddenly apparent that you inhabited a now distant seeming past, impossibly far away and inaccessible. This was cemented by the influx of consolatory emails, all so kind, from former students, colleagues and friends, which exposed a legacy of mentoring that, unbeknown to me, had characterized your career.

I admired your intellectual patience, identified with your wry humour and I hoped you knew how much I loved you. That’s all my mother and brother and I wanted to say to you in hospital - that we loved you and that you should stay with us. you weren’t the sort of person I felt comfortable expressing that kind of emotion to at any less pressing time. It was just a bit too embarrassing and we were a bit too British.

Your love for us was shown through acts not words - through books you’d recommend, emails with talks or papers you thought I’d enjoy about mathematics or Derrida or puddles in Yorkshire, and the ceaseless performance of a thousand and one small chores - whatever you thought would help us in our life. You were a perfect example of how to be good, and I miss you so so much.

I take comfort that you lived a happy life to the almost very end, a self-made man whose own father had died in your early youth, and who had built your own family life from scratch.

I hope to remember you when I cook meals from your mysterious recipes, and when I laugh uncomfortably during small talk. I’ll also remember you when I raise my eyebrows in that strained expression we both assume when talking to shop assistants and when I paw at my phone over the top of my glasses. And maybe, if I read The Cat in the Hat to children of my own or carry them up the stairs after they fall asleep on the sofa, I’ll remember your unassuming parenting, which was quietly gentle and wise.

(On display: the tendency our hair has to stick up at the back - his hidden by a hat)