VHDL framework for modeling fuzzy automata

Transcription

1 Doru Todinca Daniel Butoianu Department of Computers Politehnica University of Timisoara SYNASC 2012

2 Outline Motivation 1 Motivation Why fuzzy automata? Why a framework for modeling FA? Why VHDL? 2 Fuzzy sets and fuzzy automata VHDL implementation of FA 3 4

3 Why fuzzy automata? Why fuzzy automata? Why a framework for modeling FA? Why VHDL? Most engineering applications of fuzzy logic are based on fuzzy inference (fuzzy if-then rules) A hardware device that implements fuzzy inference is called fuzzy logic controller (FLC) No matter how complex its implementation is, a FLC s outputs depend only on its inputs, no state is considered However, many applications demand the existence of states necessity to use fuzzy automata Fuzzy automata are fuzzy extension of classic, or crisp automata

4 Crisp versus fuzzy automata Why fuzzy automata? Why a framework for modeling FA? Why VHDL? Crisp automata are used 1 for formal languages 2 for hardware applications, in almost any device from everyday life Fuzzy automata are linked to fuzzy languages, that have practical applications (text error correction, pattern recognition, etc) but there are extremely few applications of FA not related to fuzzy languages The question is: why FA are not as spread as classic automata?

5 Why fuzzy automata? Why a framework for modeling FA? Why VHDL? The problems encountered by fuzzy automata 1 There are too many types of FA: fuzzy operators (union, intersection, composition, etc) extend classic operators each classic operator can be extended in several ways too many operators it is confusing for practitioners This problem appears also in other domains of fuzzy logic, e.g. for fuzzy inference but in older domains, several operators or methods have imposed over the others: e.g. Mamdani inference, center of gravity defuzzification are more used than other similar methods. 2 The lack of controllability of FA: in certain conditions a FA cannot be moved from a certain state.

6 Why fuzzy automata? Why a framework for modeling FA? Why VHDL? Framework for modeling fuzzy automata We propose a tool that can investigate the behaviour of different types of FA The tool will be used for: investigating the behaviour of different operators used for FA finding operators that lead to more controllable FA investigating the efficiency of different techniques used for making FA more controllable (e.g. conservation of state, state normalization); The tool should be very flexible in order to accommodate FA of different types, with different sizes, and using different operators or methods to improve controllability.

7 Why VHDL? Motivation Why fuzzy automata? Why a framework for modeling FA? Why VHDL? VHDL stands for Very High Speed Integrated Circuits Hardware Description Language Being a hardware description language, VHDL can be used for: modeling and simulation automatic synthesis (e.g. on a FPGA) evaluating the hardware characteristics of the circuit: input-output delay, clock rate, number of gates, occupied surface, etc. VHDL is also a high-level programming language, which can work at different abstraction levels flexibility.

8 Fuzzy sets Motivation Fuzzy sets and fuzzy automata VHDL implementation of FA Given an universe of discourse X (a crisp set), a fuzzy set à X is the set of ordered pairs à = {(x,µã(x)) x X} where µã(x) : X [0,1] is called membership function or degree of membership. With fuzzy sets can be defined operations of: intersection: the minimum between their membership functions union: the maximum between their membership functions complement: the set having the membership function 1 µã(x) The minimum and maximum operations have been extended by t-norms and s-norms

9 Crisp automata Motivation Fuzzy sets and fuzzy automata VHDL implementation of FA Automata in general are computational systems having inputs, states, outputs a function that computes the transition from the current state to the next state (transition function) a function that computes the outputs an initial state. Automata may have also final states. The classic automata can be deterministic, non-deterministic and probabilistic.

10 Fuzzy automata Motivation Fuzzy sets and fuzzy automata VHDL implementation of FA FA are extensions of classic automata in the sense that the states, the inputs, and/or the transition functions can be fuzzy. There are different classifications of FA In Chen s classification the inputs and the states can be either crisp or fuzzy the time can be also crisp or fuzzy the time can be discrete (synchronous FA) or continuous (asynchronous FA) Here we study only discrete synchronous fuzzy automata, but we plan to extend our framework in order to include other types of FA.

11 Transition matrix Motivation Fuzzy sets and fuzzy automata VHDL implementation of FA In this work the inputs, states and outputs of the FA are fuzzy sets in the universes U, X and Y. For every input u j a transition matrix M(u j ) is defined: µ x (xt+1 1 M(u j x1 t,u j )... µ x (xt+1 m x1 t,u j ) ) =..... µ x (xt+1 1 xm t,uj )... µ x (xt+1 m xm t,uj ) Here µ x (x i t+1 xk t,u j ) is the degree of transition from state x k t to state x i t+1 when the input uj is applied (at the moment t) all µ x are fuzzy values (in the interval [0,1])

12 Conservation of state Fuzzy sets and fuzzy automata VHDL implementation of FA In the computation of the next state are taken into account not only the degree of membership of each input, but also the degrees of membership of the negated inputs. The formula for next state will be: µ x (x t+1 ) = x i u j [µ x (x t ) µ u (u t ) µ x (x t+1 x t,u t ) +µ x (x t ) µ u (u t ) µ x (x t+1 x t,u t )] and + denote algebraic sum, represents algebraic product, and µ u (u t ) is the complement of input u (at step t). The sums are for all states x i and all inputs u j.

13 VHDL implementation of FA Fuzzy sets and fuzzy automata VHDL implementation of FA We use a hierarchical system of VHDL packages: Constants: number of inputs, number of states, number of outputs Types: a fuzzy value type (fv) and arrays of fv for inputs, states, etc Functions: parametrized functions for computing fuzzy transition matrices with or without conservation of state algebraic product and sum from previous formulae are generalized by functions F 1 (membership assignment function) and F 2 (multi-membership resolution) a state normalization function. The transition matrix defining a FA and the initial state are read from a text file

14 Fuzzy sets and fuzzy automata VHDL implementation of FA VHDL implementation of FA (cont d) A fuzzy automaton is a VHDL entity that can be simulated FA s inputs can be given as signals or read from a text file FA s states, inputs and outputs can be visualized as signals and/or dumped in files. The FA can be parametrized: which operators to use, what methods for improving controllability to use, the names of input and output files, etc. Several FA can be simulated in parallel.

15 Investigations performed with our framework We aimed to study the following problems: 1 the efficiency of different operations (t-norms and s-norms) used for FA 2 the effectiveness of some methods proposed in literature in order to overcome the lack of controllability of FA: 1 conservation of state 2 normalization of state We implemented several examples of fuzzy automata from literature and tested them using our framework We used different patterns for the FA s inputs Next we present some results for a FA with 2 inputs and 4 states.

16 Example of controllable behaviour Figure : Algebraic FA with equal inputs and without conservation of state

17 Example of non-controllable FA behaviour Figure : Max-product FA with equal inputs and without conservation of state

18 Simulation results without improvement methods From the four pairs of s- and t-norms used for the functions F2 and F1, only the algebraic FA proved to be controllable For the max-min FA, the degree of membership of its states took the smallest value of the degree of membership of its inputs and after that their values could not be increased For the bounded and drastic FA, the degrees of membership of all their states went to zero and remained zero after that In consequence we decided to try to use different s-norms for function F2, but only algebraic product for F1 the max-product automaton has an uncontrollable behaviour (see figure). The automata with bounded sum used for F2 and the FA with Einstein sum for F 2 behave similarly with algebraic automaton.

19 FA with conservation of state For the max-prod FA, the degrees of membership of the states go to zero and remain zero The method did not improved the behaviour of FA with other pairs of s- and t-norms. The method worsens the behaviour of algebraic FA: the degrees of membership of their states remain very high, mostly between 0.8 and 1.0 this can be explained by the contribution given by the negated values of the degrees of the inputs: when the degree of an input is low, its negation (i.e 1 degree) is high, and the consequence is that the degrees of states remain high. The method was not efficient in our simulations.

20 FA with state normalization It means to determine the maximum value among the degree of membership of all states and to divide all the degrees of membership of the states to that maximum The method implies division operations, which are costly in hardware We tested it only for max-product automaton In our examples the method is inefficient: the degree of membership become high (close to 1) and cannot be reduced after that.

21 Conclusions Motivation We have implemented a flexible and efficient VHDL framework for the study of fuzzy automata The framework permits to test the efficiency of different functions used for state transitions of FA Our simulations show that the maximum and minimum functions (the most used in the literature) do not allow the state of the automata to be controlled by its inputs The algebraic sum and product, the bounded sum and algebraic product, and the combination of Einstein sum and algebraic product proved to be efficient We implemented two methods proposed in the literature in order to improve FA s behaviour: conservation of state and state normalization Both methods gave bad results, which is a new result

22 Future work Motivation To perform simulations and performance studies on other examples of FA To include in our framework other types of FA To perform FPGA implementation (i.e. synthesis) of several types of FA To find some interesting applications for FA.