[Paper Review] Retraction and Generalized Extension of Computing with Words

Yongzhi Cao, Mingsheng Ying+1 more|UTS ePRESS (University of Technology Sydney)|Apr 19, 2006

Advanced Algebra and Logic27 references4 citations

TL;DR

This paper introduces a generalized fuzzy automaton model for computing with words, extending Ying's formal framework by allowing arbitrary fuzzy subsets as inputs and arbitrary fuzzy transition functions. It establishes a retraction principle to map computing with words back to computing with values for crisp inputs, and a generalized extension principle to handle fuzzy inputs, proving that both traditional computing modes can be uniformly realized within the word-computing framework through algebraic consistency and compositional reasoning.

ABSTRACT

Fuzzy automata, whose input alphabet is a set of numbers or symbols, are a formal model of computing with values. Motivated by Zadeh's paradigm of computing with words rather than numbers, Ying proposed a kind of fuzzy automata, whose input alphabet consists of all fuzzy subsets of a set of symbols, as a formal model of computing with all words. In this paper, we introduce a somewhat general formal model of computing with (some special) words. The new features of the model are that the input alphabet only comprises some (not necessarily all) fuzzy subsets of a set of symbols and the fuzzy transition function can be specified arbitrarily. By employing the methodology of fuzzy control, we establish a retraction principle from computing with words to computing with values for handling crisp inputs and a generalized extension principle from computing with words to computing with all words for handling fuzzy inputs. These principles show that computing with values and computing with all words can be respectively implemented by computing with words. Some algebraic properties of retractions and generalized extensions are addressed as well.

Motivation & Objective

• To develop a formal model of computing with words that generalizes Ying's original framework by restricting the input alphabet to a subset of fuzzy subsets rather than all fuzzy subsets.
• To address the limitation of existing models by allowing arbitrary fuzzy transition functions, enhancing flexibility for real-world applications.
• To establish a retraction principle that enables computing with words to simulate traditional computing with values when inputs are crisp.
• To introduce a generalized extension principle that allows computing with words to handle fuzzy inputs, extending the scope of word-based computation.
• To prove algebraic properties of retractions and generalized extensions, ensuring consistency and compositional behavior in the formal model.

Proposed method

• Define a fuzzy automaton for computing with words (FACW) with a finite set of states, a restricted input alphabet of selected fuzzy subsets of symbols, and an arbitrary fuzzy transition function.
• Introduce the retraction principle: map a FACW to a crisp-input FACV by computing the height of the intersection between the final state distribution and the input membership function.
• Define the generalized extension principle: extend a FACW to handle fuzzy inputs by lifting the transition function to operate on fuzzy input words using max-min composition.
• Use fuzzy control methodology to formalize the retraction and extension processes, ensuring compatibility with perception-based reasoning.
• Prove that the language recognized by the retracted automaton matches the original word language under crisp inputs, and that the generalized extension preserves the original FACW behavior on fuzzy inputs.
• Employ inductive proofs based on string length to formally establish the correctness of the retraction and extension mappings using fuzzy set operations and height functions.

Experimental results

Research questions

• RQ1 How can computing with words be formally generalized to allow arbitrary fuzzy subsets as inputs, rather than requiring all fuzzy subsets?
• RQ2 Can computing with values be formally embedded within the framework of computing with words through a retraction mechanism?
• RQ3 How can fuzzy inputs be systematically handled within a computing-with-words model using a generalized extension principle?
• RQ4 What algebraic properties do retractions and generalized extensions preserve, ensuring consistency and compositional reasoning?
• RQ5 Is it possible to unify computing with values and computing with all words under a single formal model via these principles?

Key findings

• The retraction principle ensures that for any crisp input string, the language recognized by the retracted FACV matches the language recognized by the original FACW, proving that computing with values is a special case of computing with words.
• The generalized extension principle allows a FACW to process fuzzy input words by lifting the transition function to operate on fuzzy input sequences, preserving the original behavior on crisp inputs.
• The language of the generalized extension automaton is shown to be equal to the original FACW language when restricted to fuzzy inputs, confirming the correctness of the extension.
• The retraction and generalized extension mappings are proven to be consistent through inductive proofs using fuzzy set operations and height functions.
• The paper establishes that the composition of retraction and generalized extension recovers the original FACW behavior, demonstrating the closure and coherence of the formal framework.
• The proposed model generalizes Ying’s original FACAW (computing with all words) by allowing partial input alphabets and arbitrary transition functions, increasing applicability to real-world perception-based systems.