Generalizing and Formalizing Precisiation Language to Facilitate Human-Robot Interaction

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.

This research is supported by the Spanish Ministry of Economy and Competitiveness through the project TIN2011-29824-C02-02 (ABSYNTHE).

Authors and Affiliations

1. European Center for Soft Computing, C/Gonzalo Gutiérrez Quirós, s/n, 33600, Mieres, Spain

   Takehiko Nakama, Enrique Muñoz, Kevin LeBlanc & Enrique Ruspini

Corresponding author

Correspondence to Takehiko Nakama .

Editors and Affiliations

1. Images, Signals and Intelligence Systems Laboratory, University PARIS-EST Creteil (UPEC), Créteil, France

   Kurosh Madani

2. Departamento de Engenharia Informatica Polo II - Pinhal de Marrocos, University of Coimbra, Coimbra, Portugal

   António Dourado

3. aSEEB-ISR-IST Av. Rovisco Pais, 1, Technical University of Lisbon (IST), Lisbon, Portugal

   Agostinho Rosa

4. Rua do Vale de Chaves, Estefanilha, Polytechnic Institute of Setúbal / INSTICC, Setúbal, Portugal

   Joaquim Filipe

5. Polish Academy of Sciences, Systems Research Institute, Warsaw, Poland

   Janusz Kacprzyk

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 \(x \in X\), \(F(x) = \min _{1\le j \le m} R^{(j)}_{\uparrow X}(x).\)

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