1 2 Knowledge Representation and Inference Recall semantics: Work with possible worlds and an accessibility relation between them. Thus: • Soundness for modal inference • a set S of possible worlds (states); • Decision precedure for propositional S4 • 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 or false in a world s ; The set of states with the accessibility relation is called a frame . Alan Smaill KRI Jan 21 2008 Alan Smaill KRI Jan 21 2008 3 4 Modal logic K Interpretation When is a formula true for some worlds s in a structure M ? (write ( M, s ) | = p ). If we make no conditions on accessibility at all, we get the logic known as K . We claim that in K the following holds: ( M, s ) | = p iff I ( s, p ) = true K i ( p ∧ q ) → ( K i p ∧ K i q ) for atomic statements; use standard clauses for propsitional connectives, ( M, s ) | = K i p iff (( M, t ) | = p for all t such that s ≤ i t ) This means that the formula holds for every world of every frame and every The logic will be different if we impose some conditions on the relation ≤ i that interpretation: this property of a formula is called validity . we allow between possible worlds. Why is the formula valid? Take any ( M, s ) : ( M, s ) | = K i ( p ∧ q ) → ( K i p ∧ K i q ) iff [( M, s ) | = K i ( p ∧ q ) implies ( M, s ) | = ( K i p ∧ K i q )] Alan Smaill KRI Jan 21 2008 Alan Smaill KRI Jan 21 2008
5 6 Showing validity Showing non-validity So assume ( M, s ) | = K i ( p ∧ q ) and show ( M, s ) | = ( K i p ∧ K i q ) . To show that a modal formula is not valid, just find some frame and interpretation that make the formula false at some node. If ( M, s ) | = K i ( p ∧ q ) In the logic K , the formula ( K i p ) → p is not valid (write the formula as K i p → p .) then ( M, t ) | = ( p ∧ q ) for all t s.t. s ≤ i t Consider just two possible worlds w 1 , w 2 with w 1 ≤ i w 2 , and ≤ i false otherwise. so ( M, t ) | = p for all t s.t. s ≤ i t Assign p to be false in w 1 and true in w 2 . and ( M, t ) | = q for all t s.t. s ≤ i t Now check: show that ( M, w 1 ) | = K i p → p is false by showing that ie ( M, s ) | = K i p and ( M, s ) | = K i q ( M, w 1 ) | = K i p is true, and so ( M, s ) | = K i p ∧ K i q ( M, w 1 ) | = p is false. Alan Smaill KRI Jan 21 2008 Alan Smaill KRI Jan 21 2008 7 8 logic K4 logic K4 If we insist that the accessibility relation is transitive we get the logic K4 . To show this, assume that accessibility is transitive, and that ( M, a ) | = K i p ; show that ( M, a ) | = K i K i p Because the accessibility relation in the example above is transitive, we know that K i p → p is not valid in K4 either. But Transitivity gives us the validity of ( M, a ) | = K i K i p K i p → K i K i p iff ( M, b ) | = K i 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 ) | = K i p so ( M, c ) | = K i 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 Alan Smaill KRI Jan 21 2008
9 10 logic S4 Soundness of boxE If we require that accessibility is reflexive (every state is accessible to itself) as well as transitive, we get the logic S4 . ( M, a ) | = F = ⇒ H To check soundness of an inference rule using this semantics, for a given rule, eg iff not ( M, a ) | = F or ( M, a ) | = H (simplified) boxE, show that whenever a sequent matching above the line is true (at a world), then so is the formula below. and also F = ⇒ H ( M, a ) | = ✷ F = ⇒ H ✷ 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 Alan Smaill KRI Jan 21 2008 11 12 Decision procedure for S4 Decision Procedure ctd We can use the sequent rules for S4 to get a decision procedure for whether one Thus any exhaustive search of the space will give a decision procedure. (So can formula follows from others in the logic S4 . use depth first search, eg). But note that is any of these conditions fails, the search space may be infinite. The search space generated by the rules used top down is in fact finite; a full argument needs to take into account that 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. • the size of formulae reduce in any branch (counting number of logical connectives) What about preferred order of rule applications? • 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 Alan Smaill KRI Jan 21 2008
13 14 Temporal logic and possible worlds Reified time One way to consider temporal logic in the context of Kripke models is to look at If we use a modal logic to deal with time, there is no explicit mention of time in possible worlds as successive time points; the statements (just have ✷ p meaning p always true in the future). model this in one case by the natural numbers w n ≤ i w m iff n ≤ m . Another (non-modal) approach involves introducing in our representation In this model, time is: language explicit time arguments. So a proposition rain becomes a predicate with a time argument rain ( t 1 ) indicating rain at a given time; bel ( john, rain ) • linear : the points succeed each other along a single line; becomes maybe bel ( john, t 2 , rain ( t 3 )) . • 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 Alan Smaill KRI Jan 21 2008 15 16 Agents and time Belief statements Many beliefs depend on time – for example “it’s raining” implicitly refers to a What is a suitable data structure to hold an agent’s belief? Here is one way to present time. do this. So, sometimes the same agent or person will accept the statement sometimes, For a given time, the agent may believe a non-temporal statement is true , false , reject it at others, and sometimes not be in a position to decide. or unknown . We assume that beliefs are persistent , i.e. in the absence of any other information, a statement believed to be true at some time is believed to be How can we represent such beliefs? true at later times also. Alan Smaill KRI Jan 21 2008 Alan Smaill KRI Jan 21 2008
17 18 An abstract data-type Changing beliefs We cannot take rational numbers (not discrete), or integers with some maximum Use the following form to represent beliefs about Pred : (bounded). We pass over implementational problems this raises. b(A,Pred,[[Time1,t],[Time2,f],[Time3,t]]) We make a modular treatment of time if we do not specify the exact format, but meaning that: only what operations we require. Then we can plug in different formats if required (e.g. include milliseconds). agent A started to believe Pred1 at Time1, Suppose we have: believes that Pred1 turns false at Time2, and is true again for all times after Time3. add_time(Time1,Time2,Result) Before Time1, Agent1 has no belief about Pred1. less(Time1,Time2) % both assuming same format assumption : we always have that the Time s in increasing temporal order; the inRange(Time1,Time2, Test) implementation should maintain this invariant. % is Test between 2 Times? Alan Smaill KRI Jan 21 2008 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
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