Knowledge Representation and Inference Recall semantics: Work with - - PowerPoint PPT Presentation

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Knowledge Representation and Inference

• Soundness for modal inference
• Decision precedure for propositional S4

Alan Smaill KRI Jan 21 2008 2

Recall semantics:

Work with possible worlds and an accessibility relation between them. Thus:

• a set S of possible worlds (states);
• a relation between states ≤i

says that second state is accessible from the first

• an interpretation I that tells us whether a given atomic proposition p is true
• r false in a world s;

The set of states with the accessibility relation is called a frame.

Alan Smaill KRI Jan 21 2008 3

Interpretation

When is a formula true for some worlds s in a structure M? (write (M, s) | = p). (M, s) | = p iff I(s, p) = true for atomic statements; use standard clauses for propsitional connectives, (M, s) | = Ki p iff ((M, t) | = p for all t such that s ≤i t) The logic will be different if we impose some conditions on the relation ≤i that we allow between possible worlds.

Alan Smaill KRI Jan 21 2008 4

Modal logic K

If we make no conditions on accessibility at all, we get the logic known as K. We claim that in K the following holds: Ki(p ∧ q) → (Ki p ∧ Ki q) This means that the formula holds for every world of every frame and every interpretation: this property of a formula is called validity. Why is the formula valid? Take any (M, s): (M, s) | = Ki(p ∧ q) → (Ki p ∧ Ki q) iff [(M, s) | = Ki(p ∧ q) implies (M, s) | = (Ki p ∧ Ki q)]

Alan Smaill KRI Jan 21 2008

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Showing validity

So assume (M, s) | = Ki(p ∧ q) and show (M, s) | = (Ki p ∧ Ki q). If (M, s) | = Ki(p ∧ q) then (M, t) | = (p ∧ q) for all t s.t. s ≤i t so (M, t) | = p for all t s.t. s ≤i t and (M, t) | = q for all t s.t. s ≤i t ie (M, s) | = Ki p and (M, s) | = Ki q so (M, s) | = Ki p ∧ Ki q

Alan Smaill KRI Jan 21 2008 6

Showing non-validity

To show that a modal formula is not valid, just find some frame and interpretation that make the formula false at some node. In the logic K, the formula (Ki p) → p is not valid (write the formula as Ki p → p.) Consider just two possible worlds w1, w2 with w1 ≤i w2, and ≤i false otherwise. Assign p to be false in w1 and true in w2. Now check: show that (M, w1) | = Ki p → p is false by showing that (M, w1) | = Ki p is true, and (M, w1) | = p is false.

Alan Smaill KRI Jan 21 2008 7

logic K4

If we insist that the accessibility relation is transitive we get the logic K4. Because the accessibility relation in the example above is transitive, we know that Ki p → p is not valid in K4 either. Transitivity gives us the validity of Ki p → KiKi p

Alan Smaill KRI Jan 21 2008 8

logic K4

To show this, assume that accessibility is transitive, and that (M, a) | = Ki p; show that (M, a) | = KiKi p But (M, a) | = KiKi p iff (M, b) | = Ki p for all b st a ≤i b iff (M, c) | = p for all b, c st a ≤i b and b ≤i c By transitivity, for all such b, c we have a ≤i c; but we have assumed (M, a) | = Ki p so (M, c) | = Ki p for all c st a ≤i c, and the result follows. The sequent rules given in the week 3 tutorial are sound and complete for the modal logic K4.

Alan Smaill KRI Jan 21 2008

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logic S4

If we require that accessibility is reflexive (every state is accessible to itself) as well as transitive, we get the logic S4. To check soundness of an inference rule using this semantics, for a given rule, eg (simplified) boxE, show that whenever a sequent matching above the line is true (at a world), then so is the formula below. F = ⇒ H ✷F = ⇒ H

Alan Smaill KRI Jan 21 2008 10

Soundness of boxE

(M, a) | = F = ⇒ H iff not (M, a) | = F or (M, a) | = H and also (M, a) | = ✷F = ⇒ H iff not (M, a) | = ✷F or (M, a) | = H iff not ((M, b) | = F for all b st a ≤i b) or (M, a) | = H Now show that the first condition entails the second, given reflexivity of ≤i.

Alan Smaill KRI Jan 21 2008 11

Decision procedure for S4

We can use the sequent rules for S4 to get a decision procedure for whether one formula follows from others in the logic S4. The search space generated by the rules used top down is in fact finite; a full argument needs to take into account that

• the size of formulae reduce in any branch (counting number of logical

connectives)

• the proof tree branching rate is bounded (by 2)
• the rule applications completely define the subgoals of a rule application
• there are only finitely many possible rule applications for a given sequent

(bounded by the number of formulae in the sequent)

Alan Smaill KRI Jan 21 2008 12

Decision Procedure ctd

Thus any exhaustive search of the space will give a decision procedure. (So can use depth first search, eg). But note that is any of these conditions fails, the search space may be infinite. What about efficiency? We know that when there are no modal operators, there is no need to backtrack; we can build this in to the search. What about preferred order of rule applications?

Alan Smaill KRI Jan 21 2008

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Temporal logic and possible worlds

One way to consider temporal logic in the context of Kripke models is to look at possible worlds as successive time points; model this in one case by the natural numbers wn ≤i wm iff n ≤ m. In this model, time is:

• linear: the points succeed each other along a single line;
• discrete: for any point, there is a later point with no intermediate point

between these two points;

• unbounded: there is no last point.

Alan Smaill KRI Jan 21 2008 14

Reified time

If we use a modal logic to deal with time, there is no explicit mention of time in the statements (just have ✷p meaning p always true in the future). Another (non-modal) approach involves introducing in our representation language explicit time arguments. So a proposition rain becomes a predicate with a time argument rain(t1) indicating rain at a given time; bel(john, rain) becomes maybe bel(john, t2, rain(t3)).

Alan Smaill KRI Jan 21 2008 15

Agents and time

Many beliefs depend on time – for example “it’s raining” implicitly refers to a present time. So, sometimes the same agent or person will accept the statement sometimes, reject it at others, and sometimes not be in a position to decide. How can we represent such beliefs?

Alan Smaill KRI Jan 21 2008 16

Belief statements

What is a suitable data structure to hold an agent’s belief? Here is one way to do this. For a given time, the agent may believe a non-temporal statement is true, false,

• r unknown. We assume that beliefs are persistent, i.e. in the absence of any
• ther information, a statement believed to be true at some time is believed to be

true at later times also.

Alan Smaill KRI Jan 21 2008

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An abstract data-type

We cannot take rational numbers (not discrete), or integers with some maximum (bounded). We pass over implementational problems this raises. We make a modular treatment of time if we do not specify the exact format, but

• nly what operations we require. Then we can plug in different formats if

required (e.g. include milliseconds). Suppose we have: add_time(Time1,Time2,Result) less(Time1,Time2) % both assuming same format inRange(Time1,Time2, Test) % is Test between 2 Times?

Alan Smaill KRI Jan 21 2008 18

Changing beliefs

Use the following form to represent beliefs about Pred: b(A,Pred,[[Time1,t],[Time2,f],[Time3,t]]) meaning that: agent A started to believe Pred1 at Time1, believes that Pred1 turns false at Time2, and is true again for all times after Time3. Before Time1, Agent1 has no belief about Pred1. assumption: we always have that the Times in increasing temporal order; the implementation should maintain this invariant.

Alan Smaill KRI Jan 21 2008 19

Summary

• Frames and semantics for modal logic
• Validity in modal logics
• Sequent Calculus decision procedure
• Reified time and changing beliefs

Alan Smaill KRI Jan 21 2008