Intelligent Systems

    Problem-Solving Methods

Agenda

  • Motivation
  • Technical Solution
    • Development of Knowledge-based systems towards Problem-Solving Methods
    • Problem-Solving Methods
  • Illustration by a Larger Example
  • Extensions
  • Summary
  • References

    MOTIVATION

Motivation

  • In order to allow automation in the achievement of complex problems we should like a general solution with the following characteristics:

      Knowledge
      • Based on reasoning with knowledge;
      • Works with a declarative , rather than an algorithmic, representation of knowedge;
      Process
      • Represents knowledge on the problem-solving process , i.e., the dynamics of the solution;
      Reuse
      • Allows reuse of reasoning knowledge;
      • Abstracts from implementation and domain to increase reusability.

Motivating Example

  • As a motivating example we consider the task (i.e., goal) of parametric design , which can be defined over:

    • Design space – the space that contains all possible designs;

    • Requirements – a finite set of which are assumed to be provided by the user;

    • Constraints – a finite set of which model additional conditions for a valid design;

    • Preference – a function over the design space which can be used to discriminate between different solutions.

Motivating Example

  • Domain knowledge is implicit in the parametric design description in several places:

    • Design space – the concrete definition of the design space is domain specific knowledge

      [Requirements]

    • Constraints – domain knowledge concerning regularities in the domain in which the design is constructed;

      [Preference]

Motivating Example

  • Formally, a design artifact is described by a set of attribute-value pairs.

    Let A1, ..., An be a fixed set of parameters with fixed ranges R1, ..., Rn:

    • Design space is the cartesian product R1 × ... × Rn;

    • Requirements – a relation R which defines a subset of this space;

    • Constraints – a relation C which defines a subset of this space;

    • Preference – a partial function P having all possible designs as its domain and preference values as its range.

Motivating Example

Motivating Example

  • Several types of design are implicitly defined by these aspects of the problem:
    • Possible design – An element in design space (∈ R1 × ... × Rn);
    • Desired design – A design that fulfills all requirements (∈ R);
    • Valid design – A design that fulfills all constraints (∈ C);
    • Solution – A design which is desired and valid (∈ R C);
    • Optimal solution - A solution for which no other solution which has a higher preference value exists
      (any solution x such that ∀y ∈ R1 × ... × Rn . P(y) ≤ P(x)).
  • There is also the possible extension:
    • Acceptable solution - Given a threshold t, an acceptable solution is any solution x such that P(x) > t.

Motivating Example

Motivating Example

  • An inefficient naive approach to parametric design is called generate & test , which depends on the following inferences:

    • generate – requires knowledge that describes what constitutes a possible design (i.e., the design space);

    • R-test – requires knowledge that describes which possible designs are desired (i.e. the user requirements);

    • C-test – requires knowledge that describes which possible designs are valid (i.e., the domain constraints);

    • select – requires knowledge that evaluates solutions (i.e., knowledge that describes what constitutes a preferred design).

Motivating Example

Motivating Example

  • This naive solution can also be represented as:
      possible design := generate all ;
      valid design := C-test (possible design);
      desired design := R-test (possible design);
      solution := valid design desired design;
      optimal solution := select (solution)
  • Using the definition of acceptable solution this can be made somewhat more efficient as:
      repeat
        possible design := generate one ;
        valid design := C-test (possible design);
        desired design := R-test (possible design);
        solution := valid design desired design;
        acceptable solution := select (solution)
      until ø ≠ acceptable solution

Lessons from Example

  • generate & test has the following characteristics:
    • it separates the different types of knowledge;
    • it is not efficient (all possible designs are generated);
    • It may not terminate if the design space is infinite.
  • From the literature on expert systems [Stefik et al., 1983]:
    • “an important issue is the distribution of knowledge between the generator and the tester: putting as much knowledge as possible into the generator often leads to a more efficient search.”
  • A much more clever strategy is therefore to use these knowledge types to guide the generation of possible designs.
  • However to do so requires strong assumptions about the domain knowledge.

    TECHNICAL SOLUTIONS

    DEVELOPMENT OF KNOWLEDGE-BASED SYSTEMS TOWARDS PROBLEM-SOLVING METHODS

Development

  1. General Problem Solver

  2. Knowledge-is-power hypothesis

  3. Knowledge levels

        3a. Newell’s 3 levels of knowledge

        3b. Brachman’s 5 levels of knowledge

  4. Problem Solving Methods

1. General Problem Solver

  • The General Problem Solver (GPS) is a universal problem solving approach.

  • GPS is the first approach that makes the distinction between knowledge of problems domains and how to solve problems

  • GPS is domain and task independent approach.

  • GPS does not put any restrictions both on the domain knowledge and on the task.

  • Examples of GPS are: automated theorem proving and generic search methods

Automated theorem proving

  • Automatic theorem provers are GPS for which every problem can be expressed as logical inference
  • Automated theorem proving is about proving of mathematical theorems by a computer program

Generic Search Methods

  • Generic Search Methods are GPS for which every problem can be expressed as search

  • One particular example of a Generic Search Method is the A* algorithm.

  • A* works for problems that can be represented as a state space i.e. a graph of states. Initial conditions of the problem are represented as start state , goal conditions are represented as end state

  • A* is an informed search or heuristic search approach that uses the estimation function:

      f(n)=g(n)+h(n)

    • g(n) the cost to get from the star state to current state n

    • h(n) estimated cost to get from current state n to end state

    • f(n) estimated total cost from start state through current state n to the end state

See Lecture 5

1. General Problem Solver (1)

  • However, GPS has a set of limitations :

    • It works in theory but in practice works only on toy problems (e.g. Tower of Hanoi)

    • Could not solve real-world problems because search was easily lost in the combinatorial explosion of intermediate states

2. Knowledge-is-power hypothesis

    Knowledge-is-power hypothesis , also called the Knowledge Principle was formulated by E.A. Feigenbaum in 1977:

    “knowledge of the specific task domain in which the program is to do its problem solving was more important as a source of power for competent problem solving than the reasoning method employed” [Feigenbaum, 1977]

2. Knowledge-is-power hypothesis (1)

  • The Knowledge-is-power hypothesis shifted the focus on how to build intelligent systems from inference to the knowledge base .

  • Problem solving is guided by experiential, qualitative, heuristic knowledge.

  • The meaning of intelligence as knowledge is the common meaning in English world.

  • The Central Intelligence Agency (CIA) defines intelligence as knowledge.

  • The Knowledge-is-power hypothesis lead to the emergence of a new filed i.e. expert systems and a new profession i.e. knowledge engineer

3. Knowledge levels

      3a. Newell’s 3 levels of knowledge

      3b. Brachman’s 5 levels of knowledge

3a. Newell’s 3 levels of knowledge [Newell, 1981]

  • In his work from 1981, Newell tried to answer questions such as:

    • How can knowledge be characterised?

    • What is the relation of this characterisation and the representation of knowledge?

    • What is characteristic about a system which holds knowledge?

  • Newell distinguished 3 levels in the context of knowledge representation:

    • Knowledge Level

    • Logical Level

    • Implementation Level

3a. Newell’s 3 levels of knowledge (1)

    • Knowledge Level

      • The most abstract level of representing knowledge.

      • Concerns the total knowledge contained in the Knowledge Base

      • Example:

        The automated DB-Information system knows that a trip from Innsbruck to Vienna costs 120€ 

3a. Newell’s 3 levels of knowledge (2)

    • Logical Level

      • Encoding of knowledge in a formal language.

      • Example:

        Price(Innsbruck, Vienna, 120)

3a. Newell’s 3 levels of knowledge (3)

    • Implementation Level

      • The internal representation of the sentences.

      • Example:
        • As a String “Price(Innsbruck, Vienna, 120)”
        • As a value in a matrix

3b. Brachman’s 5 Levels of Knowledge [Brachman, 1979]

  • Brachman defines 5 levels for different types of representations.
  • Levels interpret the transition from data to knowledge.
  • Each level corresponds to an explicit set of primitives offered to the knowledge engineer.
  • Ordering of knowledge levels from simple/abstract to complex/concrete:
    • Implementational Level
    • Logical Level
    • Epistemological Level
    • Conceptual Level
    • Linguistic Level

3b. Brachman’s 5 Levels of Knowledge (1)

    • Implementational Level

      • The primitives are pointers and memory cells.

      • Allows the construction of data structures with no a priori semantics

3b. Brachman’s 5 Levels of Knowledge (2)

    • Logical Level

      • The primitives are logical predicates, operators, and propositions.

      • An index is available to structure primitives.

      • A formal semantic is given to primitives in terms of relations among objects in the real world

      • No particular assumption is made however as to the nature of such relations

3b. Brachman’s 5 Levels of Knowledge (3)

    • Epistemological Level

      • The primitives are concept types and structuring relations.

      • Structuring relations provide structure in a network of conceptual types or units. (i.e. inheritance: conceptual units, conceptual sub-units)

      • The epistemological level links formal structure to conceptual units

      • It contains structural connections in our knowledge needed to justify conceptual inferences.

3b. Brachman’s 5 Levels of Knowledge (4)

    • Conceptual Level

      • The primitives are conceptual relations, primitive objects and actions.

      • The primitives have a definite cognitive interpretation, corresponding to language-independent concepts like elementary actions or thematic roles

3b. Brachman’s 5 Levels of Knowledge (5)

    • Linguistic Level

      • The primitives are words, and (lingustic) expressions.

      • The primitives are associated directly to nouns and verbs of a specific natural language

      • Arbitrary relations and nodes that exist in a domain

4. Problem Solving Methods

  • Problem Solving Methods (PSM) abstract from details of the implementation of the reasoning process.

  • Characteristics of PSM [Birmingham&Klinker, 1993] :
    • A PSM specifies which inference actions have to be carried out for solving a given task.
    • A PSM determines the sequence in which these actions have to be activated.
    • Knowledge roles determine which role the domain knowledge plays in each inference action.

    PROBLEM-SOLVING METHODS

Technical Solution Overview




  • What are problem-solving methods (PSMs) ?
    • “Reasoning strategies that gain efficiency through assumptions.
      [Fensel, 2000]

Technical Solution Overview

  • Problem-solving methods achieve an efficient realization of functionality by making assumptions .
  • Assumptions put restrictions on the context of the problem-solving method, such as the domain knowledge and the possible inputs of the method or the precise definition of the functionality.
  • Assumptions play two roles:
    • they formulate requirements on reasoning support that is assumed by PSMs;
    • they put restrictions on the reasoning support that is provided by PSMs.
  • In consequence, assumptions link PSMs with the domain knowledge they use and tasks they are applied to.

Technical Solution Overview

Example Revisited

  • We consider again the task of parametric design.

  • A more efficient method is named propose & revise and depends on the following inferences:
    • propose – derives an initial design based on the requirements;
    • C-test – as before;
    • revise – tries to improve an incorrect design based on the feedback of the C-test step.

  • In other words, instead of proposing a complete design which is then repaired, we can also incrementally develop a design and repair it at each step where a constraint violation occurs.

Example Revisited

Example Revisited

  • A parameter which should receive a value in the next propose step is nondeterministically selected:
    • The selection process does not make further assumptions about knowledge that could guide this second selection step
    • The implicit assumption is that this selection does not a
      affect the performance of the problem solving process and the quality of its result
    • These are very strong assumptions because to improve performance, heuristic methods are definitely needed
    • At any time there is either precisely one applicable propose rule or one user input to derive the value of the selected parameter
    • A parameter should not depend on itself (no recursive derivation rules)

Example Revisited

  • revise is decomposed into:
    • select-violation - nondeterministically selects a constraint violation from those detected by C-test; implicit assumption is that this selection does not influence the performance of the problem solving method and the quality of the result; strong assumption again
    • derive-fixes - computes the set of all possible fix combinations that could possibly resolve the selected constraint violation; each combination must be finite
    • select-fix - selects a fix combination, guided by a cost-function
    • apply-fix - applies a fix combination

Example Revisited

  • Test – propose & revise does not require an explicit R-test, the method assumes that:
    • propose derives only desired designs;
    • revise delivers designs that are desired or that requirement violations that it does not x must be accepted.

  • Selection – does not contain such a process concerning user preferences:
    • It assumes that the propose step and the revise step deliver acceptable (or optimal) solutions or that the functionality of the task is reduced to finding an arbitrary solution.

Description of Problem-Solving Methods

  • The main elements of a specification in the PSM framework are:
    • the task – specifies the goals that should be solved in order to solve a given problem. A second part of a task specification is the definition of requirements on domain knowledge;

    • the problem-solving method – describes an reasoning steps to perform a task as an operational specification separately from a description of the competence, and a description of the requirements on domain knowledge;

    • the domain model – usually ontology-based description in three parts, a meta-level characterisation of properties, the domain knowledge, and (external) assumptions of the domain model;

    • the adapter – maps the different terminologies of the task definition, problem-solving method and domain model. Moreover, gives further requirements and assumptions needed to relate the competence of the PSM with the functionality of the task.

Description of Problem-Solving Methods

Description of Problem-Solving Methods

  • Several proof obligations follow the conceptual model of such a specification of a knowledge-based system:
    • PO-i : the consistency of the task definition to ensure that a model exists, otherwise one could define an unsolvable problem;
    • PO-ii : that the operational specification of the PSM describes a terminating process which has the competence as specified;
    • PO-iii : the internal consistency of the domain knowledge and domain model, also that the assumptions on domain knowledge implies its meta-level characterisation;
    • PO-iv : relationships between the specification elements –
      1. the requirements of the adapter imply the knowledge requirements of the PSM and task,
      2. the adapter’s additional requirements on domain knowledge and assumption guarantee that the competence of the PSM is strong enough for task,
      3. The requirements of the adapter are implied by the meta knowledge of the domain model.

PSM Framework Example

  • To illustrate the description of PSMs we consider search
  • Local search can be refined into other versions using adapters, specifically:
    • Hill-climbing : a local search algorithm which stops when it has found a local optimum , based on recursively considering the successors to a start object, selecting better at each stage;
    • Set-Minimizer : finds a minimal but still correct subset of a given set, with respect to hill-climbing –
      • generic ‘object’ becomes a set,
      • ‘successor’ relationship is hard-wired,
      • preference is implicit;
    • Abductive Diagnosis : receives a set of observations as input and delivers a complete (explains all input data) and parsimonious (no subset of hypotheses explains all observations) explanation .

PSM Framework Example

  • Local search can be represented in the following operational specification:

  • operational specification local search
      output := local search(input);
      local search(X)
        begin
          currents := select1(X);
          output := recursion(currents);
        end
      recursion(X)
        begin
        successors := generate(X);
        new := select2(X, successors);
          if X = new
          then output := select3(X);
          else recursion(new);
        end



    select1(x) ⊆ x


    x ∈ generate(y) ↔ x ∈ input ∧ ∃z.(z ∈ y ∧ successor(z, x))


    select2(y, y’) ⊆ y ∪ y’


    ¬∃x, z . (z ∈ (y ∪ y’) \ select2(y, y’) ∧ x < z)
    y ∪ y’ ≠ ∅ → ∃x.(x ∈ select2(y, y’))


    select3(y) ⊆ y


    ¬∃x, z . (z ∈ (y \ select3(y)) ∧ x ∈ select3(y)
    ∧ x < z)
    y ≠ ∅ → ∃x.(x ∈ select3(y))
    endoperational spec

PSM Framework Example

  • Adapters between refinement levels can be represented as follows:
      PSM refinement adapter set-minimizer -> abduction-method
        correct(x) = complete(x);
        input = {h | h is hypothesis};
        H ⊆ H’ → explain(H) ⊆ explain(H’)
      endPSM refinement adapter
      PSM refinement adapter hill-climbing -> set-minimizer
        correct(input);
        select1(x) = {x};
        successor(x, y) ↔ ∃z . (z ∈ x ∧ y = x \ {z});
        x < y ↔ correct(y) ∧ y ⊂ x
      endPSM refinement adapter
      PSM refinement adapter local-search -> hill-climbing
        |select1(x)| = 1;
        ¬∃z . (z ∈ y’ ∧ y < z) → select2({y}, y’) = {y};
        |select2({y}, y’)| = 1
      endPSM refinement adapter

    ILLUSTRATION BY A LARGER EXAMPLE

Sisyphus-I

  • The Sisyphus-I office allocation problem was the first of a series of test applications for approaches to building knowledge-based systems.

  • The aim is to automate the problem-solving behaviour of ‘Siggi’, a hypothetical domain expert .

  • Specifically Siggi is required to make assignments of the staff of a research group ‘YQT’ to offices.

  • The 4-page problem statement describes the office layout and relevant data about the 15 group members.

  • A ‘protocol’ is provided, describing the steps Siggi takes.

Sisyphus-I Office Layout

Sisyphus-I Protocol

Sisyphus-I YQT Members Data

  • From the protocol can be drawn data such as: 

Sisyphus-I PSM-Based Approach

    • The approach used rests on mapping from task concepts to method concepts as shown [Motta, 1999] :

Sisyphus-I Values Ranges

  • From the protocol can be drawn value ranges, based on given justifications in the protocol:

Sisyphus-I Requirements and Constraints

  • From the protocol can be drawn the following requirements and constraints:

Sisyphus-I Preferences

  • From the protocol can be drawn the following preferences:

Sisyphus-I Cost Function

  • Cost function produces a 4-dimensional vector,

    < n1, n2, n3, n4 > , where:

    • n1 measures the distance between the room of the head of group and that of the secretaries;

    • n2 measures the distance between the manager’s room and the rooms of the head of group and the secretaries;

    • n3 measures the distance between the heads of projects and the head of group and secretaries;

    • n4 provides a measure of the ‘project synergy’ afforded by a solution.

Sisyphus-I Cost Function (cntd.)

  • A design model in the Sisyphus-I domain d1, with cost function
    < n11, n12, n13, n14 > is cheaper than a design model d2, with cost
    < n21, n22, n23, n24 > , iff one or more of the following conditions are satisfied:

    • n11 < n21

    • n11 = n21 and n12 < n22

    • n11 = n21 and n12 = n22 and n13 < n23

    • n11 = n21 and n12 = n22 and n13 = n23 and n14 < n24

Sisyphus-I PSM-based Solutions

  • Solving by Gen-design-psm (Generic Model for Parametric Design), which enhances simple depth-first search via:
    • Focus selection – a DSR strategy analyses
      • Value ranges associated with each parameter,
      • The most constrained parameter in the current design model ;
    • Operator selection – operator chosen according to given ordering.
  • Solving by HC-design (Hill climbing) – same competence as Gen-design-psm, but much less efficient (measured at 10%, as compared to 78%).
  • Solving by A*-design, based on heuristic function using estimated cost function – better competence (optimal solution), but comparable efficiency to HC-design.

    EXTENSIONS

Application to Web Service Models

  • A standard language was developed for the description of PSMs called the Unified Problem-Solving Method Language (UPML) [Fensel et al, 2002]

  • UPML was applied to the modelling of Web Services in the Web Service Modelling Framework (WSMF) [Fensel & Bussler, 2002]

  • The WSMF approach was encoded in the Web Service Modeling Ontology (WSMO) – wherein ontology-based models are builts for goals (~tasks), services (~methods) and these are linked by mediators (~adapters) [Fensel et al., 2007]

  • WSMO was encoded in a family of ontology languages, WSML, and an open-source implementation was carried out in WSMX.

    SUMMARY

Summary

    • Problem-solving methods offer a means to structure and reuse elements of knowledge-based systems by abstracting from their domain.

    • Efficiency is achieved by introducing assumptions that either restrict the size of the problem or that postulate strong requirements on the available domain knowledge.

    • Adapters link tasks and PSMs to domains, allowing reuse of both, and express refinements between PSMs, allowing libraries to be built of these.

    REFERENCES

References

  • Mandatory reading:
    • [Fensel, 2000] “Problem-Solving Methods: Understanding, Description, Development and Reuse”, D. Fensel, Springer LNAI 1791, 2000

  • Further reading:
    • [Motta, 1999] “Reusable Components for Knowledge Modelling: Case Studies in Parametric Design”, E. Motta, IOS Press, 1999
    • [Schreiber et al., 1994] “CommonKADS: A Comprehensive Methodology for KBS Development”, A. Th. Schreiber, B. Wielinga and J. M. Akkermans, W. Van de Vedle, and
    • R. De Hoog, IEEE Expert , 9(6):28–37, 1994
    • [Stefik et al., 1983] “Basic Concepts for Building Expert Systems”, M. Stefik, J. Aitkins, R. Balzer, J. Benoit, L. Birnbaum, F. Hayes-Roth, and E. Sacerdoti in Building Expert Systems , Addison-Wesley, 1983

References

    • [Smith & Lowry, 1990] “Algorithm Theories and Design Tactics”, D. R. Smith and M. R. Lowry in Science of Computer Programming , 14:305–321Addison-Wesley, 1990

    • [Fensel et al, 2002] “The Unified Problem-solving Method Development Language UPML”, D. Fensel, E. Motta, R. Benjamins, M. Crubezy, S. Decker, M. Gaspari, R. Groenboom, W. Grosso, F. van Harmelen, M. Musen, E. Plaza, G. Schreiber, R. Studer, B. Wielinga, Knowledge and Information Systems, 5(1), 83–131, 2002

    • [Fensel & Bussler, 2002] “The Web Service Modeling Framework (WSMF)”, D. Fensel and C. Bussler, Electronic Commerce Research and Applications, 1(2), 2002

    • [Fensel et al., 2007] “Enabling Semantic Web Services:The Web Service Modeling Ontology (WSMO)”, D. Fensel, H. Lausen, A. Polleres, J. de Bruijn, M. Stollberg, D. Roman, and J. Domingue, Springer, 2007

References

    • [Brachman, 1979] “On the Epistemological Status of Semantic Networks” R. J. Brachman. In: N.V. Findler (ed.): Associative Networks: Representation and Use of Knowledge by Computers. New York: Academic Press, 1979, 3-50.

    • [Birmingham&Klinker, 1993] “Knowledge Acquisition Tools with Explicit Problem-Solving Methods”, W. Birmingham and G. Klinker. The Knowledge Engineering Review 8, 1 (1993), 5-25

    • [Feigenbaum, 1977] “The Art of Artificial Intelligence: Themes and Case Studies of Knowledge Engineering,” E.A. Feigenbaum. Proceedings of the International Joint Conference on Artificial Intelligence, Cambridge, MA, 1977

    • [Newell, 1981] “The Knowledge Level” Newell, AI Magazine 2 (2), 1981, p. 1-20

References

  • Wikipedia and other links:

    • http://en.wikipedia.org/wiki/Problem_solving

    • http://www.wsmo.org/

    • http://www.wsmo.org/wsml/

    • http://www.wsmx.org/

    • http://www.wsmo.org/papers/publications/wsmf.paper.pdf

    Questions?