[[!toc levels=2]] # Doing things with monads ## Extended application: Groenendijk, Stokhof and Veltman's *Coreference and Modality* GSV are interested in developing and establishing a reasonable theory of discourse update. One way of looking at this paper is like this: GSV = GS + V That is, Groenendijk and Stokhof have a well-known theory of dynamic semantics, and Veltman has a well-known theory of epistemic modality, and this fragment brings both of those strands together into a single system. We will be interested in this paper both from a theoretical point of view and from a practical engineering point of view. On the theoretical level, these scholars are proposing a strategy for managing the connection between variables and the objects they designate in way that is flexible enough to be useful for describing natural language. The main way they attempt to do this is by inserting an extra level in between the variable and the object: instead of having an assignment function that maps variables directly onto objects, GSV provide *pegs*: variables map onto pegs, and pegs map onto objects. We'll discuss in considerable detail what pegs allow us to do, since it is highly relevant to one of the main applications of the course, namely, reference and coreference. What are pegs? The term harks back to a paper by Landman called `Pegs and Alecs'. There pegs are simply hooks for hanging properties on. Pegs are supposed to be as anonymous as possible. Think of hanging your coat on a physical peg: you don't care which peg it is, only that there are enough pegs for everyone's coat to hang from. Likewise, for the pegs of GSV, all that matters is that there are enough of them. (Incidentally, there is nothing in Gronendijk and Stokhof's original DPL paper that corresponds naturally to pegs; but in their Dynamic Montague Grammar paper, pegs serve a purpose similar to discourse referents there, though the connection is not simple.) On an engineering level, the fact that GSV are combining anaphora and bound quantification with epistemic quantification means that they are gluing together related but distinct subsystems into a single fragment. These subsystems naturally cleave into separate layers in a way that is obscured in the paper. We will argue in detail that re-engineering GSV using monads will lead to a cleaner system that does all of the same theoretical work. Empirical targets: on the anaphoric side, GSV want to On the epistemic side, GSV aim to account for asymmetries such as It might be raining. It's not raining. #It's not raining. It might be raining. ## Basics There are a lot of formal details in the paper in advance of the empirical discussion. Here are the ones that matter: type var = string type peg = int type refsys = var -> peg type ent = Alice | Bob | Carl type assignment = peg -> ent So in order to get from a variable to an object, we have to compose a refsys `r` with an assignment `g`. For instance, we might have r (g ("x")) = Alice. type pred = string type world = pred -> ent -> bool type pegcount = int type poss = world * pegcount * refsys * assignment type infostate = [poss] Worlds in general settle all matters of fact in the world. In particular, they determine the extensions of predicates and relations. In this discussion, we'll (crudely) approximate worlds by making them a function from predicates such as "man" to a function mapping each entity to a boolean. As we'll see, indefinites as a side effect increase the number of pegs by one. GSV assume that we can determine what integer the next unused peg corresponds to by examining the range of the refsys function. We'll make things easy on ourselves by simply tracking the total number of used pegs in a counter called `pegcount`. So information states track both facts about the world (e.g., which objects count as a man), and facts about the discourse (e.g., how many pegs have been used). The formal language the fragment interprets is Predicate Calculus with equality, existential and universal quantification, and one unary modality (box and diamond, corresponding to epistemic necessity and epistemic possibility). Terms in this language are either individuals such as Alice or Bob, or else variables. So in general, the referent of a term can depend on a possibility: ref(i, t) = t if t is an individual, and g(r(t)) if t is a variable, where i = (w,n,r,g) Here are the main clauses for update (their definition 3.1). Following GSV, we'll write `update(s, φ)` (the update of information state `s` with the information in φ) as `s[φ]`. s[P(t)] = {i in s | w(P)(ref(i,t))} So `man(x)` is the set of live possibilities `i = (w,r,g)` in s such that the set of men in `w` given by `w(man)` maps the object referred to by `x`, namely, `r(g("x"))`, to `true`. That is, update with "man(x)" discards all possibilities in which "x" fails to refer to a man. s[t1 = t2] = {i in s | ref(i,t1) = ref(i,t2)} s[φ and ψ] = s[φ][ψ] When updating with a conjunction, first update with the left conjunct, then update with the right conjunct. Existential quantification requires adding a new peg to the set of discourse referents. s[∃xφ] = {(w, n+1, r[x->n], g[n->a]) | (w,n,r,g) in s and a in ent}[φ] Here's the recipe: for every possibility (w,n,r,g) in s, and for every entity a in the domain of discourse, construct a new possibility with the same world w, an incrementd peg count n+1, and a new r and g adjusted in such a way that the variable x refers to the object a. Note that this recipe does not examine φ. This means that this analysis treats the formula prefix `∃x` as if it were a meaningful constituent independent of φ. Negation is natural enough: s[neg φ] = {i | {i}[φ] = {}} If updating φ with the information state that contains only the possibility i returns the empty information state, then not φ is true with respect to i. In GSV, disjunction, the conditional, and the universals are defined in terms of negation and the other connectives. Exercise: assume that there are two entities in the domain of discourse, Alice and Bob. Assume that Alice is a woman, and Bob is a man. Show the following computations: 1. {}[∃x.person(x)] 2. {}[∃x.man(x)] 3. {}[∃x∃y.person(x) and person(y)] 4. {}[∃x∃y.x=x] 5. {}[∃x∃y.x=y]