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Willem Labuschagne & Laura vd Westhuizen Information templates — a useful data structure Presented by Willem Labuschagne & Laura van der Westhuizen, reporting research contributed to by Johannes Heidema Tommie Meyer Dave Ferguson Outline Evolution of the idea of an information template subjunctive conditionals nonmonotonic logic AGM belief revision semantic information theory Application: logics of knowledge and belief Application: equipping agents with affect

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Willem Labuschagne & Laura vd Westhuizen Subjunctive conditionals Material conditional: If Oswald did not kill Kennedy, then someone else did. Subjunctive conditional: If Oswald had not killed Kennedy, then someone else would have. Pre-semantic analysis (Chisholm 1955) Suppose we have a dry match under ordinary conditions. If it were struck (p), it would light (q). The reason p q is true is that there are physical laws r regarding the chemical structure of the match and circumstances c (e.g. the match being dry) such that p r c entails q. Problem: circumstances c must contain whatever conditions are necessary — the match being dry, sufficient oxygen being present, etc. But what is etc?

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Willem Labuschagne & Laura vd Westhuizen Subjunctive conditionals — semantics 1 Stalnaker (1968) Idea:p q is true iff it is the case that if the agent adds p to its beliefs and modifiesthe set of beliefs to make them consistent with p, but makes the modification as small as possible, then the result entails q Possible worlds: p q is satisfied at w iff q is satisfied at w, where w is the model of p which is most closely similar to w Problem: From uniqueness of w it follows that at every world w, ( ) ( ) is satisfied. But take = ‘The temperature outside is not 10 degrees’ and = ‘The temperature outside is 20 degrees’.

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Willem Labuschagne & Laura vd Westhuizen Subjunctive conditionals — semantics 2 Lewis (1973) p q is satisfied at w iff q is satisfied at w, for every w that is a model of p and belongs to the subset of models most closely similar to w Similarity is a total preorder on the set of worlds. Example Light-Fan: p = ‘The light is on’, q = ‘The fan is on’ States of the system: W = {11, 10, 01, 00} Suppose actual state w is 11 (light and fan both on). Similarity: 0110 00 11 = w

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Willem Labuschagne & Laura vd Westhuizen Subjunctive conditionals — semantics 3 The attachment of a similarity relation to each w may be visualised as a system of spheres: 0110 00 11 = w 11 10 01 00 Objections to the limit assumption led Lewis and others (e.g. Lindström & Rabinowicz 1991) to drop the linearity assumption on systems of spheres. This is a mistake, because the generalised consequence principle fails: If is a set of sentences (not necessarily finite) and for each , is satisfied at w, and entails , then it should be the case that is satisfied at w.

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Willem Labuschagne & Laura vd Westhuizen Nonmonotonic logic Rational consequence (Lehmann & Magidor 1992) Given a strict modular partial order on worlds, e.g. 0110 00 11 |~ iff is satisfied at all minimal models of . For example, (p q) |~ (p q). |~ is not in general monotonic or transitive, but satisfies: rational monotony: if |~ and not |~ then |~ the generalised consequence principle: if is a set of sentences (not necessarily finite) and for each , |~ , and |= , then |~ .

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Willem Labuschagne & Laura vd Westhuizen A functional view A strict partial order < on W is modular iff there is a function f : W N such that s < t iff f(s) < f(t). Hence we may visualise < as a linearly ordered partition: 0110 00 11 10 01 00 Of course we may also visualise < as concentric spheres: 11 10 01 00

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Willem Labuschagne & Laura vd Westhuizen AGM belief revision Belief revision semantics (Grove 1988) Given a total preorder order on worlds, e.g. 0110 00 11 the revised set K* is the set of all sentences true in the minimal models of . 11 10 01 00 and a set K of beliefs, where K = Th({11}), Functional view (Spohn 1988) an ordinal conditional function g : W N can be visualised as a ‘generalised’ ordered partition in which empty boxes may appear — these are intended to give a notion of distance whereby changes can be undone

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Willem Labuschagne & Laura vd Westhuizen Semantic information theory Semantic content (Carnap & Bar-Hillel 1953): An agent’s information can be represented by the worlds the agent is able to rule out. For example, suppose an agent observing the Light-Fan sees the light is on but cannot tell whether the fan is on. This information can be viewed as a dichotomous ordered partition in which the top box is distinguished (i.e. contains the worlds that are ruled out): 11 10 00 01 Declaratively the agent’s information is represented by sentences having the excluded worlds as nonmodels — i.e. having the worlds in the bottom box as models, just as in belief revision.

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Willem Labuschagne & Laura vd Westhuizen Information templates 1 Visually: If there are n worlds in W, an information template is a stack of n+1 boxes, with the uppermost box distinguished (for worlds that are ruled out): 11 00 01 The definite information represented by this template is that the worlds 01 and 00 can be ruled out. The agent’s knowledge is Th({11, 10}). The indefinite information is that the world 10 is less normal than 11. The agent’s set of defeasible beliefs is Th({11}). 10

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Willem Labuschagne & Laura vd Westhuizen Information templates 2 Functionally: Given n worlds in W, an information template is a function f : W {0, 1, …, n} and is in normal form iff for each j

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Willem Labuschagne & Laura vd Westhuizen Epistemic doxastic logic 1 Let L be the modal propositional language generated from the atoms p 1, p 2, …, p m with modal operators [0], [1], …, [n] where n = 2 m. Let W the set of all valuations v : {p 1, …, p m } {0, 1}. An interpretation of L is a pair (W, F) where F assigns to each w W an information template F(w). For example: agent can see whether light is on and has default rule ‘if the light is on, then normally the fan is on’ 11 00 01 10 00 01 11 10 F(11) = F(10) = F(01) = F(00) =

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Willem Labuschagne & Laura vd Westhuizen Epistemic doxastic logic 2 A sentence L is satisfied at w W iff is one of p 1, p 2, …, p m and valuation w makes true is a Boolean combination of shorter sentences and is satisfied according to the usual criteria = [j] and is satisfied at every w such that F(w)(w) j. We define j = [j] and so w satisfies = j iff is satisfied at some wsuch that F(w)(w) j. 11 00 01 10 If w = 11 and F(11) = then w satisfies [0](p q) so p q is a defeasible belief (at w) w satisfies [3]p so p is knowledge (at w)

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Willem Labuschagne & Laura vd Westhuizen Epistemic doxastic logic 3 Information template F(w) gives rise to a total preorder: u w v iff F(w)(u) F(w)(v) strict modular partial order: u < w v iff F(w)(u) < F(w)(v) These induce entrenchment orderings on L: e.g. is more entrenched than iff for every v satisfying there is some u satisfying with u w v i.e. excludes every world v at least as strongly as some world is excluded by . Theorem: The entrenchment orderings are compatible with the modal operators: is more entrenched than (at w) iff w satisfies [j] →[j] for all j, where → is the usual material conditional.

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Willem Labuschagne & Laura vd Westhuizen Epistemic doxastic logic 4 Various counterparts of the usual axiom schemes may be formulated, for example: K:([j] [j]( )) [j] for each j n T:[n-1] 4:[j] [n-1][j] for each j n 5: [j] [n-1] [j] for each j n Collectively these schemes characterise the class of agents for whom mutually accessible worlds have the same templates (where to say that u is accessible from w means that F(w)(u) < n). Theorem: Interpretation (W, F) is a model of KT45 iff at every w W w is accessible from itself if u is accessible from w then F(u) = F(w). Corollary: In a KT45 model, accessibility is an equivalence relation.

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Willem Labuschagne & Laura vd Westhuizen Epistemic doxastic logic 5 Problem: Rational consequence is not transitive. Definition: At a world w, the entrenchment of is the greatest j such that [j] is satisfied at w, and the disbelief in is the least j such that j is satisfied at w. Example: Belief set Cn(p q) and template F(w) below: 00 10 01 11 DegreeEntrenchmentDisbelief 0 p q, p q, q, p q All beliefs 1 p, q p p q, p+q, q, p q 2 pqpq p q, p 3 p p p q 4 p p Rational transitivity: If |~ and |~ , and is not more disbelieved than , then |~ .

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Willem Labuschagne & Laura vd Westhuizen Affective agents 1 Emotions play 3 roles in agents: Cutting down the search space — the agent may prefer some possibilities and avoid exploring others because of emotional valencies attached to them. Sustaining conative focus — an emotion acts to suppress other foci for attention and effort, e.g. Minsky’s story of agency Work exploiting Anger to stop Sleep (Society of Mind p.42). Changing goals or methods —in the absence of progress towards a goal, the analog of frustration can trigger a change of attack; emotion-analogs may also change the goal entirely, e.g. an agent doing some computation may, upon becoming aware of the imminent exhaustion of some resource, switch to ensuring the preservation of the most important data. The first role is played at the object-level, the remaining two at the meta-level.

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Willem Labuschagne & Laura vd Westhuizen Affective agents 2 Emotional valencies: An agent wants to find a day care provider for his small child. He considers various possibilities, rejecting most because of negative valency: one family care provider had huge dogs nanny referral services wanted huge fees up front. Some possibilities drew further investigation because of positive valency: “I know people who are very happy with family care providers.” Problems: One needs more than two valencies in order to prioritise. Also, one must beware of the numerical mania — to attach numbers is to invite the unjustified assumption that arithmetical operations are meaningful and that a notion of distance may be employed.

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Willem Labuschagne & Laura vd Westhuizen Affective agents 3 Templates (in normal form) provide a distance-free representation of graduated bivalency. The bottom box represents the most positive possibilities. Less positive cases occur higher up, and cases which are definitely unacceptable (and should not be investigated any further) are placed in the top box. This avoids the situation with negative-zero-positive representations, in which the assumption is that a step in the positive direction is ‘the same size’ or neutralised by a step in the negative direction. It avoids the problem of ‘where to draw the line’ in threshold-based representations. The excluded possibilities are those which can be definitely ruled out, and there are no borderline cases to quibble about. Needed: suitable merging operations so that templates representing typicality and emotional valency can be combined.

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Willem Labuschagne & Laura vd Westhuizen Selected Bibliography On subjunctive conditionals: Lewis: Counterfactuals Harvard U. Press 1973 and Pollock: Subjunctive Reasoning D. Reidel 1976 On semantic information: Carnap & Bar-Hillel: Semantic information, British journal for the Philosophy of Science 4:147-157 1953 On nonmonotonic logic: Lehmann & Magidor; What does a conditional knowledge base entail? Artificial Intelligence 55:1-60 1992 On semantics of belief revision and entrenchment: Meyer, Labuschagne & Heidema: Refined epistemic entrenchment, JoLLI 9:237-259 2000 Affective agents: Picard: Affective Computing, MIT Press 1997

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1 Knowledge Based Systems (CM0377) Lecture 4 (Last modified 5th February 2001)

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