1. Models -1 Scientists often describe what they do as constructing models. Understanding scientific reasoning requires understanding something about models and how they are used in science. Scientists often describe what they do as constructing models. Understanding scientific reasoning requires understanding something about models and how they are used in science. There are at least 3 kinds of models: There are at least 3 kinds of models: –scale: e.g. model airplane –analog: e.g. conventional city maps –theoretical: e.g. Newtonian physics equations

2. Models -2 Models need to be put in correspondence with reality, through hypotheses and interpretations. Models need to be put in correspondence with reality, through hypotheses and interpretations. A model may predict something that is not confirmed, in which case the model is incorrect. A model may predict something that is not confirmed, in which case the model is incorrect. A model may fail to predict something it should be able to, in which case it is incomplete. A model may fail to predict something it should be able to, in which case it is incomplete. Like other mal-functioning artefacts, mistaken models can be diagnosed. Like other mal-functioning artefacts, mistaken models can be diagnosed.

3. Model-based diagnosis - 1 Diagnosis is concerned with the development of algorithms and techniques that can determine whether the behaviour of a system (or artefact) is correct. The artefact may be a theory. Diagnosis is concerned with the development of algorithms and techniques that can determine whether the behaviour of a system (or artefact) is correct. The artefact may be a theory. If the system is not functioning correctly, the algorithm should be able to determine, as accurately as possible, which part of the system is failing, and the kind of fault it is facing. If the system is not functioning correctly, the algorithm should be able to determine, as accurately as possible, which part of the system is failing, and the kind of fault it is facing. The computation is based on observations which provide information on the current behaviour. The computation is based on observations which provide information on the current behaviour.

4. Model-based diagnosis - 2 Model-based diagnosis is an example of abductive reasoning using a model of the system: Model-based diagnosis is an example of abductive reasoning using a model of the system:

5. Model-based diagnosis - 3 A model describes the behaviour of the system, or artefact. The model can itself be the artefact. A model describes the behaviour of the system, or artefact. The model can itself be the artefact. It an abstraction of the behaviour of the system and can be incomplete. The faulty behaviour may be little-known, and the fault model might not be represented. If the model is a program: debugging. It an abstraction of the behaviour of the system and can be incomplete. The faulty behaviour may be little-known, and the fault model might not be represented. If the model is a program: debugging. Given the observations, the diagnoser simulates the system using the model, and compares the observations actually made to the observations predicted by the simulation. Given the observations, the diagnoser simulates the system using the model, and compares the observations actually made to the observations predicted by the simulation.

6. Model-based diagnosis - 4 The modelling can be expressed by the rules (where Ab is the Abnormality predicate) : The modelling can be expressed by the rules (where Ab is the Abnormality predicate) : If the behaviour of the system is not abnormal (i.e. normal), then the internal (unobservable) behaviour will be Int1 and the observable one Obs1. If the behaviour of the system is not abnormal (i.e. normal), then the internal (unobservable) behaviour will be Int1 and the observable one Obs1. Otherwise, the internal behaviour will be Int2 and the observable behaviour Obs2. Otherwise, the internal behaviour will be Int2 and the observable behaviour Obs2. Given the observations Obs, the problem is to determine whether the system behaviour is normal or not ( ¬ Ab(S) or Ab(S) ). This is an example of abductive reasoning. Given the observations Obs, the problem is to determine whether the system behaviour is normal or not ( ¬ Ab(S) or Ab(S) ). This is an example of abductive reasoning.

7. Falsifiability - 1 In science and philosophy of science, falsifiability, contingency, and defeasibility are roughly equivalent terms referring to the property of empirical statements that they must admit of logical counterexamples. In science and philosophy of science, falsifiability, contingency, and defeasibility are roughly equivalent terms referring to the property of empirical statements that they must admit of logical counterexamples. This stands in contradistinction to formal and mathematical statements that may be tautologies, that is, universally true by dint of definitions, axioms, and proofs. This stands in contradistinction to formal and mathematical statements that may be tautologies, that is, universally true by dint of definitions, axioms, and proofs.

8. Falsifiability - 2 Some philosophers and scientists, most notably Karl Popper, have asserted that no empirical hypothesis, proposition, or theory can be considered scientific if it does not admit the possibility of a contrary case. Some philosophers and scientists, most notably Karl Popper, have asserted that no empirical hypothesis, proposition, or theory can be considered scientific if it does not admit the possibility of a contrary case. For example, the proposition "all swans are white" would be falsified by observing a black swan, which would in turn depend on there being a black swan somewhere in existence. For example, the proposition "all swans are white" would be falsified by observing a black swan, which would in turn depend on there being a black swan somewhere in existence.

9. Falsifiability - 3 A falsifiable proposition or theory must define in some way what is, or will be, forbidden by that proposition or theory. A falsifiable proposition or theory must define in some way what is, or will be, forbidden by that proposition or theory. For example, the existence of a black swan is forbidden by the proposition in question. The possibility in principle of observing a black swan as a counterexample to the general proposition is sufficient to qualify the proposition as falsifiable. For example, the existence of a black swan is forbidden by the proposition in question. The possibility in principle of observing a black swan as a counterexample to the general proposition is sufficient to qualify the proposition as falsifiable.

10. Falsifiability - 4 The falsification of statements occurs through modus tollens, via some observation. The falsification of statements occurs through modus tollens, via some observation. Suppose some universal statement U implies an observation O : Suppose some universal statement U implies an observation O : U → O U → O An observation conflicting with O, however, is made: An observation conflicting with O, however, is made: ¬ O ¬ O So by modus tollens: So by modus tollens: ¬ U ¬ U

11. Falsifiability - 5 It is always possible to the universal statement or the existential statement so that falsification does not occur. It is always possible to revise the universal statement or the existential statement so that falsification does not occur. On hearing that a black swan has been observed in Australia, one might introduce the ad hoc hypothesis, " all swans are white except those found in Australia ". On hearing that a black swan has been observed in Australia, one might introduce the ad hoc hypothesis, " all swans are white except those found in Australia ". The universal statement is defeasible through exceptions. And there may be exceptions to the exceptions. The universal statement is defeasible through exceptions. And there may be exceptions to the exceptions.

12. Belief Revision- 1 Belief revision is the process of changing beliefs to take into account a new piece of information. Belief revision is the process of changing beliefs to take into account a new piece of information. The logical formalization of belief revision is researched in philosophy, in databases, and in artificial intelligence for the design of rational agents. The logical formalization of belief revision is researched in philosophy, in databases, and in artificial intelligence for the design of rational agents. What makes belief revision non-trivial is that several different ways for performing this operation may be possible. What makes belief revision non-trivial is that several different ways for performing this operation may be possible.

13. Belief Revision- 2 E.g., the current knowledge includes the 3 facts “A is true”, “B is true” and “if A and B are true then C is true”. E.g., the current knowledge includes the 3 facts “A is true”, “B is true” and “if A and B are true then C is true”. The introduction of the new information “C is false” can be done preserving consistency only by removing at least one of the 3 facts. In this case, there are at least 3 different ways for performing revision. The introduction of the new information “C is false” can be done preserving consistency only by removing at least one of the 3 facts. In this case, there are at least 3 different ways for performing revision. In general, there may be several different ways for changing knowledge. In general, there may be several different ways for changing knowledge.

14. Belief Revision- 3 Two kinds of change are usually distinguished: Two kinds of change are usually distinguished: Update. New information is about the present, while the old beliefs refer to the past; update is the operation of changing the old beliefs to take into account the change. Update. New information is about the present, while the old beliefs refer to the past; update is the operation of changing the old beliefs to take into account the change. Revision. Both the old beliefs and the new information refer to the same situation; an inconsistency between them is explained by the possibility of old information being less reliable than the new one; revision is the process of inserting the new information into the set of old beliefs without generating an inconsistency. Revision. Both the old beliefs and the new information refer to the same situation; an inconsistency between them is explained by the possibility of old information being less reliable than the new one; revision is the process of inserting the new information into the set of old beliefs without generating an inconsistency.

15. Belief Revision- 4 The main assumption of belief revision is that of minimal change: the knowledge before and after the change should be as similar as possible. The main assumption of belief revision is that of minimal change: the knowledge before and after the change should be as similar as possible. In the case of update, this principle formalizes the assumption of inertia. In the case of update, this principle formalizes the assumption of inertia. In the case of revision, this principle enforces as much information as possible to be preserved by the change. In the case of revision, this principle enforces as much information as possible to be preserved by the change.

16. Logic Program Revision The problem: The problem: –A LP represents consistent incomplete knowledge; –New factual information comes. –How to incorporate the new information? The solution: The solution: –Add the new facts to the program; –If the union is consistent this is the result; –Otherwise restore consistency to the union. The new problem: The new problem: –How to restore consistency to an inconsistent program?

17. Simple revision example - 1 P:flies(X)  bird(X), not ab(X). bird(a) . ab(X)  penguin(X). Sois true. Next, we learn. is consistent,is false, is. Nothing needs to be done. So flies(a) is true. Next, we learn penguin(a). P  {penguin(a)} is consistent, flies(a) is false, not ab(a) is defeated. Nothing needs to be done. We learn instead.is We learn instead ¬flies(a). flies(a) is rebutted. is inconsistent. What to do? P  {¬flies(a)} is inconsistent. What to do? Since the inconsistency rests on the assumption, revise that assumption, e.g. by adding the fact, thereby obtaining a new program. Since the inconsistency rests on the assumption not ab(a), revise that assumption, e.g. by adding the fact ab(a), thereby obtaining a new program P’.