Generalizing and Formalizing Precisiation Language to Facilitate Human-Robot Interaction

Abstract

We develop a formal logic as a generalized precisiation language. This formal logic can serve as a middle ground between the natural-language-based mode of human communication and the low-level mode of machine communication. Syntactic structures in natural language are incorporated in the syntax of the formal logic. As regards the semantics, we establish the formal logic as a many-valued logic. We present examples that illustrate how our formal logic can facilitate human-robot interaction.

Appendix

We describe three operations on fuzzy relations that are used in determining the truth conditions of atomic propositions in our formal logic: projection, cylindric extension, and cylindric closure. First, we establish notation. Let $X_1, X_2,\ldots , X_n$ be sets, and let $X_1 \times X_2 \times \cdots \times X_n$ denote their Cartesian product. We will also denote the Cartesian product by $\times _{i \in \mathbb {N}_n} X_i$, where $\mathbb {N}_n$ denotes the set of integers 1 through n. A fuzzy relation on $\times _{i \in \mathbb {N}_n} X_i$ is a function from the Cartesian product to a totally ordered set, which is called a valuation set. In our formulation, the unit interval [0, 1] is used as a valuation set. Each n-tuple $(x_1, x_2,\ldots , x_n)$ in $X_1 \times X_2 \times \cdots \times X_n$ (thus $x_i \in X_i$ for each $i \in \mathbb {N}_n$) will also be denoted by $(x_i\ |\ i \in \mathbb {N}_n)$. Let $I \subset \mathbb {N}_n$. A tuple $y:=(y_i\ |\ i \in I)$ in $Y:=\times _{i \in I} X_i$ is said to be a sub-tuple of $x:=(x_i\ |\ i \in \mathbb {N}_n)$ in $\times _{i \in \mathbb {N}_n} X_i$ if $y_i = x_i$ for each $i \in I$, and we write $y \prec x$ to indicate that y is a sub-tuple of x.

Let $X := \times _{i\in \mathbb {N}_n} X_i$ and $Y := \times _{i \in I} X_i$ for some $I \subset \mathbb {N}_n$. Suppose that $R: X \rightarrow [0, 1]$ is a fuzzy relation on X. Then a fuzzy relation $R': Y \rightarrow [0, 1]$ is called the projection of R on Y if for each $y \in Y$, we have $R'(y) = \max _{x \in X\ :\ y \prec x} R(x).$ We let $R_{\downarrow Y}$ denote the projection of R on Y.

We continue with $X := \times _{i\in \mathbb {N}_n} X_i$ and $Y := \times _{i \in I} X_i$ ($I \subset \mathbb {N}_n$). Let $F: Y \rightarrow [0,1]$ be a fuzzy relation on Y. A fuzzy relation $F': X \rightarrow [0, 1]$ is said to be the cylindric extension of F to X if for all $x \in X$, we have $F'(x) = F(y),$ where y is the tuple in Y such that $y \prec x$. We let $F_{\uparrow X}$ denote the cylindric extension of F to X. The cylindric extension $F_{\uparrow X}$ of a fuzzy relation $F: Y\rightarrow [0, 1]$ is the “largest” fuzzy relation on X such that its projection on Y equals F; if we let $\mathscr {R}$ denote the set of all fuzzy relations $R': X \rightarrow [0, 1]$ such that $R'_{\downarrow Y} = F$, then for all $x \in X$, we have $F_{\uparrow X}(x) = \max \{R'(x)\ |\ R' \in \mathscr {R}\}.$

For each j, let $Y_j := \times _{i\in I_j} X_i$, where $I_j \subset \mathbb {N}_n$. Let $R^{(j)}: Y_j \rightarrow [0, 1]$ denote a fuzzy relation on $Y_j$. Then a fuzzy relation $F: X \rightarrow [0, 1]$ is called the cylindric closure of $R^{(1)}, R^{(2)},\ldots , R^{(m)}$ on X if for each $x \in X$, $F(x) = \min _{1\le j \le m} R^{(j)}_{\uparrow X}(x).$