The common ground

Here, we make good on the assumption mentioned earlier that common grounds amount to probability distributions over indices of some kind. In general, we will allow the meanings of expressions to be determined by indices in the following way. For any constants, e.g.,

\[ \begin{align*} \ct{see} &: ι → e → e → t \\ \ct{ling} &: ι → e → t \end{align*} \]

etc., there are other constants

\[ \begin{align*} \updct{see} &: (e → e → t) → ι → ι \\ \updct{ling} &: (e → t) → ι → ι \end{align*} \]

which may update some index $i$ with a particular value. Thus our theory of indices is effectively a theory of states and locations: any given index represents a kind of state; meanwhile, constants such as $\ct{see}$ and $\ct{ling}$ represent different locations associated with that state. For example, given some index $i$, $\updct{see}(p)(i)$ is a new index just like $i$, but where the value stored at the location $\ct{see}$ has been overwritten by $p$. As a result, our constants should satisfy equations like the following:

\[ \begin{align*} \ct{see}(\updct{see}(p)(i)) &= p \\ \ct{see}(\updct{ling}(p)(i)) &= \ct{see}(i) \\[2mm] \ct{ling}(\updct{ling}(p)(i)) &= p \\ \ct{ling}(\updct{see}(p)(i)) &= \ct{ling}(i) \end{align*} \]

That is, when $\ct{see}$ encounters an index which has been updated at its associated location, it grabs the value that the index has been updated with. If it encounters an index which has been updated at a different location, it keeps looking. (Similarly, for $\ct{ling}$.)

Finally, we define a common ground to be a probability distribution over indices.

Definition: a common ground is a probabilistic program of type $\P ι$.

Here, again, $ι$ is understood to be a variable over types: its type doesn’t really matter, as long as it can be understood as supporting the theory of states and locations described just above. We further define a constant representing a starting index, which we call ‘$\ct{@}$’.

\[\ct{@} : ι\]

Let’s briefly consider a concrete example. One way of defining a common ground is by encoding a distribution over heights for some entity; say, Jo. The following common ground updates the value stored for the constant $\ct{height} : ι → e → r$:

\[ \begin{array}[t]{l} h ∼ \abbr{Normal}(0, 1) \\ \pure{\updct{height}(λx.h)(\ct{@})} \end{array} \]

This common ground encodes uncertainty about Jo’s height by associating it with a normal distribution centered at 0 and with a standard deviation of 1. Note that because we are considering only one individual—Jo—we can update the height value globally. If we wish to describe a common ground that encodes uncertainty about the heights of more than individual—say, Jo and Bo—we can make the function with which indices are updated a bit more sophisticated:

\[ \begin{array}[t]{l} h_{j} ∼ \abbr{Normal}(0, 1) \\ h_{b} ∼ \abbr{Normal}(0, 1) \\ \pure{\updct{height}(λx.\ite(x = \ct{j}, h_{j}, h_{b}))(\ct{@})} \end{array} \]

Here, $\ite$ should be understood as satisfying the following two equations:

\[ \begin{align*} \ite(\True, x, y) &= x \\ \ite(\False, x, y) &= y \end{align*} \]

Thus the common ground in (5) updates the starting index with a value for $\ct{height}$ consisting of a function that returns $h_{j}$ on the argument $\ct{j}$ (i.e., Jo) and $h_{b}$ otherwise (i.e., when the argument is $\ct{b}$, i.e., Bo).

--- title: "The common ground" bibliography: ../../pds.bib format: html: css: ../styles.css html-math-method: mathjax mathjax-config: loader: {load: ['[tex]/bussproofs','[tex]/bbox','[tex]/colorbox']} tex: packages: {'[+]': ['bussproofs','bbox','colorbox']} --- ::: {.hidden} $$ \newcommand{\expr}[3]{\begin{array}{c} #1 \\ \bbox[lightblue,5px]{#2} \end{array} ⊢ #3} \newcommand{\ct}[1]{\bbox[font-size: 0.8em]{\mathsf{#1}}} \newcommand{\updct}[1]{\ct{upd\_#1}} \newcommand{\abbr}[1]{\bbox[transform: scale(0.95)]{\mathtt{#1}}} \newcommand{\pure}[1]{\bbox[border: 1px solid orange]{\bbox[border: 4px solid transparent]{#1}}} \newcommand{\return}[1]{\bbox[border: 1px solid black]{\bbox[border: 4px solid transparent]{#1}}} \def\P{\mathtt{P}} \def\Q{\mathtt{Q}} \def\True{\ct{T}} \def\False{\ct{F}} \def\ite{\ct{if\_then\_else}} \def\Do{\abbr{do}} $$ ::: Here, we make good on the assumption mentioned earlier that common grounds amount to probability distributions over indices of some kind. In general, we will allow the meanings of expressions to be determined by indices in the following way. For any constants, e.g., $$ \begin{align*} \ct{see} &: ι → e → e → t \\ \ct{ling} &: ι → e → t \end{align*} $$ etc., there are other constants $$ \begin{align*} \updct{see} &: (e → e → t) → ι → ι \\ \updct{ling} &: (e → t) → ι → ι \end{align*} $$ which may update some index $i$ with a particular value. Thus our theory of indices is effectively a theory of *states and locations*: any given index represents a kind of state; meanwhile, constants such as $\ct{see}$ and $\ct{ling}$ represent different locations associated with that state. For example, given some index $i$, $\updct{see}(p)(i)$ is a new index just like $i$, but where the value stored at the location $\ct{see}$ has been overwritten by $p$. As a result, our constants should satisfy equations like the following: (@ex-indices-eqs) $$ \begin{align*} \ct{see}(\updct{see}(p)(i)) &= p \\ \ct{see}(\updct{ling}(p)(i)) &= \ct{see}(i) \\[2mm] \ct{ling}(\updct{ling}(p)(i)) &= p \\ \ct{ling}(\updct{see}(p)(i)) &= \ct{ling}(i) \end{align*} $$ That is, when $\ct{see}$ encounters an index which has been updated at its associated location, it grabs the value that the index has been updated with. If it encounters an index which has been updated at a different location, it keeps looking. (Similarly, for $\ct{ling}$.) Finally, we define a common ground to be a probability distribution over indices. (@ex-common-ground) Definition: a *common ground* is a probabilistic program of type $\P ι$. Here, again, $ι$ is understood to be a variable over types: its type doesn't really matter, as long as it can be understood as supporting the theory of states and locations described just above. We further define a constant representing a *starting* index, which we call '$\ct{@}$'. (@ex-starting-index) $$\ct{@} : ι$$ Let's briefly consider a concrete example. One way of defining a common ground is by encoding a distribution over heights for some entity; say, Jo. The following common ground updates the value stored for the constant $\ct{height} : ι → e → r$: (@ex-jo-cg) $$ \begin{array}[t]{l} h ∼ \abbr{Normal}(0, 1) \\ \pure{\updct{height}(λx.h)(\ct{@})} \end{array} $$ This common ground encodes uncertainty about Jo's height by associating it with a normal distribution centered at 0 and with a standard deviation of 1. Note that because we are considering only one individual---Jo---we can update the height value globally. If we wish to describe a common ground that encodes uncertainty about the heights of more than individual---say, Jo and Bo---we can make the function with which indices are updated a bit more sophisticated: (@ex-jo-bo-cg) $$ \begin{array}[t]{l} h_{j} ∼ \abbr{Normal}(0, 1) \\ h_{b} ∼ \abbr{Normal}(0, 1) \\ \pure{\updct{height}(λx.\ite(x = \ct{j}, h_{j}, h_{b}))(\ct{@})} \end{array} $$ Here, $\ite$ should be understood as satisfying the following two equations: (@ex-ite) $$ \begin{align*} \ite(\True, x, y) &= x \\ \ite(\False, x, y) &= y \end{align*} $$ Thus the common ground in (@ex-jo-bo-cg) updates the starting index with a value for $\ct{height}$ consisting of a function that returns $h_{j}$ on the argument $\ct{j}$ (i.e., Jo) and $h_{b}$ otherwise (i.e., when the argument is $\ct{b}$, i.e., Bo).