First-Order Diagnosis by Propositional Reasoning: A Representation-Based Approach∗

We present an approach to diagnosis that first compiles a first-order system description into a propositional representation, and then solves the diagnostic problem as a linear programming instance. The compilation method employs techniques from relevance reasoning and may exploit structural features of the system. To solve diagnostic problems, we use cost-based hypothetical reasoning as an intermediate problem representation. The diagnostic engine is based on a 0-1 integer linear programming method that generates a near-optimal diagnosis in polynomial time. A probabilistic interpretation of the solution diagnoses is possible as well. Preliminary empirical results indicate that our approach has great potential to efficiently solve diagnosis problems traditionally represented in first-order Horn logic. Introduction and Motivation A long-standing problem of model-based diagnosis is the complexity underlying the diagnostic reasoning task. Recently, it was shown that the efficiency of the diagnostic engine is highly sensitive to structural properties of the system description. For instance, Stumptner and Wotawa (1997) show that diagnosis achieves high speed for tree-structured systems. If the system description is not already in a form that can be processed efficiently, Darwiche (1998) proposes to compile the system description to a sentence called consequence, which has provably good computational behavior and guarantees on its size. Another property that gained much less attention is a representational property of system descriptions, i.e., the expressivity of the descriptions. This is due to the fact that most approaches to diagnosis only consider propositional representations in the first place. However, many domains are more naturally described in a more expressive language. In this paper, we allow for system descriptions expressed in first-order Horn logic (without function symbols). In order to preserve the advantageous property of ‘propositionality’, we compile the first-order description ∗ This paper is a modified version of Prendinger and Ishizuka (1999b), with focus on the reformation phase. to a propositional one. Our approach is similar in spirit to the work of Williams and Nayak (1997), which starts out with a temporal logic specification of a system, and produces an essentially propositional representation after compilation (by prime implicant generation). By contrast, we start with a first-order specification, and our compilation phase does not rely on the possibly expensive generation of prime implicants. To achieve efficient propositional representations, techniques from relevance reasoning (Levy, Fikes, & Sagiv 1997) are used to filter out the part of the system that is ‘relevant’ for certain observation types, and to economically instantiate variables by appropriate constants (usually denoting values of system attributes). As a result, we may retain the compactness of firstorder descriptions, while being able to process an efficient propositional theory. Given a propositional system description, we will use the Networked Bubble Propagation (NBP) mechanism (Ohsawa & Ishizuka 1997), a high-speed hypothetical (‘abductive’) reasoner, as the diagnostic engine. Focusing on least-expensive (most-probable) diagnoses is realized by assigning a numerical weight to each hypothesis (possible system fault). The NBP mechanism tries to find an optimal solution, i.e., a diagnosis with a minimal sum of individual faults’ weights, and actually generates a near-optimal solution in polynomial time of approximately O(n), where n the number of hypotheses in the problem formulation. Since the efficiency of the NBP mechanism relies on methods from 0-1 integer linear programming, diagnosis problems will be translated to instances of linear programming problems. Williams and Nayak (1996) follow a similar approach, by recasting their model-based configuration manager as a combinatorial optimization problem. The rest of the paper is organized as follows. In the following section, we explain the notions of ‘diagnostic problem’ and ‘diagnosis’. Then we introduce a general framework for extracting information relevant to finding diagnoses, relative to certain observation types and the structure of the system. Moreover, we discuss a rather sophisticated instantiation method based on relevance reasoning. Next, we explain how hypothetical reasoning problems (corresponding to diagnostic problems) can be seen as problems of integer linear programming. We also report on some preliminary experimental results obtained in testing our approach. Finally, we briefly discuss related work and draw some conclusions. Definition of Diagnosis A diagnostic problem is characterized by a set of observations to be explained, given a behavioral model of some system (device). The behavioral model of a system describes its normal and/or faulty behavior. A solution to a diagnostic problem (a diagnosis for short) is a set of hypotheses which, if assumed, would ‘explain’ the observations. The following definition borrowed from Console and Torasso (1991) subsumes both abductive and consistency-based definitions of diagnosis. Integrity constraints are not present in the original formulation. Definition 1 A diagnostic problem DP is a quadruple DP = 〈T, I, CXT, 〈O, O−〉〉, such that • T is a set of Horn clauses that contains no cycles, i.e., a state must not be a direct or indirect cause of itself. T is called a behavioral model. Base predicates in T denote either contextual (‘boundary’, input) conditions or states that can be assumed as hypotheses. We use H to denote the set of hypotheses. • I is a set of inconsistency constraints, i.e., a set of Horn clauses of the form ⊥ ← q1(X̄1) ∧ ... ∧ qn(X̄n) where each qi denotes a context condition or a hypothesis, and the symbol ⊥ denotes the logical constant falsum. • CXT is a set of variable-free (ground) atoms that denote context conditions. • O denotes the set of observations that have to be covered by the solution; O− is the set of observations that ‘contradict’ (or ‘conflict with’) the observations. Console and Torasso (1991) observe that a diagnostic problem (as defined above) has the same structure as a hypothetical reasoning problem (HRP). Consequently, the notion of ‘diagnosis’ can be explained by giving a definition of a solution to a HRP. The following definition extends their definition. Definition 2 Let HRP = 〈T, I, CXT, 〈O+,O−〉〉 be a hypothetical reasoning problem, and H a set of hypotheses. A set H ⊆ H is a solution hypotheses set for HRP if and only if • for each m ∈ O: T ∪ CXT ∪H ` m; • T ∪ CXT ∪H ∪ O− 6` ⊥; • I ∪ CXT ∪H 6` ⊥. The first condition says that elements from O must be covered by the solution, whereas the second condition says that O− has to be consistent with the solution. We also account for the case where certain solution sets are not admissible, given the inconsistency constraints and system inputs (third condition). This condition is not present in (Console & Torasso 1991). function factorize(T ) return a partition P of T . Input theory T and P = {{Q}}, where Q is the query.