LECTURE 06 Logical Agent Course : T0264 – Artificial Intelligence 10/30/2019 Course : T0264 – Artificial Intelligence Year : 2013 LECTURE 06 Logical Agent

T0264 - Artificial Intelligence Learning Outcomes At the end of this session, students will be able to: LO 2 : Explain various intelligent search algorithms to solve the problems LO 3 : Explain how to use knowledge representation in reasoning purpose T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Outline Knowledge-Based Agents Logic in General Propositional Logic (PL) : A Very Simple Logic Proportional Theorem Proving (Resolution Algorithm) Forward and Backward Algorithm Summary T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Knowledge-Based Agents Knowledge base = set of sentences in a formal language Declarative approach to building an agent (or other system): Tell it what it needs to know Then it can Ask itself what to do - answers should follow from the KB Agents can be viewed at the knowledge level i.e., what they know, regardless of how implemented Or at the implementation level i.e., data structures in KB and algorithms that manipulate them T0264 - Artificial Intelligence

Tugas Presentasi I Logical Agents & Wumpuss World ANDREAN HARRY RAMADHAN DIONDY KUSUMA RIVKY CHANDRA WIENA MARCELINA WILSON GUNAWAN ANDRE IVAN T0264 - Artificial Intelligence

A Simple Knowledge-Based Agent Agents A Simple Knowledge-Based Agent The agent must be able to: Represent states, actions, etc. Incorporate new percepts Update internal representations of the world Deduce hidden properties of the world Deduce appropriate actions T0264 - Artificial Intelligence

Wumpus World PEAS Description Knowledge-Based Agents Wumpus World PEAS Description Performance measure gold +1000, death -1000 -1 per step, -10 for using the arrow Environment Squares adjacent to wumpus are smelly Squares adjacent to pit are breezy Glitter if gold is in the same square Shooting kills wumpus if you are facing it Shooting uses up the only arrow Grabbing picks up gold if in same square Releasing drops the gold in same square Sensors: Stench, Breeze, Glitter, Bump, Scream Actuators: Left turn, Right turn, Forward, Grab, Release, Shoot T0264 - Artificial Intelligence

Wumpus World Characterization Knowledge-Based Agents Wumpus World Characterization Fully Observable No – only local perception Deterministic Yes – outcomes exactly specified Episodic No – sequential at the level of actions Static Yes – Wumpus and Pits do not move Discrete Yes Single-agent? Yes – Wumpus is essentially a natural feature T0264 - Artificial Intelligence

Exploring a Wumpus World Knowledge-Based Agents Exploring a Wumpus World A = Agent B = Breeze G = Glitter, Gold OK = Safe square P = Pit S = Stench V = Visited W = Wumpus T0264 - Artificial Intelligence

Exploring a Wumpus World Knowledge-Based Agents Exploring a Wumpus World T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Logic In General Logics are formal languages for representing information such that conclusions can be drawn Syntax defines the sentences in the language Semantics define the "meaning" of sentences; i.e., define truth of a sentence in a world E.g., the language of arithmetic x+2 ≥ y is a sentence; x2+y > { } is not a sentence x+2 ≥ y is true if the number x+2 is no less than the number y x+2 ≥ y is true in a world where x = 7, y = 1 x+2 ≥ y is false in a world where x = 0, y = 6 T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Logic In General Entailment Entailment means that one thing follows from another: KB ╞ α Knowledge base KB entails sentence α if and only if α is true in all worlds where KB is true E.g., the KB containing “the Giants won” and “the Reds won” entails “Either the Giants won or the Reds won” E.g., x+y = 4 entails 4 = x+y Entailment is a relationship between sentences (i.e., syntax) that is based on semantics T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Logic In General Models Logicians typically think in terms of models, which are formally structured worlds with respect to which truth can be evaluated We say m is a model of a sentence α if α is true in m M(α) is the set of all models of α Then KB ╞ α iff M(KB)  M(α) E.g. KB = Giants won and Reds won α = Giants won T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Logic In General Wumpus Models KB = wumpus-world rules + observations T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Logic In General Wumpus Models KB = wumpus-world rules + observations α1 = "[1,2] is safe", KB ╞ α1, proved by model checking KB = wumpus-world rules + observations T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Logic In General Wumpus Models KB = wumpus-world rules + observations α2 = "[2,2] is safe", KB ╞ α2 T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Logic In General Inference KB ├i α = sentence α can be derived from KB by procedure i Soundness: i is sound if whenever KB ├i α, it is also true that KB╞ α Completeness: i is complete if whenever KB╞ α, it is also true that KB ├i α Preview: we will define a logic (first-order logic) which is expressive enough to say almost anything of interest, and for which there exists a sound and complete inference procedure. That is, the procedure will answer any question whose answer follows from what is known by the KB. T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Logic In General Sentences are physical configuration of agent, and reasoning is a process of constructing new physical configurations from old ones. New configurations represent aspects of the world that actually follow from the aspect that the old configurations represent T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Propositional Logic: A Very Simple Logic Standard Logic Symbols  = For all ; [e.g : every one, every body, any time, etc]  = There exists ; [e.g : some one, some time, etc]  = Implication ; [ if … then ….]  = Equivalent ; biconditional [if … and … only … if …]  = Not ; negation  = OR ; disjunction  = AND ; conjunction T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Propositional Logic: A Very Simple Logic Syntax Propositional logic is the simplest logic – illustrates basic ideas The proposition symbols S1, S2 etc are sentences If S is a sentence, S is a sentence (negation) If S1 and S2 are sentences, S1  S2 is a sentence (conjunction) If S1 and S2 are sentences, S1  S2 is a sentence (disjunction) If S1 and S2 are sentences, S1  S2 is a sentence (implication) If S1 and S2 are sentences, S1  S2 is a sentence (biconditional) T0264 - Artificial Intelligence

Propositional Logic: A Very Simple Logic Semantics True is true in every model and False is false in every model The truth value of every other proposition symbol must be specified directly in the model. For complex sentences, there are five rules for any sub sentences P and Q in any model m :  P is true iff P is false in m.  P  Q is true iff both P and Q true in m.  P  Q is true iff either P or Q is true in m.  P  Q is true unles P is true and Q is false in m. i.e., is false iff P is true and Q is false  P  Q is true iff P and Q are both true or both false in m. i.e., P Q is true and Q  P is true Simple recursive process evaluates an arbitrary sentence, e.g., P1,2  (P2,2  P3,1) = true  (true  false) = true  true = true T0264 - Artificial Intelligence

Truth Tables for Connectives Propositional Logic: A Very Simple Logic Truth Tables for Connectives T0264 - Artificial Intelligence

A Truth-Table Enumeration Algorithm Propositional Logic: A Very Simple Logic A Truth-Table Enumeration Algorithm Depth-first enumeration of all models is sound and complete For n symbols, time complexity is O(2n), space complexity is O(n) T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Propositional Logic: A Very Simple Logic Logical Equivalences Two sentences are logically equivalent iff true in same models: α ≡ ß iff α╞ β and β╞ α T0264 - Artificial Intelligence

Propositional Theorem Proving (Resolution Algorithm) Inference procedure based on resolution work by using the principle of proof by contradiction. That is, to show that KB├ = , we show that (KB   ) is unsatisfied. We do this by contradiction. Therefore: First : (KB   ) is convert into CNF (normal clause form) Second : show clauses obtained by resolving pairs in the row T0264 - Artificial Intelligence

Propositional Theorem Proving (Resolution Algorithm) Convert all the proposition logic to normal clause form (CNF). Negate P and convert the result to clause form. Add it to the set of clause in step Repeat until either a contradiction is found or no progress can be made, or a predetermined amount of effort has been expended. Select two clause. Call these the parent clauses. T0264 - Artificial Intelligence

Propositional Theorem Proving (Resolution Algorithm) Algorithm: (cont’d) Resolve them together. The resolvent will be the disjunction of all of the literals of both parent clauses with appropriate substitutions performed and with the following exception: If there is one pairs of literals T1 and T2 such that one of the parent clauses contains T1 and the other contains  T2 and if T1 and T2 are unifiable, then neither T1 nor  T2 should appear in the resolvent. If there is more than one pair of complementary literals, only one pair should be omitted from the resolvent. If the resolvent is the empty clause, then a contradiction has been found. If it is not, then add it to the set of clause available to the procedure. T0264 - Artificial Intelligence

Simple Pseudo Code Resolution algorithm for proportional logic Propositional Theorem Proving (Resolution Algorithm) Simple Pseudo Code Resolution algorithm for proportional logic T0264 - Artificial Intelligence

Conversion to Normal Clause Form (CNF Algorithm) Propositional Theorem Proving (Resolution Algorithm) Conversion to Normal Clause Form (CNF Algorithm) Eliminate , using: a  b = a  b. Reduce the scope of each  to a single term, using de Morgan’s laws: (p) = p (ab) = a  b (ab) = a  b x : P(x) =x : P(x) x : P(x) = x : P(x) 3. Standardize variables For sentences like (∀x P(x)) ∨ (∃x Q(x)) which use the same variable name twice, change the name of one of the variables. This avoids confusion later when we drop the quantifiers. For example, from ∀x [∃y Animal(y) ∧ ¬Loves(x, y)] ∨ [∃y Loves(y, x)]. we obtain: ∀x [∃y Animal(y) ∧ ¬Loves(x, y)] ∨ [∃z Loves(z, x)]. T0264 - Artificial Intelligence

Convert to Normal Clause Form (cont’d) Propositional Theorem Proving (Resolution Algorithm) Convert to Normal Clause Form (cont’d) Move all quantifiers to the left of the formula without changing their relative order. Eliminate existential quantifiers by inserting Skolem functions. ∃x P(x) into P(A), where A is a new constant Drop universal quantifiers Convert the matrix into a conjunction of disjoints, using associativity and distributivity (distribute ORs over ANDs) T0264 - Artificial Intelligence

Example : Conversion to Normal Clause Form Propositional Theorem Proving (Resolution Algorithm) Example : Conversion to Normal Clause Form Sentence: “Every body who know Hitler, either like Hitler or think that anyone who killed some one is crazy” Proportional Logic is : x : [body(x)  know(x, Hitler)] [like(x, Hitler) (y:z: killed(y, z)  crazy(x, y)] T0264 - Artificial Intelligence

Applied of Clause Form Algorithm Propositional Theorem Proving (Resolution Algorithm) Applied of Clause Form Algorithm x : [body(x)  know(x, Hitler)] [like(x, Hitler) (y:z: killed(y, z)  crazy(x, y)] x : [body(x)  know(x, Hitler)]  [like(x, Hitler) (y: (z: killed(y, z))  crazy(x, y))] Ok ! x : y : z : [body(x)  know(x, Hitler)]  [like(x, Hitler) (killed(y, z))  crazy(x, y)] [body(x)  know(x, Hitler)]  [like(x, Hitler) (killed(y, z))  crazy(x, y)] body(x)  know(x, Hitler)  like(x, Hitler) killed(y, z)  crazy(x, y) T0264 - Artificial Intelligence

Conversion to Normal Form Propositional Theorem Proving (Resolution Algorithm) Conversion to Normal Form T0264 - Artificial Intelligence

Propositional Theorem Proving (Resolution Algorithm) Conversion to Normal Form T0264 - Artificial Intelligence

Forward and Backward Chaining Algorithm Forward and Backward Chaining Idea: fire any rule whose premises are satisfied in the KB, add its conclusion to the KB, until query is found T0264 - Artificial Intelligence

Forward Chaining Algorithm Forward and Backward Algorithm Forward Chaining Algorithm Forward chaining is sound and complete for Horn KB T0264 - Artificial Intelligence

Forward Chaining Example Forward and Backward Algorithm Forward Chaining Example T0264 - Artificial Intelligence

Forward Chaining Example Forward and Backward Algorithm Forward Chaining Example T0264 - Artificial Intelligence

Forward Chaining Example Forward and Backward Algorithm Forward Chaining Example T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Forward and Backward Algorithm Proof of Completeness FC derives every atomic sentence that is entailed by KB FC reaches a fixed point where no new atomic sentences are derived Consider the final state as a model m, assigning true/false to symbols Every clause in the original KB is true in m a1  …  ak  b Hence m is a model of KB If KB╞ q, q is true in every model of KB, including m T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Forward and Backward Algorithm Backward Chaining Idea: work backwards from the query q: to prove q by BC, check if q is known already, or prove by BC all premises of some rule concluding q Avoid loops: check if new sub goal is already on the goal stack Avoid repeated work: check if new sub goal has already been proved true, or has already failed T0264 - Artificial Intelligence

Backward Chaining Example Forward and Backward Algorithm Backward Chaining Example T0264 - Artificial Intelligence

Backward Chaining Example Forward and Backward Algorithm Backward Chaining Example T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Forward and Backward Algorithm Backward Chaining Example T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Forward and Backward Algorithm Backward Chaining Example T0264 - Artificial Intelligence

T0264 - Artificial Intelligence Summary Logical agents apply inference to a knowledge base to derive new information and make decisions Basic concepts of logic: syntax: formal structure of sentences semantics: truth of sentences in the models entailment: necessary truth of one sentence given another inference: deriving sentences from other sentences soundness: derivations produce only entailed sentences completeness: derivations can produce all entailed sentences Resolution is complete for propositional logic Forward, backward chaining are linear-time, complete for Horn clauses T0264 - Artificial Intelligence

T0264 - Artificial Intelligence References Stuart Russell, Peter Norvig,. 2010. Artificial intelligence : a modern approach. PE. New Jersey. ISBN:9780132071482, Chapter 7 Elaine Rich, Kevin Knight, Shivashankar B. Nair. 2010. Artificial Intelligence. MHE. New York. , Chapter 5 Propositional Logic: http://logic.stanford.edu/classes/cs157/2009/notes/chap02.html The Resolution Method: http://www.doc.ic.ac.uk/~sgc/teaching/pre2012/v231/lecture8.html T0264 - Artificial Intelligence

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