Modelling and Simulation in Engineering

Volume 2018 (2018), Article ID 7842402, 13 pages

https://doi.org/10.1155/2018/7842402

## Qualitative Reasoning for Quantitative Simulation

Correspondence should be addressed to Mehmet Fatih Hocaoğlu

Received 11 May 2017; Revised 5 August 2017; Accepted 5 September 2017; Published 1 January 2018

Academic Editor: Luis Carlos Rabelo

Copyright © 2018 Mehmet Fatih Hocaoğlu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Qualitative simulation is a well-known reasoning technique that involves the use of simulation technologies. Reasoning is made to determine qualitative values and change directions of system variables, and it is done for each time point and time interval following the time point. Qualitative variables possess continuous qualitative value sets that are discretized by landmark points. Qualitative simulation uses qualitative time representation and its quantitative value is of no interest. The main purpose of this study was to develop a technique to determine time steps for a quantitative simulation under guidance of qualitative information. The proposed technique determined time advances using qualitative and quantitative information together to obtain a robust time step as wide as possible for simulation time advances. For this purpose, sign algebraic properties and derivation roots of quantitative equations and qualitative variable values with their change directions were used to compute time advances. In the approach, qualitative simulation determined landmark points to be advanced, and quantitative simulation calculated the duration required. Using the proposed algorithm, the simulation is advanced instead of iterating simulation time for a predefined time step and checking whether or not there is any activity in the interval, directly to the time points that are qualitatively different.

#### 1. Introduction

Qualitative simulation (QS) is defined as a reasoning technique using simulation, and it generates possible value and their change directions that are defined as state on time axis for continuous event systems, using qualitative information of the system being simulated, and the states following each other in time point and time interval sequences constitute behavior trees [1, 2]. Qualitative differential equations (QDEs) utilized by QS represent a function family, not a specific function; they accommodate a set of functions beneath. Time axis in QS is constituted by time points and interval between time points. Qualitative state descriptions are computed for both time points and intervals, and they constitute a simulation trajectory. Because of the incomplete information used by QS, each qualitative variable yields next value set rather than a single value. As a result of this, simulation execution gives possible alternative trajectories.

The main motivation of this study is to develop a simulation time management technique that advances simulation steps to system discontinuity points that are discovered by qualitative reasoning and sign algebra. In contrast to the fixed time advance mechanism for continuous event simulation, this approach maps continuous event simulation into discrete event simulation by discovering events and discontinuity points using qualitative information with quantitative functional analysis. It is known that every time step in continuous event time management may possibly not compute a qualitatively distinct state vector. As seen in the bathtub example given in Section 5, although different numerical values for each time point in a time interval are calculated, the time interval represents the same qualitative state vector, because all numerical values between and are mapped to the same qualitative interval. For example, qualitative meaning is that water level is between zero and full and it is increasing () (different cases for tub problem are given in Table 2). Until a qualitative landmark point is reached, qualitative interpretations and mathematical properties (sign of derivatives, change direction) for the time interval remain the same.

In this study, a time advance algorithm is developed based on qualitative state vectors and qualitative algebra. The main idea is to calculate a time step as wide as possible to reach the next qualitative state vector point (named as landmark points) that is close to the current state value of simulation model depending on its change in direction instead of reaching the qualitative value by many small time steps that the quantitative equations are solved for. In some sense, this involves mapping a continuous space into a discrete one and allowing it to reach a state point directly, instead of traversing from one state to another by small time steps.

In Section 2, qualitative simulation is summarized and Section 3 gives brief information about the solution procedures for continuous event simulation with a time management taxonomy. While Section 4 gives fundamentals of the proposed qualitative time management approach, the approach is clarified by an example in Section 5. In the last section, advantages and disadvantages of the proposed solution are discussed.

#### 2. Qualitative Simulation

Qualitative simulation (QS) is accepted as one of the qualitative reasoning (QR) techniques which are utilized to model human reasoning approach. The aim of QS is to generate possible future qualitative behaviors of any kind of system. During behavior generation, initial states, qualitative constraint set, and qualitative value set that are associated with qualitative state variables of the system are handled as an input set.

Several approaches and ontologies are proposed to accomplish this purpose by researchers. Device centered ontology [3], process centered ontology [4], and constraint based approach [2] are well-known methods. QSIM algorithm, as a constraint based approach, is one of the most widely used algorithms in the developed techniques [1]. In constraint based qualitative simulation, qualitative variable values are represented as a discrete value set consisting of landmark points. A landmark point represents a qualitatively distinct model state. QS is performed from time point to time interval and continued. In Table 1, qualitative value transition rules from time point to time interval (P) and time interval to time point (I) are depicted. As seen in the table, a qualitative value of variable is represented by a value or an interval with a change direction. The QSIM representation was developed, in part, to make the abstraction relationship between the qualitative representations and the theory of differential equations explicit and precise. Qualitative differential equation (QDE) is a tuple of four elements, , each of which will be defined below [2], and they are used to filter out spurious behaviors.(i) is a set of variables, each of which is a “reasonable” function of time.(ii) is a set of quantity spaces, one for each variable in .(iii) is a set of constraints applied to the variables in . Each variable in must appear in some constraint.(iv) is a set of transitions which are rules defining the boundary of the domain of applicability of the QDE.