Automatic Diagnosis with Constraint Solver

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International Journal of Information and Computer Science (IJICS) Volume 4, 2015 doi: 10.14355/ijics.2015.04.005

Automatic Diagnosis with Constraint Solver Xuenong Zhang*1, Wei Lu2 Department of E-commerce, GuangDong Pharmaceutical University, Guangzhou, China zxnxlq@163.com

*1

Abstract A constraint-based model of diagnostic problem is proposed and the classical diagnostic problem can be transformed to a problem of constraint solving. Then, a depth first search with branch and bound approach to reduce the complexity of traveling through the diagnostic space is proposed. Its soundness and completeness is proved. And its complexity is also analyzed.

Keywords Automatic Diagnosis; Constraint Solving; Diagnostic Model

Introduction Generally, a diagnostic problem is characterized by a set of observations to account for the differences between the system’s actual behaviors and its expected behaviors. Because of its generality and its dramatic importance in many application domains, automated diagnosis has been receiving considerable attention in the field of AI research.

The classical diagnostic method is built on the well-known consistency-based theory [1] and the classical model describes the system's structure and behavior usually by the first-order language. This theory was extended by de Kleer et al. [2] in 1992. Another logical theory of diagnosis, developed by Console et al. [3] and others, was called abductive diagnosis. They used logical implications from causes to effects to represent causal knowledge. The diagnostic process is formulized as reasoning from effects to causes.

The computational complexity of diagnosis is one of the well-known problems that need to be tackled in order to deploy the real-world applications of Model-based diagnosis. Several relevant contributions can be found in the literatures [4, 5].

In recently years, some effective constraint solvers [6] have been proposed. For utilizing constraint solving technology, we propose a constraint-based model of diagnostic problem and a diagnostic algorithm in this paper.

Constraint-Based Model In this section we introduce the diagnostic model based on constraint.

Definition 1 Diagnostic Problem Diagnostic problem is a triple (SD, COMPS, OBS) where: (1) SD is a function ℘: COMPS →R, where R is a set of constraints corresponding to components. Var(R)= Varinter(R) + Varobs(R) is the set of constraint variables, and Varinter(R) is the set of internal variables, Varobs(R) is the set of external variables; (2) COMPS is a set of components; (3) OBS is a set of observes.

Definition 2 Diagnosis Space SH=2COMPS is called diagnosis space, and diagnostic hypothesis h∈SH is a set of components.

Definition 3 Observe Space Observe space SOBS=dom(v1) ×…× dom(vi) is a set of all possible observes, where vi∈Varobs(R), dom(vi) is the domain of

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variable vi.

Definition 4 Diagnostic Solution A solution D of a diagnostic problem (SD, COMPS, OBS) is a diagnostic hypothesis h where the constraint problem {℘ (a1),…, ℘ (an) | ai∈COMPS-h} ∪OBS is satiable.

Theorem 1 If a diagnostic hypothesis h is not a solution of a diagnostic problem (SD, COMPS, OBS), any h′⊂h is also not a solution.

Proof: If a diagnostic hypothesis h is not a solution of a diagnostic problem (SD, COMPS, OBS), the constraint problem {℘ (a1),…, ℘ (an) | ai∈COMPS-h} ∪OBS is not satiable. Therefore, {℘ (a1),…, ℘ (an) | ai∈COMPS- h′} ∪OBS is not satiable because of h′⊂h.

Definition 5 Minimal Diagnostic Solution A solution D is minimal if and only if not exist any solution D′⊂D. The intuitive mean of diagnostic solution is: because components in set D are abnormal, their outputs are propagated by normal components and lead to the abnormal output of system.

On the basis of above, we can search minimal diagnostic solutions with a constraint solver.

Searching All Minimal Diagnostic Solutions In this section, the idea of searching all minimal diagnostic solutions will be described with a simple example at first. Then, we will present the algorithm and prove its soundness and completeness.

Example 1 The poly-box system depicted in Fig. 1 contains five components: M1, M2 and M3 are multipliers, and A1 and A2 are adders. The system's inputs are: A = 2, C = 2, E = 2, B =3, D = 3, F = 3; its outputs are: G =10, H =12. There are four minimal diagnostic solutions: {(A1), {M1}, {M2, A2} ,{M2, M3}.

FIG 1: POLY-BOX SYSTEM

For finding all minimal diagnostic solutions, a naïve approach works as follow circularly: (1) Generate a hypothesis h; (2) Test if ℘ (a1),…, ℘ (an) | ai∈COMPS-h} ∪OBS is satiable by a constraint solver; (3) If it is satiable then add h to solution and delete the supersets of h.

Because the diagnostic space includes 2COPMS hypothesizes, it is important to choose the candidate hypothesis. On the basis of theorem 1 and definition 5, there are three facts which can be utilized for searching minimal diagnostic solutions: (1) all subsets of a minimal diagnostic solution are not minimal diagnostic solutions; (2) all supersets of a minimal diagnostic solution are not minimal diagnostic solutions; (3) all subsets of a diagnostic hypothesis which is not a diagnostic solution are not diagnostic solutions.

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International Journal of Information and Computer Science (IJICS) Volume 4, 2015

With these facts, we will propose a depth first search with branch and bound approach to reduce the complexity of traveling through the diagnostic space.

Algorithm 1 diagnosis (hypothesis, work-list, stack, prune-list) lw:=work-list; s:=stack; pl:=prune-list; s.push(hypothesis); while stack is not empty do { h:= s.pop(); if h is not marked then { mark h; if {℘ (a1),…, ℘ (an) | ai∈COMPS-h} ∪OBS is satiable then { removes all supersets of h from the solution; if no subset of h is in solution then insert h to solution and lw for all subsets h′ where |h′|=|h |-1 do { if h′ is not a subset of any member in the prune-list pl then { if lw=∅ then s.push(h′) else { S:=lw.first; if S⊆ h′ then {for all subsets h′′ where |h′′|=| h′ |-1 and S⊄ h′′ do diagnosis (h′′, lw/{S}, s, pl)} else diagnosis (h′, lw/{S}, s, pl) } } } else {removes all subsets of h from pl; if no superset of h is in pl then insert h to the pl; } } } } Theorem 2 Algorithm 1 is sound and complete.

Proof: Because any hypothesis h inserted to solution is satisfied with℘ (a1),…, ℘ (an) | ai∈COMPS-h} ∪OBS and there is not a subset of h in solution, algorithm 1 is sound.

According to algorithm 1, all satiable hypothesizes are divided into three parts: (1) not be inserted to solution, which is the superset of a member of solution; (2) be removed from solution, which is also the superset of a new member of solution; (3) minimal diagnostic solution. Therefore, all minimal diagnostic solutions will be found. In another word, algorithm 1 is complete.

Example 2 For finding all minimal diagnostic solutions of the problem in example 1, all tested hypothesizes in the diagnostic space are depicted in Fig. 2. The underlined hypothesis is tested by constraint solver. Algorithm 1 only chooses 10 candidates from 32 hypothesizes to test and finds 4 minimal diagnostic solutions.

Complexity Analysis of Algorithm The algorithm goes over through the hypothesis h′′ where |h′′|=| h′ |-1 and s⊄ h′′. There are at most |S|-1 such hypothesizes. For each one, the algorithm is recursively called with next m in lw as long as lw is not empty.

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Obviously, S is bounded by the number of components n and lw is bounded by |solution|. Therefore, the algorithm will be invoked n |solution| times at most by itself. In each time of the invoked algorithm, it will add at most one hypothesis to the stack. Thus, it provides at most n |solution|+1 trial hypothesizes until S is added to solution.

In the first time, only COMPS is in the stack and solution is empty and any S satisfied with ℘ (a1),…, ℘ (an) | ai∈COMPS-S} ∪OBS is a subset of COPMS. Therefore, there are at most |solution| n |solution|+1 hypothesizes will be tested by constraint solver.

Therefore, the runtime complexity is O(|solution| n |solution|+1).

FIG 2: DIAGNOSTIC SPACE AND TESTED HYPOTHESIZES

Conclusions For utilizing the effective constraint solver, a constraint-based model of diagnostic problem is proposed, and the classical diagnostic problem can be transformed to a problem of constraint solving. On the basis of above, invoking a constraint solver, a depth first search with branch and bound approach to reduce the complexity of traveling through the diagnostic space is proposed. It can search all minimal diagnostic solutions effectively.

ACKNOWLEDGMENT

This paper is supported by Guangdong Natural Science Foundation (Grant No: S2011010002511).

REFERENCES

[1] R. Reiter, A theory of diagnosis from first principles. Artificial Intelligence, 1987, 32(1);57-96.

[2] J. de Kleer, A.K. Mackworth, R. Reiter, Characterizing diagnoses and systems, Artificial Intelligence, 1992, 52:197-222.

[3] Console L., Torasso P., A spectrum of logical definitions of model-based diagnosis. Computational Intelligence, 1991,7(3):133-141.

[4] Yannick Pencole. A formal framework for the decentralised diagnosis of large scale discrete event systems and its application to telecommunication networks. Artificial Intelligence, 2005, 164: 121-170.

[5] Luigi Portinale, Diego Magro, Pietro Torasso. Multi-modal diagnosis combining case-based and model-based reasoning:a formal and experimental analysis. Artificial Intelligence 2004,158:109–153.

[6] Ji xiaohui, Zhang jian. Constraint processing. ACTA Automatica Sinica, 2007,33(2):125-131.

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