Models -1 Scientists often describe what they do as constructing models. Understanding scientific reasoning requires understanding something about models

Models -1Models -1

ScientistsScientists often describe what they do as often describe what they do as

constructing constructing modelsmodels. Understanding . Understanding

scientific reasoning requires understanding scientific reasoning requires understanding

something about models and how they are something about models and how they are

used in science.used in science.

There are at least 3 kinds of models:There are at least 3 kinds of models:

– scalescale: : e.g. model airplanee.g. model airplane

– analoganalog: : e.g. conventional city mapse.g. conventional city maps

– theoreticaltheoretical: : e.g. Newtonian physics equations e.g. Newtonian physics equations

Models -2Models -2

ModelsModels need to be put in need to be put in correspondencecorrespondence with with

realityreality, through , through hypotheseshypotheses and and interpretationsinterpretations..

A model may A model may predictpredict something that is something that is not not

confirmedconfirmed, in which case the model is , in which case the model is incorrectincorrect..

A model may A model may fail to predictfail to predict something it should something it should

be able to, in which case it is be able to, in which case it is incompleteincomplete..

Like other mal-functioning Like other mal-functioning artefactsartefacts, mistaken , mistaken

models can be models can be diagnoseddiagnosed. .

Model-based diagnosis - 1Model-based diagnosis - 1

DiagnosisDiagnosis is concerned with the development of is concerned with the development of algorithms and techniques that can determine algorithms and techniques that can determine whether the whether the behaviourbehaviour of a system (or of a system (or artefactartefact) is ) is correct. The artefact may be a correct. The artefact may be a theorytheory..

If the system is not If the system is not functioningfunctioning correctly, the correctly, the algorithm should be able to determine, as algorithm should be able to determine, as accurately as possible, which part of the system accurately as possible, which part of the system is failing, and the kind of is failing, and the kind of faultfault it is facing. it is facing.

The computation is based on The computation is based on observationsobservations which which provide information on the current behaviour. provide information on the current behaviour.

Model-based diagnosis - 2Model-based diagnosis - 2 Model-based diagnosis is an example of Model-based diagnosis is an example of

abductiveabductive reasoningreasoning using a using a modelmodel of the of the system:system:

Model-based diagnosis - 3Model-based diagnosis - 3

A A modelmodel describes the behaviour of the system, or describes the behaviour of the system, or artefact. The model can itself artefact. The model can itself be be the artefact.the artefact.

It an abstraction of the It an abstraction of the behaviourbehaviour of the system of the system and can be incomplete. The and can be incomplete. The faulty behaviourfaulty behaviour may may be little-known, and the be little-known, and the fault modelfault model might not be might not be represented. If the model is a program: represented. If the model is a program: debuggingdebugging..

Given the Given the observationsobservations, the diagnoser, the diagnoser simulates simulates the system using the model, and compares the the system using the model, and compares the observations actually made to the observations observations actually made to the observations predictedpredicted by the simulation. by the simulation.

Model-based diagnosis - 4Model-based diagnosis - 4 The modelling can be expressed by the rules The modelling can be expressed by the rules

(where (where AbAb is the is the AbAbnormality predicate)normality predicate)::

If the behaviour of the system is not If the behaviour of the system is not abnormal (i.e. normal), then the internal abnormal (i.e. normal), then the internal (unobservable) behaviour will be (unobservable) behaviour will be Int1Int1 and the and the observable one observable one Obs1Obs1..

Otherwise, the internal behaviour will be Otherwise, the internal behaviour will be Int2Int2 and the observable behaviour and the observable behaviour Obs2Obs2..

Given the observations Given the observations ObsObs, the problem is to , the problem is to determine whether the system behaviour is determine whether the system behaviour is normal or not (normal or not (¬ ¬ Ab(S)Ab(S) or or Ab(S) Ab(S) ). This is an ). This is an example of example of abductive reasoningabductive reasoning..

Falsifiability - 1Falsifiability - 1

In science and philosophy of science, In science and philosophy of science,

falsifiabilityfalsifiability, , contingencycontingency, and , and defeasibilitydefeasibility

are roughly equivalent terms referring to are roughly equivalent terms referring to

the property of empirical statements that the property of empirical statements that

they must admit of logical they must admit of logical counterexamplescounterexamples..

This stands in contradistinction to formal This stands in contradistinction to formal

and mathematical statements that may be and mathematical statements that may be

tautologies, that is, universally true by dint tautologies, that is, universally true by dint

of definitions, axioms, and proofs.of definitions, axioms, and proofs.


Some philosophers and scientists, most Some philosophers and scientists, most

notably notably Karl PopperKarl Popper, have asserted that no , have asserted that no

empirical hypothesis, proposition, or theory empirical hypothesis, proposition, or theory

can be considered scientific if it does not can be considered scientific if it does not

admit the possibility of a contrary case.admit the possibility of a contrary case.

For example, the proposition For example, the proposition "all swans are "all swans are

white"white" would be falsified by observing a would be falsified by observing a

black swan, which would in turn depend on black swan, which would in turn depend on

there being a black swan somewhere in there being a black swan somewhere in

existence.existence.


A falsifiable proposition or theory must define A falsifiable proposition or theory must define

in some way what is, or will be, forbidden by in some way what is, or will be, forbidden by

that proposition or theory.that proposition or theory.

For example, the existence of a black swan is For example, the existence of a black swan is

forbidden by the proposition in question. The forbidden by the proposition in question. The

possibility in principle of observing a black possibility in principle of observing a black

swan as a counterexample to the general swan as a counterexample to the general

proposition is sufficient to qualify the proposition is sufficient to qualify the

proposition as falsifiable.proposition as falsifiable.


The falsification of statements occurs The falsification of statements occurs through through modus tollensmodus tollens, via some , via some observation.observation.

Suppose some universal statement Suppose some universal statement UU implies an observation implies an observation O O ::

U U →→ OO An observation conflicting with An observation conflicting with O O , however, , however,

is made:is made: ¬ ¬ OO

So by modus tollens:So by modus tollens:

¬ ¬ UU


It is always possible to It is always possible to revise the universal the universal statement or the existential statement so statement or the existential statement so that falsification does not occur.that falsification does not occur.

On hearing that a black swan has been On hearing that a black swan has been observed in Australia, one might introduce observed in Australia, one might introduce the the ad hocad hoc hypothesis, hypothesis, ""all swans are all swans are white except those found in Australiawhite except those found in Australia""..

The universal statement isThe universal statement is defeasible defeasible through through exceptionsexceptions. And there may be . And there may be exceptions to the exceptions.exceptions to the exceptions.

Belief Revision- 1Belief Revision- 1

Belief revisionBelief revision is the process of changing is the process of changing beliefs to take into account a new piece of beliefs to take into account a new piece of information.information.

The logical formalization of belief revision is The logical formalization of belief revision is researched in philosophy, in databases, and in researched in philosophy, in databases, and in artificial intelligence for the design of rational artificial intelligence for the design of rational agents.agents.

What makes belief revision non-trivial is that What makes belief revision non-trivial is that several different ways for performing this several different ways for performing this operation may be possible.operation may be possible.


E.g., the current knowledge includes the 3 E.g., the current knowledge includes the 3 facts “facts “AA is true is true”, “”, “BB is true is true” and “” and “if if AA and and BB are true then are true then CC is true is true”.”.

The introduction of the new information “The introduction of the new information “CC is falseis false” can be done preserving ” can be done preserving consistency only by removing at least one consistency only by removing at least one of the 3 facts. In this case, there are at least of the 3 facts. In this case, there are at least 3 different ways for performing revision.3 different ways for performing revision.

In general, there may be several different In general, there may be several different ways for changing knowledge.ways for changing knowledge.

Belief Revision- 3Belief Revision- 3 Two kinds of Two kinds of changechange are usually distinguished: are usually distinguished: Update.Update. New information is about the present, New information is about the present,

while the old beliefs refer to the past; update while the old beliefs refer to the past; update is the operation of changing the old beliefs to is the operation of changing the old beliefs to take into account the change. take into account the change.

Revision.Revision. Both the old beliefs and the new Both the old beliefs and the new information refer to the same situation; an information refer to the same situation; an inconsistency between them is explained by inconsistency between them is explained by the possibility of old information being less the possibility of old information being less reliable than the new one; revision is the reliable than the new one; revision is the process of inserting the new information into process of inserting the new information into the set of old beliefs without generating an the set of old beliefs without generating an inconsistency. inconsistency.


The main assumption of belief revision is that The main assumption of belief revision is that

of minimal change: the knowledge before and of minimal change: the knowledge before and

after the change should be as similar as after the change should be as similar as

possible.possible.

In the case of update, this principle In the case of update, this principle

formalizes the assumption of inertia.formalizes the assumption of inertia.

In the case of revision, this principle enforces In the case of revision, this principle enforces

as much information as possible to be as much information as possible to be

preserved by the change.preserved by the change.

Logic Program RevisionLogic Program Revision

The problem:The problem:– A LP represents A LP represents

consistent incomplete consistent incomplete knowledge;knowledge;

– New New factualfactual information comes.information comes.

– How to incorporate How to incorporate the new the new information?information?

The solution:The solution:– Add the new facts to Add the new facts to

the program;the program;– If the union is If the union is

consistent this is the consistent this is the result;result;

– Otherwise Otherwise restore restore consistencyconsistency to the to the union.union.

The new problem:The new problem:– How to restore consistency to an How to restore consistency to an

inconsistent program?inconsistent program?

Simple revision example - 1Simple revision example - 1P: flies(X) bird(X), not ab(X). bird(a) .

ab(X) penguin(X).

SoSo flies(a) is true. Next, we learnis true. Next, we learn penguin(a).. P {penguin(a)} is consistent,is consistent, flies(a) is is false,false, not ab(a) isis defeated. Nothing needs to . Nothing needs to be done.be done. We learn insteadWe learn instead ¬flies(a).. flies(a) isis rebutted.

P {¬flies(a)} is inconsistent. What to do?is inconsistent. What to do?

Since the inconsistency rests on the Since the inconsistency rests on the assumptionassumption not ab(a), revise that assumption, , revise that assumption, e.g. by adding the facte.g. by adding the fact ab(a), thereby , thereby obtaining a new programobtaining a new program P’..

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Models -1 Scientists often describe what they do as constructing models. Understanding scientific reasoning requires understanding something about models