a compositional reasoning system for executing nonmonotonic theories of reasoning

A Compositional Reasoning System forExecuting Nonmonotonic Theories ofReasoningJ. Engelfriet, J. Treur*Vrije Universiteit Amsterdam, Department of Artificial Intelligence,De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands

In this paper the framework DESIRE for the design of compositional reasoning systems andmulti-agent systems was applied to build a generic nonmonotonic reasoning system. Theoutcome is a general reasoning system that can be used to model different nonmonotonicreasoning formalisms and that can be executed by a generic execution mechanism. The mainadvantages of using DESIRE (for example, compared to a direct implementation in a program-ming language such as PROLOG) are that the design is generic and has a transparent compo-sitional structure, and the explicit declarative specification of both the static and dynamic aspectsof the nonmonotonic reasoning processes, including their control. © 2003 Wiley Periodicals, Inc.

1. INTRODUCTION

An agent that is reasoning about the world often needs to draw (defeasible)conclusions that are not logically entailed by its (incomplete) knowledge about theworld. Nonmonotonic reasoning systems can be used to model these reasoningpatterns. Implementations of agents that can reason in a defeasible manner need toinclude an implemented nonmonotonic reasoning system. In different applications,agents may need different types of nonmonotonic reasoning. Therefore, it is usefulto develop a generic reasoning system that covers different types of nonmonotonicreasoning (as opposed to implementations for one specific nonmonotonic formal-ism as in, e.g., Refs. 1, 2, or 3). Moreover, the development of such a genericreasoning system can be made in a transparent manner if a central role is playedby an implementation-independent design specification based on current softwareengineering principles such as compositionality and information hiding.

Reasoning can be seen as an activity of an agent taking place in time. Theagent starts with a set of initial beliefs to which it applies some (nonmonotonic)rules to arrive at a new state, in which it has more knowledge. In this new state, theagent again may apply rules to arrive at the next state. Viewing the agent from the

*Author to whom all correspondence should be addressed: e-mail: [email protected].

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 18, 593–607 (2003)© 2003 Wiley Periodicals, Inc. Published online in Wiley InterScience(www.interscience.wiley.com). • DOI 10.1002/int.10108

outside, we can look at the knowledge of the agent at all points in time. Thus, moreformally, we can describe the behavior of this agent by a reasoning trace or atemporal model that describes what the agent knows at the different points in time.Because (nonmonotonic) reasoning may be nondeterministic (the agent may havea choice of which rules to apply), we should allow multiple traces that start withthe same set of initial beliefs. If we want to describe the reasoning of the agentexhaustively, we may incorporate traces starting with any set of initial beliefs. Tospecify such sets of temporal models (or traces), temporal logic can be used. Thisgeneral temporal view of reasoning was discussed in Ref. 4, where we introduceda formal notion of a temporal model and a temporal specification language suitedto specify sets of temporal models. In this article, we present a design andspecification of an executable nonmonotonic reasoning system to implement thisgeneric approach to nonmonotonic reasoning. For any specification of a theory ofnonmonotonic reasoning, given as an input to the system, the possible reasoningtraces are generated.

The system was developed using the compositional modeling environmentDESIRE (framework for DEsign and Specification of Interacting REasoning com-ponents; see Refs. 5, 6, and 7) developed at the AI Department of the VrijeUniversity Amsterdam. This environment for the development of compositionalreasoning systems and multiagent systems has been used successfully to designvarious types of reasoning systems and multiagent systems and applications inparticular domains. The DESIRE software environment offers a graphical editor,an implementation generator, and an execution environment to execute specifica-tions automatically. In its use to build complex reasoning systems, DESIRE can beviewed as an advanced theorem proving environment in which both the knowledgeand the control of the reasoning can be specified in an explicit, declarative, andcompositional manner.

In this study, in Section 2 we briefly summarize the generic approachintroduced in Ref. 4. In Section 3 a very brief introduction to DESIRE is presented.In Section 4 the design of the nonmonotonic reasoning system is discussed. Anexample trace of the system is described in Section 5, and Section 6 givesconclusions and suggestions for further research.

2. TEMPORAL SPECIFICATION OFNONMONOTONIC REASONING

In this section, we briefly review the main notions introduced in Ref. 4. Thebasic language in which the agent expresses its beliefs will be propositional logic.Semantically, a state is a set of propositional models, representing the set of beliefsconsisting of the formulas true in all models in the state.

DEFINITION 2.1. (temporal model)

(i) An information state is a nonempty set of propositional models. A proposi-tional formula � is true in an information state M, denoted M �, if m �,for all m � M. For two information states M and N, we say M contains moreinformation than N, denoted N � M if M � N.

594 ENGELFRIET AND TREUR

(ii) A temporal model is a sequence (�s)s�IN, where each �s is an informationstate.

(iii) A temporal model � is conservative if �s � �s�1 for all s � IN.(iv) The refinement ordering � on temporal models is defined by:

� � � N @ s: �s � �s and �0 � �0

Note that when N � M, for all propositional formulas � we have N �f M �,so that indeed M contains more information than N. The conservativity of atemporal model means that the agent never forgets anything it has previouslydeduced. The temporal language we will introduce contains the operators H0, F,G, and C, which refer, respectively, to knowledge at point 0, knowledge at somepoint in the future, at all points in the future, and to knowledge in the current timepoint. The formal semantics of these operators are as follows.

DEFINITION 2.2. (temporal interpretation)

(a) For a propositional formulae �,

��, s� F�N ? t � IN, t � s and �t �

��, s� G�N @ t � IN t � sf�t �

��, s� C�N�s �

��, s� H0�N�0 �

(b) For a temporal formula �, (�, s) ¬� N it is not the case that (�, s) �(c) For a set A of temporal formula, (�, s) �A N for all � � A: (�, s) �(d) A formula � is true in a model �, denoted � �, if for all s � IN: (�, s) �(e) A set of formulas T is true in a model �, denoted � T, if for all � � T, �

�. We call � a model of T

In a specification, we allow only a certain format in the temporal formulas,which correspond to the application of a rule by the agent.

DEFINITION 2.3. (reasoning theories)

(a) A formula is called a (nonmonotonic) reasoning formula if it is of the form� ∧ � ∧ � ∧ � 3 G, where

� � � {H0� � A} for a set of propositional formulae A� � � {¬H0�� B} for a set of propositional formulae B� � � {¬F�� C} for a set of propositional formulae C� � � {C � � D} for a set of propositional formulae D � is a propositional formula

(b) A set Th of reasoning formulas is called a theory of reasoning.

NONMONOTONIC THEORIES OF REASONING 595

In a temporal model of a theory of reasoning, all (nonmonotonic) rules that areapplicable at some point in time actually have been applied by the agent. But wealso want to make sure that the agent knows nothing more than what it can deduce.So we will look at models of a theory (the rules have been applied) in which theknowledge of the agent over time is minimal (the agent knows nothing more).

DEFINITION 2.4. (minimal temporal model). A temporal model � is called aminimal model of a theory Th if it is a model of Th and for any model � of Th,if � � � then � � �.

The minimal models of a theory describe the reasoning process of an agentspecified by this theory. The compositional reasoning system we will describe inSection 4 will find these minimal models by “executing” the theory. Therefore, ourapproach can be seen as an executable modal (nonmonotonic) logic (see Ref. 8).

We will give an example of such a theory. In Ref. 9 it was established thatthere exists a faithful translation of Reiter’s default logic into the temporallanguage introduced previously. The minimal models of the translation correspondto extensions of the theory. We will not go into the details of the translation, butrather give an example. Let the following default theory �W, D� be given: W � {a,d, b3 ¬c} and D � {(a: b)/b, (d: c)/c, (b: ¬c)/e}. The atoms a and d will beinitial facts. Formally, the formula b 3 ¬c also would be a fact, but because thegeneric reasoning system to be described does not perform general propositionalreasoning (it uses a subset of natural deduction called chaining), we will translateit into the following two rules that describe the application of the formula:

(1) Cb 3 G¬ c(2) Cc 3 G¬ b

The default rules are translated as follows (See (3)–(5)):

(1) Cb 3 G¬ c(2) Cc 3 G¬ b(3) Ca ∧ ¬F¬b 3 Gb(4) Cd ∧ ¬F¬c 3 Gc(5) Cb ∧ ¬Fc 3 Ge

The idea behind this translation is that we should add the conclusion of a defaultrule like (b: ¬c)/e when b has been derived, and ¬c remains consistent throughoutour further reasoning, which means we should never derive c in the future. Theminimal models of the resulting theory are shown in Fig. 1. In this figure, a 1 meansthe corresponding atoms have been derived, a 0 means its negation has beenderived, and a u means that neither the atoms nor its negation have been derived.The model M corresponds to the extension Cn({a, b, ¬c, d, e, b 3 ¬c}), andthe model N corresponds to the extension Cn({a, ¬b, c, d, b3 ¬c}). The readerfamiliar with default logic may check that these are (the only) extensions of �W,D�. We will use this example in Section 5, where an example trace of the generic


reasoning system is given. In this same fashion, one can translate a logic programto a temporal theory, yielding the stable semantics.

3. A SPECIFICATION FRAMEWORK FORCOMPOSITIONAL SYSTEMS

In the framework DESIRE knowledge of

(1) A task hierarchy(2) Information exchange(3) Sequencing of tasks(4) Task delegation(5) Knowledge structures

are explicitly modeled and specified. Each of these types of knowledge is discussedin the following subsections.

3.1. Task Composition

To model and specify a composition of tasks, knowledge of the followingtypes is required

● A task hierarchy● Information a task requires as input● Information a task produces as a result of task performance● Meta-object relations between tasks

Within a task hierarchy, composed and primitive tasks are distinguished; in contrastto primitive tasks, composed tasks are tasks for which subtasks are identified.Subtasks, in turn, can be either composed or primitive. Tasks are directly relatedto components; composed tasks are specified as composed components and prim-itive tasks are specified as primitive components. Primitive components are exe-cuted by a simple classical deduction system, based on the inference relationchaining (modus ponens and conjunction introduction).

Information required/produced by a task is defined by input and outputsignatures (or information types) of a component. The signatures used to name theinformation are defined in a predicate logic with a hierarchically ordered sort

Figure 1. Minimal models.


structure (order-sorted predicate logic). Units of information are represented bythe ground atoms defined in the signature.

The role information plays within reasoning is indicated by the level of anatom within a signature; different (meta)levels may be distinguished. In a two-levelsituation the lowest level is termed object level information, and the second levelis termed meta-level information. Meta-level information contains informationabout object level information and reasoning processes, e.g., for which atoms thevalues are still unknown (epistemic information). Often, more than two levels ofinformation and reasoning occur, resulting in meta-meta-information and reason-ing.

3.2. Information Exchange Between Tasks

Information exchange between tasks is specified as information links betweencomponents. Each information link relates the output of one component A to theinput of another component B by specifying which truth value of a specific outputatom of A is linked with which truth value of a specific input atom of B. Atoms canbe renamed; each component can be specified in its own language, independent ofother components. The conditions for activation of information links are explicitlyspecified as task control knowledge, a kind of knowledge of the sequencing oftasks.

3.3. Sequencing of Tasks

Task sequencing is explicitly specified within components as task controlknowledge. Task control knowledge includes not only knowledge of which tasksshould be activated when and how, but also knowledge of the goals associated withtask activation and the extent to which goals should be derived. These aspects arespecified as component and link activation together with task control foci andextent to define the component’s goals. Components are, in principle, black boxesto the task control of an encompassing component; task control is based purely oninformation about the success and/or failure of component reasoning. Reasoning ofa component is considered to have been successful with respect to its task controlfocus if it has reached the goals specified by this task control focus to the extentspecified (e.g., any or every).

3.4. Delegation of Tasks

During a modeling process, a task as a whole is modeled. In the course of themodeling process, decisions are made as to which tasks are (to be) performed bywhich agent. This process results in the delegation of tasks to the parties involvedin task execution.

3.5. Knowledge Structures

During design, an appropriate structure for domain knowledge must bedevised. The meaning of the concepts used to describe a domain and the relations


between concepts and groups of concepts are determined. Concepts are required toidentify objects distinguished in a domain (domain-oriented ontology), but also toexpress the methods and strategies used to perform a task (task-oriented ontology).Within primitive components, concepts and relations between concepts are definedin hierarchies and rules (based on order-sorted predicate logic). In a specification,references to appropriate knowledge structures (specified elsewhere) suffice; com-positional knowledge structures are composed by reference to other knowledgestructures.

4. A GENERIC COMPOSITIONAL NONMONOTONICREASONING SYSTEM

The nonmonotonic reasoning task of finding minimal models of a reasoningtheory is modeled as a composition of two subtasks; the first subtask generates allpossible continuations of the reasoning trace, and the second subtask selects acontinuation. Within the first task, it is determined which rules are applicable.Applicable rules are the rules

� ∧ � ∧ � ∧ � 3 G

where

� � �{H0� � A} for a set of propositional formula A� � �{¬H0�� B} for a set of propositional formula B� � �{¬F�� C} for a set of propositional formula C� � �{C � � D} for a set of propositional formula D � is a propositional formula

for which the conditions that refer to the past and present (i.e., �, �, and �) arefulfilled. Next, for each of the applicable rules, for the future-directed conditions �two possibilities are generated:

● Either the conditions that refer to the future will be fulfilled in the reasoning trace thatis generated, or

● These future-directed conditions will not be fulfilled.

In the first case, the rule will contribute its conclusion to the reasoning process andwe have to make sure in the future that the future conditions are indeed fulfilled(we add constraints to ensure this). In the second case, no explicit contribution willbe made by the rule. However, by the subsequent generation of the reasoning traceit will have to be guaranteed that the future-directed conditions indeed will beviolated (and we again add constraints to ensure this). In this sense, an impliciteffect on the reasoning trace occurs: all traces that do not contradict theseconditions will be rejected (see Ref. 10). Note that we do not try to execute thefuture condition; we merely guess whether it will be fulfilled. In this respect, theserules are not of the form “declarative past implies imperative future” of Ref. 11.

The design of the compositional nonmonotonic reasoning system has beenspecified in DESIRE, according to the five types of knowledge discussed in Section3. Five levels of abstraction are distinguished in the task hierarchy (see Fig. 2).


4.1. Top Level of the System

At the highest level, the system consists of four components. During thereasoning process, in the component maintain_current_state, the facts are repre-sented that have been derived. The component maintain_history stores relevantaspects of the reasoning process in order to perform belief revision if required. Thereasoning is performed by the component generate_possible_continuations, whichgenerates the possible next steps of the reasoning trace and select_continuation,which chooses one of these possibilities. By this selection the actual next step inthe reasoning trace is determined. In Fig. 3 the information exchange at the toplevel of the system is depicted. (In this figure and the following figures, we haveleft out some links that go from a component to itself. For instance, such links aresometimes used to model a closed-world assumption.)

4.2. Generate Possible Continuations

Within the component generate_possible_continuations two subcomponentsare distinguished: test_applicability_of_rules and determine_combinations (seeFig. 4 for the information flow at this level).

The former component is primitive and determines the rules that are appli-cable in the current state of the reasoning process. Its knowledge base consists ofrules, e.g., of the form

If posH0condition (L: literals, R: rules)

Figure 2. Complete task hierarchy of the system.


And not initial_fact (L: literals)Then not applicable (R: rules)

If currentcondition (L: literals, R: rules)And not current_fact (L: literals)Then not applicable (R: rules)

By application of a form of the closed-world assumption, the applicable rules arederived. The second component determines for each of the applicable rules twopossibilities: it is assumed that either (�) the conditions of the rule that refer to thefuture will be fulfilled in the reasoning trace that is generated or (�) thesefuture-directed conditions will not be fulfilled.

The component determine_combinations is rather simple (see Fig. 5). The appli-cable rules are treated one by one and combinations are constructed. Both subcomponentsare primitive. The knowledge base of the first subcomponent just consists of one rule:

If applicable (R: rules)And not covered (R: rules)

Figure 3. Information exchange at the top level of the system.

Figure 4. Information exchange within “generate possible continuations.”


Then infocus (R: rules)

The knowledge base of the second subcomponent consists of the following tworules:

If old_combination (C: combinations)And infocus (R: rules)And future dependent (R: rules)Then new_combination (app (C: combinations, tup(R: rules, neg)))

If old_combination (C: combinations)And infocus (R: rules)Then new_combination (app(C: combinations, tup(R: rules, pos)))

These rules build new combinations from old combinations and the rule in focus.The predicate app (for append) is used to build up a list, and the predicate tup (fortuple) is used to make tuples. Only rules with a part that refers to the future (futuredependent) are allowed not to be applied (meaning that tup(R: rules, neg) mayoccur in a combination).

Figure 6. Information exchange within select continuation.

Figure 5. Information exchange within determine combinations.


4.3. Select Continuation

Within the component select_continuation, focus combinations are chosenone by one and processed (see Fig. 6). Processing a combination is performed byfirst making an interpretation of the information represented by a combination andsubsequently checking on consistency against the current facts and checking theconstraints (see Figs. 7 and 8). The knowledge base of the primitive componentcheck_consistency consists of three rules, an example of which is

If next (A: atoms)And current (neg(A: atoms))Then inconsistency

5. EXAMPLE TRACE

In this section, we will give an example of the execution of our generic reasoningsystem. The theory of reasoning is the translation of the example default theory ofSection 2. When this theory is given to the generic reasoning system, the following willhappen (in this list, on the left we will indicate the literals that are derived, on the rightwe will indicate the component in which it is derived, in brackets; not everythingderived in every component is listed and no information links are listed).

Figure 7. Information exchange within process combinations.

Figure 8. Information exchange within check combinations.


The minimal model displayed corresponds to the extension based on the literals {a,b, ¬c, d, e}. When the user clicks “no,” the execution will stop. Otherwise aretrieve will be performed by maintain_history, after which choose_combinationwill choose the combination in which r5 is not applied (but r1 is). Eventually thiswill lead to a violated constraint, so another retrieve will occur. Then, choose_c-ombination will take the last combination (for r3 and r4), in which r3 is not appliedand r4 is. Ultimately, the second minimal model will be found and displayed(corresponding to the literals {a, ¬b, c, d}). If the user wants to search for anotherminimal model, none will be found, and a message indicating this will be dis-played, after which execution ends.

6. DISCUSSION

In this study the framework DESIRE for the design of compositional reason-ing systems and multiagent systems was applied to build a generic nonmonotonicreasoning system. The outcome is a general reasoning system that can be used tomodel different nonmonotonic reasoning formalisms (see, for instance, Ref. 12),and that can be executed by a generic execution mechanism. The main advantagesof using DESIRE (compared with a direct implementation in a programminglanguage such as PROLOG) are

Figure 9. User interaction.


● The design is generic and has a transparent compositional structure; it is easily readable,modifiable, and reusable. The generic nonmonotonic reasoning system is easily usableas a component in agents that are specified in DESIRE. For example, if an agent isdesigned that has default knowledge, then the generic reasoning system can be includedas one of the agent’s components and the representation of the agent’s default knowl-edge can be translated to the temporal representation of the generic reasoning system.

● Explicit declarative specification of both the static and the dynamic aspects of thenonmonotonic reasoning processes, including their control. The current system gener-ates one or all reasoning traces that are possible without any specific guidance. However,a number of approaches to nonmonotonic reasoning have been developed that inaddition use explicit knowledge about priorities between nonmonotonic rules (e.g.,Refs. 13 and 14). This knowledge can be incorporated easily within the component“select_continuation,” in particular, within its subcomponent “choose_combina-tion.”

Even though the efficiency of the reasoning system can be improved (by addingknowledge to prevent generating possible continuations that can be seen easily toviolate constraints, by adding heuristic knowledge in the selection of continuations,etc.), implementations for specific nonmonotonic formalisms (e.g., for defaultlogic, see Refs. 2, 3, 15 or 16) often can be made more efficient. Generally, theylack the ability to handle different kinds of nonmonotonic reasoning and theextendibility of our approach. Also, they cannot handle dynamic queries (e.g., hasa literal been derived before time point 3).

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a compositional reasoning system for executing nonmonotonic theories of reasoning

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