KRR. Example. Example (formalization) Example (cont.) Knowledge representation & reasoning

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1 Klassische hemen der Computerwissenschaften Artificial Intelligence Knowledge representation & reasoning Prof. Dr. ranz Wotawa KRR Example Use logic to represent knowledge (e.g., the theory) Represent important aspects of a problem! Separate different kinds of knowledge (background knowledge, case-specific knowledge,..) Describe knowledge from Causes to Effects Use logical reasoning to derive new knowledge orce? But also: Example (cont.) Example (formalization) ormalization in propositional logic: equalorce midposition lowerorce downposition higherorce upposition break downposition 1

2 Abduction: Abductive vs. deductive reasoning Reasoning from Effects to Causes rom downposition follows either break or lowerorce or a definition have a look at previous slides Deduction: Reasoning from Causes to Effects rom break follows downposition using the definition of logical consequences Monotonic vs. Non-monotonic reasoning Monotonic: rom new knowledge we can derive more knowledge If we add downposition shouldhappen to the theory, we can derive shouldhappen when assuming lowerorce. Non-Monotonic: Adding new knowledge leads to a decrease of knowledge that can be derived Non-monotonic reasoning - Inconsistencies What happens when we add: (midposition downposition) (midposition upposition) (upposition downposition) to the theory together with the following observations? higherorce downposition he theory becomes inconsistent!!! Consequences of inconsistencies Everything can be derived! We cannot derive anything meaningful anymore! Solution Change the theorie Use a different schema for deriving new knowledge Allow for removing knowledge in order to retain consistency! New heory: New formalization equalorce break midposition lowerorce break downposition higherorce break upposition break downposition (midposition downposition) (midposition upposition) (upposition downposition) 2

3 Default reasoning Idea In sentences we have some kind of hidden assumptions, i.e., defaults! Example: equalorce break midposition Assumption that the system is not broken, i.e., the system is working as expected. Idea (cont.) Make hidden assumptions explicit (as always in AI) Introduce defaults, i.e., logical sentences that can only be used unless a contradiction can be derived. Definition Default Definition (Default): A default δ is any configuration of the form A : B 1,,B n C where A, B 1,,B n, C are logical sentences (A: prerequisite, B 1,..,B n : justifications, C: consequent). We also write (A : B 1,,B n C) to express defaults. Semantics of defaults Informal semantics: If A is true and it is not contradictory to assume B 1,,B n, then we can conclude C. Given two sets of sentences S and and a default δ. By applying δ to with respect to S (i.e., δ S ), we mean that if A and B 1,, B n S, then C. S is a set of axioms of and all consequences of previously used defaults. Default theories Definition (Default heory): A default theory is a pair (R,Δ) where R is a set of logical sentences and Δ is a set of defaults. Example: Δ = { (bird : fly fly) } and R = { penguin, penguin bird, penguin fly } Using (R,Δ) and the fact penguin we cannot longer derive a contradiction! 3

4 How to handle defaults ake all axioms of the theory R and apply defaults. We define this idea by introducing the smallest set of beliefs Λ (R,Δ) (S) that we can derive from the given knowledge and S: 1. R Λ (R,Δ) (S) 2. h(λ (R,Δ) (S)) = Λ (R,Δ) (S) 3. If δ Δ, then δ S Λ (R,Δ) (S) Extensions Definition: A set E of sentences is an extension for a default theory (R,Δ) iff E is a fixed point of the operator Λ (R,Δ) (i.e., Λ (R,Δ) (E) = E) Example: he default theory ({R,Q},{(R: P P), (Q:P P)}) has two extensions E 1 ={P} and E 2 ={ P} Extensions describe POSSIBLE WORLDS Goals / Problems Compute all extensions of default theories his computation is very complex hus we restrict the default theories to comprise only facts, rules (i.e. implications), and normal defaults. A default is said to be normal iff it is of the form (A : B B). References Philippe Besnard, An Introduction to Default Logic, Springer, Assumption-based reasoning Assumption-based reasoning Use special propositions, i.e., assumptions that can be either true or false. Use assumptions to retain consistency of theories Afterwards extensions and explanations for inconsistencies can be given 4

5 he coffee machine 1. Request request 2. Water water 3. Beans beans 4. request water beans coffee 5. coffee request 6. coffee water 7. coffee beans 8. no_coffee request water no_beans 9. no_coffee request beans no_water 10.no_coffee beans water no_request 11.beans no_beans 12.water no_water 13.request no_request 14.coffee no_coffee 15.no_coffee Coffee Coffee-ilter Coffee-Machine Coffee-Container Coffee-Mill Water-ank Water AMS Basic ideas Assumption-based ruth Maintenance System (AMS) Retain truth Knowledge-base updates irst step for computing explanations AMS Basic ideas (cont.) Request request Proposition Assumption Represent assumptions, propositions and rules as directed graph Request coffee no_coffee request Nogood (Inconsistency, alsum) AMS node Nodes store basic information Nodes for Assumptions, propositions, and Inconsistencies (Contradictions are stored in a node called NOGOOD). Nodes store heir name A set of a set of assumptions which have to be true in order to make the node true (Environments) A set of justifications where a justification is a set of propositions which have to be true in order to make the node true. AMS node (cont.) Request request Request request (Request, {{Request}}, {(Request)}) (request, {{Request}}, {(Request)}) Name Assumption Justification Representation: Some remarks Premises (p,{{}},{()}) Assumptions (A,{{A}},{(A)}) Propositions (p,{s 1,..}, {J 1,..}) Logical interpretation of (p, {{A 11,..,A 1n(1) },..}, {(q 11,..,q 1m(1) ),..} A 11 A 1n(1) p q 11 q 1m(1) p 5

6 ask of the AMS or every rule added to the AMS construct possible new nodes and vertices. Compute the labels, i.e., the set of assumptions, for every node such that the labels are: Consistent Minimal Complete Sound Example Add rules in the given order 1. Request request 2. Water water 3. Beans beans 4. request water beans coffee 5. coffee no_coffee 6. no_coffee (AC, Premise) AMS before adding fact / premise AMS after adding fact Request {{Request}} request {{Request}} Request {{Request}} request {{Request}} Water {{Water}} water {{Water}} coffee {{Request,Water,Beans}} Water {{Water}} water {{Water}} coffee {} coffee {{Request,Water,Beans}} Beans {{Beans}} beans {{beans}} {} Beans {{Beans}} beans {{Beans}} {} {{Request,Water,Beans}} no_coffee {} no_coffee {} no_coffee {{}} What happens when.. Adding the fact request? Adding the fact beans? AMS algorithm (1) Call algorithm if justification (=rule) is added. Remark: J: q 1 q n p Antecedence Consequent When J is supplied call PROPAGAE(J,Φ,{{}}) where Φ indicates the absence of an optional antecedence node. 6

7 AMS algorithm (2) PROPAGAE ( q 1 q n p, a, I) 1. [Compute the incremental update] L = WEAVE (a, I, {q 1,,q n }) 2. [Update label and recure] UPDAE (L,p) AMS algorithm (3) UPDAE (L, p) 1. [Dedect nogoods] If p = the call NOGOOD (E) for each E L and return {} 2. [Update p s label ensuring minimality] 1. Delete every environment from L which is a superset of some label environment of p. 2. Delete every environment from the label of p which is a superset of some element of L. 3. Add every remaining environment of L to the label of p. 3. [Propagate the incremental change of p s label to its consequences] or every justification J in which p is an antecedence node call PROPAGAE (J,p,L). AMS algorithm (4) AMS algorithm (5) WEAVE (p,i,x) 1. [ermination condition] If X is empty, return I. 2. [Iterate over the antecedence nodes] Let h be the first node of the list X and R the rest. 3. [Avoid computing the full label] If h = p, return WEAVE (Φ,I,R) 4. [Incrementally construct the label] Let I be the set of all environments formed by computing the union of an environment of I and an environment of h s label. 5. [Ensure that I is minimal and contains no known inconsistency] Remove from I all duplicates, nogoods, as well as any environment subsumed by any other. 6. Return WEAVE (p, I, R). NOGOOD (E) 1. Mark E as nogood 2. Remove E and any superset from every node label AMS remarks AMS retains consistency by maintaining the node labels and the nogood. Label update for the AMS is NPcomplete! Various extensions mentioned in literature References Johan de Kleer. An assumption-based MS, Artificial Intelligence, 28: , Johan de Kleer: A general labeling algorithm for assumption-based truth maintenance. In proceedings of the AAAI, pp ,

8 Explanations Basic idea AMS retains consistency Inconsistent knowledge is stored in a NOGOOD. he labels of the NOGOOD represent all possible ways for deriving an inconsistency. Hence, the provide information about the explanation for an inconsistency. Example More general Given the nogood NOGOOD {{Water},{Beans}} his means that both Water and Beans lead to an inconsistency. Hence, we have to refill the water and the beans in order to obtain coffee. ormally: Water and Beans In order to explain consistency we have to consider all environments of the nogood node. In particular we have to take one element of each nogood and make it false. Hence, logically the nogood cannot longer be derived! NOGOOD Environments: A 11 A 1n(1) A m1 A mn(1) Selection of A 1i(1) A m i(m) and setting their truth values to false ensures consistency! Computing all selection is equivalent to computing all hitting sets (HS) of a set of sets. Hitting sets Definition: Given a set of sets. A set H x x is called a hitting set for iff the intersection of h with every set x is not empty, i.e., x H. Definition: A hitting set H is minimal if no proper subset H of H is a hitting set. Example - HS Given: = {{a,b},{a,c}} Solution: 2 minimal hitting sets {a} {b,c} 8

9 Computing hitting sets here are many algorithms described in literature, e.g., Ray Reiter, A heory of Diagnosis from irst Principles, Artificial Intelligence, 32(1):57-95, R. Greiner, B.A. Smith, R.W. Wilkerson, A Correction to the Algorithm in Reiter s heory of Diagnosis, Artificial Intelligence, 41(1):79-88, Putting it all together Idea: construct a graph from the given set of sets Computing extensions for normal defaults Given a default theory (R,Δ) where each δ Δ is of the form (X : a a). irst compute a theory = R { A X a (X : a a) Δ} where A is a unique assumable which correspond to a. Let AS be the set of all assumables. eed into an AMS ake the environments of the NOGOOD and compute the hitting sets. We know that all elements of each hitting sets have to be assigned a false truth value. In our case extensions in contrast have only positive elements. All hitting sets are explanations for an inconsistency. We can take a hitting set H and compute h( (H\AS)) which is an extension of (R,Δ). Example Given: ({r,q, np p },{(r:np np), (q:p p)}) = {r,q, np p, NP r np, P q p} he AMS delivers the NOGOOD {{P,NP}}. We obtain two hitting sets H 1 = {P} H 2 = {NP} Hence, we get two extensions E 1 = h( ({P,NP}\H 1 )) = {r,q,np,np} E 2 = h( ({P,NP}\H 2 )) = {r,q,p,p} Alternative computation using truth tables and SA {r,q, np p, NP r np, P q p} r q p np P NP heory SA-Solver Programs for proving satisfiability of propositional theories See E.g. Java-Sat Solver: SA4J 9

10 Conclusion Overview of logic-based AI focusing on some aspects of default and assumption-based reasoning Restricted to propositional logic comprising facts and rules (Horn Clause heories) Applications: Commonsense Reasoning Diagnosis Knowledge-based configuration Conclusion Modeling for AI Abductive vs. deductive reasoning Abduction: rom a theory and observations O derive explanations E, i.e., E O and E is consistent Deduction: rom a theory and facts derive new facts N, i.e., N Non-monotonic vs. monotonic reasoning 10