Knowledge Lecture 1: Lecture 2 This leads to PROLOG as an efï¬پcient inference engine for definite logic

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  • Modeling the Dynamics of Knowledge

    Jürgen Dix and Wojtek Jamroga

    Department of Computer Science Clausthal University of Technology, Germany

    ESSLLI (Dublin 2007)

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    Lecture Overview

    Lecture 1: Chapter 1: Basics about Knowledge. Lecture 2: Chapter 2: Answer Set Programming. Lecture 3: Chapter 3: Modal Logic.

    Chapter 4: Logics of Action and Time (1).

    Lecture 4: Chapter 4: Logics of Action and Time (2). Chapter 5: Knowledge in Flux.

    Lecture 5: Chapter 6: Combining Knowledge and Time. Chapter 7: Modalities in Action. Chapter 8: Where the two frameworks meet.

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    1. Basics about Logic

    Chapter 1. Basics about Logic

    Basics about Logic 1.1 Why Logic? 1.2 Sentential Logic (SL) 1.3 Examples 1.4 Calculi 1.5 First-Order Logic (FOL) 1.6 Logic Programming (LP) 1.7 References

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    1. Basics about Logic

    We make the case for using logics as a representation formalism for reasoning about the world. While almost everything can be done with logic, the formalization is often awkward and cumbersome. We illustrate this with theWumpus world and Sudoku-puzzles. We introduce two sorts of calculi for propositional logics: a Hilbert type and a resolution calculus. We introduce first-order logic (FOL) and reconsider the Wumpus world. The dynamics of the changing world can be modeled with the terms: they enable us to explicitly denote the situation we are in and to reason about it: McCarthy's situation calculus.

    Jürgen Dix and Wojtek Jamroga � ESSLLI (Dublin 2007) 4/456

  • 1. Basics about Logic

    One of the main features is to ask queries of the form Dφpxq to a theory T . We expect not just a “yes/no” answer, but an instantiation of the variable x. This can be achieved with the resolution calculus for FOL. While resolution is much more efficiently implementable (compared to Hilbert-type calculi), the search space is still huge. Thus it was suggested to restrict resolution and apply it to a smaller class of formulae: Horn clauses. This leads to PROLOG as an efficient inference engine for definite logic programs. However, full declarativeness is lost and PROLOG is not (yet) the answer: emphasis is put on computing answer substitutions.

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    1. Basics about Logic 1. Why Logic?

    1.1 Why Logic?

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    1. Basics about Logic 1. Why Logic?

    Why logic at all?

    framework for thinking about systems, makes one realise hidden assumptions, . . . and then we can: investigate them, accept or reject them, relax some of them and still use a part of the formal and conceptual machinery,

    Jürgen Dix and Wojtek Jamroga � ESSLLI (Dublin 2007) 7/456

    1. Basics about Logic 1. Why Logic?

    Why logic at all?

    verification: check specification against implementation, executable specification,

    planning as model checking

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  • 1. Basics about Logic 1. Why Logic?

    Symbolic AI: Symbolic representation, e.g. sentential or first order logic. Agent as a theorem prover.

    Traditional: Theory about agents. Implementation as stepwise process (Software Engineering) over many abstractions.

    Symbolic AI: View the theory itself as executable specification. Internal state: Knowledge Base (KB), often simply called D (database).

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    1. Basics about Logic 2. Sentential Logic (SL)

    1.2 Sentential Logic (SL)

    Jürgen Dix and Wojtek Jamroga � ESSLLI (Dublin 2007) 10/456

    1. Basics about Logic 2. Sentential Logic (SL)

    Definition 1.1 (Sentential Logic LSL, Language L LSL)

    The language LSL of propositional (or sentential) logic consists of

    l and J: the constants falsum and verum, p,q,r,x1,x2, . . .xn, . . .: a countable set AT of SL-constants, , ^, _,Ñ: the sentential connectives ( is unary, all others are binary operators), p, q: the parentheses to help readability.

    In most cases we consider only a finite set of SL-constants. They define a language L LSL. The set of L-formulae FmlL is defined inductively.

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    1. Basics about Logic 2. Sentential Logic (SL)

    Definition 1.2 (Semantics, Valuation, Model)

    A valuation v for a language L LSL is a mapping from the set of SL-constants defined by L into the set ttrue, falseu with vplq � false, vpJq � true. Each valuation v can be uniquely extended to a function v̄ : FmlL Ñ ttrue, falseu so that:

    v̄p pq � "

    true, if v̄ppq � false, false, if v̄ppq � true.

    v̄pϕ^ γq � "

    true, if v̄pϕq � true and v̄pγq � true, false, else

    v̄pϕ_ γq � "

    true, if v̄pϕq � true or v̄pγq � true, false, else

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  • 1. Basics about Logic 2. Sentential Logic (SL)

    Definition (continued) v̄pϕÑ γq �"

    true, if v̄pϕq � false or (v̄pϕq � true and v̄pγq � trueq, false, else

    Thus each valuation v uniquely defines a v̄. We call v̄ L-structure. A structure determines for each formula if it is true or false. If a formula φ is true in structure v̄ we also say Av is a model of φ. From now on we will speak of models, structures and valuations synonymously.

    Semantics The process of mapping a set of L-formulae into ttrue, falseu is called semantics.

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    1. Basics about Logic 2. Sentential Logic (SL)

    Definition 1.3 (Validity of a Formula, Tautology)

    1 A formula ϕ P FmlL holds under the valuation v if v̄pϕq � true. We also write v̄ |ù ϕ or simply v |ù ϕ. v̄ is a model of ϕ.

    2 A theory is a set of formulae: T FmlL . v satisfies T if v̄pϕq � true for all ϕ T . We write v |ù T .

    3 A L-formula ϕ is called L-tautology if for all possible valuations v in L v |ù ϕ holds.

    From now on we suppress the language L, because it is obvious from context. Nevertheless it needs to be carefully defined.

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    1. Basics about Logic 2. Sentential Logic (SL)

    Definition 1.4 (Consequence Set CnpT q)

    A formula ϕ follows from T if for all models v of T (i.e. v |ù T ) also v |ù ϕ holds. We write: T |ù ϕ. We call

    CnLpT q �def tϕ P FmlL : T |ù ϕu,

    or simply CnpT q, the semantic consequence operator.

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    1. Basics about Logic 2. Sentential Logic (SL)

    Lemma 1.5 (Properties of CnpT q) The semantic consequence operator has the following properties: 1 T -extension: T CnpT q, 2 Monotony: T T 1 ñ CnpT q CnpT 1q, 3 Closure: CnpCnpT qq �CnpT q.

    Lemma 1.6 (ϕ R Cn(T)) ϕ RCnpT q if and only if there is a model v with

    v |ù T and v̄pϕq � false.

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  • 1. Basics about Logic 2. Sentential Logic (SL)

    Definition 1.7 (MOD(T), CnpUq)

    If T FmlL then we denote withMODpT q the set of all L-structures A which are models of T :

    MODpT q �def tA : A |ù Tu.

    If U is a set of models, we consider all those sentences, which are valid in all models of U. We call this set CnpUq:

    CnpUq �def tϕ P FmlL : @v PU : v̄pϕq � trueu.

    MOD is obviously dual to Cn:

    CnpMODpT qq �CnpT q, MODpCnpT qq �MODpT q.

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    1. Basics about Logic 2. Sentential Logic (SL)

    Definition 1.8 (Completeness of a Theory T )

    T is called complete if for each formula ϕ P Fml: T |ù ϕ or T |ù ϕ holds.

    Attention: Do not mix up this last condition with the property of a valuation (model) v: each model is complete in the above sense.

    Jürgen Dix and Wojtek Jamroga � ESSLLI (Dublin 2007) 18/456

    1. Basics about Logic 2. Sentential Logic (SL)

    Definition 1.9 (Consistency of a Theory)

    T is called consistent if there is a valuation (model) v with v̄pϕq � true for all ϕ P T .

    Lemma 1.10 (Ex Falso Quodlibet) T is consistent if and only if CnpT q � FmlL .

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    1. Basics about Logic 3. Examples

    1.3 Examples

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  • 1. Basics about Logic 3. Examples

    Wumpus World

    Breeze Breeze

    Breeze

    Breeze Breeze

    Stench

    Stench

    Breeze PIT

    PIT

    PIT

    1 2 3 4

    1

    2

    3

    4

    START

    Gold �

    Stench

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    1. Basics about Logic 3. Examples

    A �

    B �

    G

    P �

    S

    W �

    = Agent = Breeze = Glitter, Gold

    = Pit = Stench

    = Wumpus

    OK = Safe square

    V �

    = Visited

    A �

    OK

    1,1 2,1 3,1 4,1

    1,2 2,2 3,2 4,2

    1,3 2,3 3,3 4,3

    1,4 2,4 3,4 4,4

    OKOK B

    P? �

    P? �

    A �

    OK OK