TP2623 : Knowledge Representation & Reasoning

TP2623 : Knowledge Representation & Reasoning Topic 2 : Propositional Logic By : Shereena Arif Room : T2/8, Blok H Email : shereen@ftsm.ukm.my / shereen.ma@gmail.com

Learning Outcomes At the end of this topic, students will: • Understand the concept of knowledge-bases and propositional logic • Able to construct knowledge base according to propositional logic representation. TP2623, shereen@ftsm.ukm.my

Introduction • What is ‘Knowledge Representation & Reasoning’ ? • Also known as KRR. • Is a sub-area in AI- Artifical Intelligence. • Fundamental goal  to represent knowledge in a manner that facilitatesinferencing(i.e. drawing conclusions) from knowledge. • It analyzes how to formally think – • how to use a symbol system to represent a domain of discourse (that which can be talked about), along with functions that allow inference (formalized reasoning) about the objects TP2623, shereen@ftsm.ukm.my

These concepts are central to the entire field of AI: • The representation of knowledge • The reasoning processes that bring knowledge to life • The importance of KR&R : • To artificial agents  successful behavior that would be very hard to achieve otherwise. • E.g : permainancatur. • In partially observable situation • E.g : doktordanpesakit • Understanding natural language • E.g : Hamzahtertarikdengansebentukcincindidalamkotakpamerandanberhasratmembelinya. • Flexibility to changes in the environment by updating the relevant knowledge. TP2623, shereen@ftsm.ukm.my

Knowledge-Based Agents • The central component of a knowledge-based agent is its knowledge base, KB. • Informally, a KB is a set of sentences related, but not identical to natural languages. • Each sentence is expressed in a language called knowledge representation language. • How to : • Add new sentences into KB  TELL • Query about something already known  ASK TP2623, shereen@ftsm.ukm.my

Both tasks may involved inference  deriving new sentences from old ones. • In logical agents, inference must obey the requirement : • When one ASKs a question of the KB, • The answer should follow from what has been told (or TELLed) to the KB previously. TP2623, shereen@ftsm.ukm.my

function KB-AGENT (percept) returns an action static :KB, a knowledge base t, a counter, initially 0, indicating time TELL (KB, MAKE-PERCEPT-SENTENCE(percept, t)) action ASK(KB, MAKE-ACTION-QUERY(t)) TELL(KB, MAKE-ACTION-SENTENCE(action, t)) t t + 1 returnaction • Agent takes percept as input and returns an action. • Agent maintains a knowledge base, KB, which may initially contain some background knowledge. TP2623, shereen@ftsm.ukm.my

Each time the agent program is called it does two things. • It TELLs the knowledge base what it perceives. • It ASKs the knowledge base what action it should perform. • In the process of answering, extensive reasoning may be done about the current state, about outcomes of possible actions, and so on. • Once the action is chosen, the agent records its choice with TELL and executes the action. TP2623, shereen@ftsm.ukm.my

The wumpus world • The wumpus world is a cave consisting of rooms connected by passageways. • Lurking somewhere in the cave is the wumpus, a beast that eats anyone who enters its room • The wumpus can be shot by an agent, but the agent has only one arrow. • Some rooms contain bottomless pits that will trap anyone who wanders into these room (except the wumpus, who are too big). • Why ??.. The possibility of finding a heap of gold. • Wumpus world  an excellent testbed environment for intelligent agents. TP2623, shereen@ftsm.ukm.my

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 • Shoot kills wumpus if you are facing it • Shooting uses up the only arrow • Grab picks up gold if in same square • Release drops the gold in same square • Sensors: The percept [Stench, Breeze, Glitter, Bump, Scream] • Actuators: Left turn, Right turn, Forward, Grab, Release, Shoot TP2623, shereen@ftsm.ukm.my

Exploring a wumpus world [None, None, None, None, None] [None, Breeze, None, None, None] TP2623, shereen@ftsm.ukm.my

[Stench, None, None, None, None] [Stench, Breeze, Glitter, None, None] TP2623, shereen@ftsm.ukm.my

From the fact that there was no stench or breeze in [1,1], the agent can infer that [1,2] and [2,1] are free of dangers. • From the fact that it is still alive, it can infer than [1,1] is also OK. • A cautious agent will only move into a square that it knows is OK. • Let us suppose the agent moves to [2,1], giving us (b) in page 11. • The agent detects a breeze in [2,1], so there must be a pit in a neighbouring square, either [2,2] or [3,1]. (The notation P? indicates a possible pit.) • The pit cannot be in [1,1] because the agent has already been there and did not fall in. • At this point, there is only one known square that is OK and has not yet been visited. • So the prudent agent turns back and goes to [1,1] and then [1,2], giving us (a) in page 12 below. TP2623, shereen@ftsm.ukm.my

The stench in [1,2] means that there must be a wumpus nearby. • But the wumpus cannot be in [1,1] and it cannot be in [2,2] (or the agent would have detected a stench when it was in [2,1]. • Therefore, the agent can infer that the wumpus is in [1,3]. • The notation W! indicates this. • Moreover, the lack of a Breeze in [1,2] implies that there is no pit in the [2,2]. • Yet, we already inferred that there must be a pit in either [2,2] or [3,1], so this means it must be in [3,1]. • This is fairly a difficult inference because it combines knowledge gained at different times in different places. • When the agent moved to [2,3], it detects a glitter, which means that the gold must be there. TP2623, shereen@ftsm.ukm.my

In each case where the agent draws a conclusion from the available information, that conclusion is guaranteed to be correct if the available information is correct  fundamental property of reasoning. • In the rest of discussion, we will learn how to build logical agents that can represent the necessary information and draw the conclusions that were described. TP2623, shereen@ftsm.ukm.my

Logic in general • Logics are formal languages for representing information so 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 to each possible world • E.g x+y =4 is true if x=y=2, but false in x=y=1 • 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 TP2623, shereen@ftsm.ukm.my

Entailment • Entailment means that one thing follows from another: KB ╞ α • Knowledge base KB entails sentence α if and only if in all worlds where KB is true, α is also true • Another way to say this is if KB is true, then αmust also be true • E.g., the KB containing “the RumahKuning won” and “the RumahMerah won” entails “Either the RumahKuning won or the RumahMerah won” • E.g., x+y = 4 entails 4 = x+y • Entailment is a relationship between sentences (i.e., syntax) that is based on semantics TP2623, shereen@ftsm.ukm.my

Entailment in the wumpus world Situation after detecting nothing in [1,1], moving right, breeze in [2,1] Consider possible models for KB assuming only pits 3 Boolean choices  8 possible models 4 3 2 1 1 2 3 4 TP2623, shereen@ftsm.ukm.my

Wumpus models TP2623, shereen@ftsm.ukm.my

Wumpus models • KB = wumpus-world rules + observations TP2623, shereen@ftsm.ukm.my

Wumpus models • KB = wumpus-world rules + observations • α1 = "[1,2] is safe", KB ╞ α1, proved by model checking • In every model in which KB is true,α1 is also true TP2623, shereen@ftsm.ukm.my

Wumpus models • KB = wumpus-world rules + observations TP2623, shereen@ftsm.ukm.my

Wumpus models • KB = wumpus-world rules + observations • In some models in which KB is true, α2 is false • α2 = "[2,2] is safe", KB ⊭α2 • The agent cannot conclude that [2,2] is safe. • The example shows how definition of entailment can be used to derive conclusions  logical inference TP2623, shereen@ftsm.ukm.my

Inference • Entailment & inference : • KB is a haystack & α is the needle • Entailment  needle in the haystack • Inference  how to find the needle • If an inference i can deriveα from KB : KB ├i α • “α can be derived from KB by procedure i” • “i derives α from KB” • Soundness/Truth preserving: • An inference algo is sound if it derives only entailed sentences • i is sound if whenever KB ├i α, it is also true that KB╞ α • an unsound inference procedure makes things up as it goes along  announces the discovery nonexistent needles. • Model checking (if exist) is a sound procedure • Completeness • An inference algo is complete if it can derive any sentence that is entailed • i is complete if whenever KB╞ α, it is also true that KB ├i α TP2623, shereen@ftsm.ukm.my

Recap : • Described a reasoning process whose conclusions are guaranteed to be true in any world in which the premises are true. • “if KB is true in the real world, then any sentence α is also true in the real world” • Sentences  physical configurations of the agent • Reasoning  a process of constructing new physical configurations from the old ones. • Logical reasoning  ensures that the new ones actually follow the old ones. • 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. TP2623, shereen@ftsm.ukm.my

Propositional Logic or Calculus • Propositional logic • A very simple logic • Also called as Boolean logic. • Syntax & Semantics  The way in which the truth of sentences is determined. • Syntax • Defines allowable sentences.Two types of sentences : • Atomic sentences • Complex sentences TP2623, shereen@ftsm.ukm.my

Atomic sentences (indivisible) • consist of proposition symbol. • Each symbol stands for a proposition that can be true or false  P, Q, R etc. • Ex : W1,3  Wumpus is in [1,3] • Two proposition symbols with fixed meaning : True & False. • Complex sentences • Constructed from simpler sentences. • The sentences connected by logical connectives. • , , , ,  TP2623, shereen@ftsm.ukm.my

Logical Connectives •  • Not (negation) • Ex :  W1,3  negation of W1,3 •  • And (conjunction) • A  B  A and B •  • Or (disjunction) • A  B  A or B TP2623, shereen@ftsm.ukm.my

Logical Connectives •  • Implies also known as rules or if-then statements. • Conditional of A and B • Implication of B given A • Sometimes written as → or ⊃ • Ex : (W1,3  P3,1 )   W2,2 • Premise/Antecedent : (W1,3  P3,1 ) • Conclusion/Consequent :  W2,2 • Ex : A  B • If A, then B • A implies B • B when A • A only if B • A is antecedent, B is consequent TP2623, shereen@ftsm.ukm.my

Propositional Logic Syntax ORValid Sentences • Logical constants by themselves • T or F • Propositional symbol by itself • P or Q • Parents around a sentence • (P) • Sentences can be combined with logical connectives • (P Q) • Order of precedence (highest to lowest) •      • So P  Q  R  S is the same as ((P)  (Q  R))  S TP2623, shereen@ftsm.ukm.my

Propositional logic: Syntax • Propositional logic is the simplest logic – illustrates basic ideas • The proposition symbols P1, P2 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) TP2623, shereen@ftsm.ukm.my

Example Expressions • A • F • A • A A • A F • AB • (A B) (B) TP2623, shereen@ftsm.ukm.my

Propositional logic: Semantics • Semantics  rules for determining the truth of a sentence with respect to a particular model • Each model specifies true/false for each proposition symbol • Ex : P1,2 P2,2 P3,1 false true false • m1 = {P1,2 = false, P2,2= true, P3,1 = false} TP2623, shereen@ftsm.ukm.my

Propositional logic: Semantics • The semantics for propositional logic must specify how to compute the truth value of any sentence, given a model. • Sentences are constructed from atomic sentences and the 5 connectives : • Atomic sentences  true in every model or false? • Complex sentences  rules, summarized in a truth table. TP2623, shereen@ftsm.ukm.my

Propositional logic: Semantics Rules for evaluating truth with respect to a model m: S is true iff S is false S1 S2 is true iff S1 is true and S2 is true S1 S2 is true iff S1is true or S2 is true S1 S2 is true iff S1 is false or S2 is true i.e., is false iff S1 is true andS2 is false S1 S2 is true iff S1S2 is true andS2S1 is true Simple recursive process evaluates an arbitrary sentence, e.g., P1,2  (P2,2 P3,1) = true (true  false) = true true = true TP2623, shereen@ftsm.ukm.my

Truth Table P Q P P  Q P  Q P  Q P  Q F F T F F T T F T T F T T F T F F F T F F T T F T T T T TP2623, shereen@ftsm.ukm.my

Solving Problems with Truth Tables(Enumeration Method) • Problem • If Ali starts playing with his Playstation, then he will stay up late. If he stays up late, he will miss his morning class. He was in class. • Did he start playing his playstation? • Propositions • P1: Ali plays playstation • P2: Ali stays up late • P3: Ali attends class • Premises • P1 -> P2 • P2 -> -P3 • P3 Answer TP2623, shereen@ftsm.ukm.my

Truth Tables • Truth tables are effective, but impractical • n propositions requires a table with 2n rows • More efficient methods of proving conclusions from premises exist • E.g. Wang’s Algorithm TP2623, shereen@ftsm.ukm.my

Some Exercises Using Truth Table • (T  F)  T • Answer: • (F  F)  T • Answer: • ((F  T)  F)  ((T  F)) • Answer: TP2623, shereen@ftsm.ukm.my

Things we can say with Propositional Calculus • The sky is grey = S1 • It will rain = R1 • I have three arms = A1 • If S1 and R1 are T, then The sky is grey  It will rain is also T • If S1 is T and A1 is F then The sky is grey I have three arms is T TP2623, shereen@ftsm.ukm.my

Combining expressions in Propositional Calculus • (A1 B1) C1 • (A2 ( C1 B4)  C3 • (A4 B2) C2 Is equivalent to • ((A1 B1) C1) ((A2 ( C1 B4) C3) ((A4 B2) C2) TP2623, shereen@ftsm.ukm.my

Logical equivalence • Two sentences are logically equivalent} iff true in same models: α ≡ ß iff α╞ β and β╞ α TP2623, shereen@ftsm.ukm.my

Validity and satisfiability A sentence is valid if it is true in all models, also known as tautology e.g., True, A A, A  A, (A  (A  B))  B Validity is connected to inference via the Deduction Theorem: KB ╞ α if and only if (KB α) is valid A sentence is satisfiable if it is true in some model e.g., A  B, C e.g, if sentence α is true in a model m m satisfies α, or m is a model of α A sentence is unsatisfiable if it is true in no models e.g., AA α is valid iff α is unsatisfiable Satisfiability is connected to inference via the following: KB ╞ α if and only if (KBα) is unsatisfiable TP2623, shereen@ftsm.ukm.my

Entailment: Propositional Inference(Enumeration Method) Let  = AB and KB = (AB)(B C) Is it the case that KB ╞ ? Check all possible models -  must be true wherever KB is true A B C AB B C KB  F F F F F T F T F F T T T F F T F T T T F T T T Answer TP2623, shereen@ftsm.ukm.my

The Method of Deduction

Intro • Previously we used the truth table to test arguments for validity • Here we develop an alternative approach  the method of deduction • We can use rules of inference and rules of replacement • Covers standard patterns of inference that can be applied to derive chains of conclusions that lead to the desired goal  inference rules TP2623, shereen@ftsm.ukm.my

Inference & Replacement with Propositional Calculus • Modus Ponens (Implication elimination) Assume P  Q, Given P ----------------- Then Q • For example: • If the statement “If it is raining, then you get wet” is true, and the statement “It is raining” is true, Then you can infer that “you get wet.” TP2623, shereen@ftsm.ukm.my

Inference & Replacement with Propositional Calculus • Disjunctive Syllogism (or introduction) Assume P -------------- Then P Q • For example: • If the statement “It is raining” is true, Then you can infer that the compound statement “It is raining” OR <anything> will be true TP2623, shereen@ftsm.ukm.my

TP2623 : Knowledge Representation & Reasoning