Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · –...
Transcript of Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · –...
![Page 1: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/1.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
1
Chapter 7Logical Agents
CS 461 – Artificial IntelligencePinar Duygulu
Bilkent University, Spring 2008
Slides are mostly adapted from AIMA and MIT Open Courseware
![Page 2: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/2.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
2
Outline
• Knowledge-based agents• Wumpus world• Logic in general - models and entailment• Propositional (Boolean) logic• Equivalence, validity, satisfiability• Inference rules and theorem proving
– forward chaining– backward chaining– resolution
![Page 3: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/3.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
3
Introduction
• The representation of knowledge and the reasoning processes that bring knowledge to life are central to entire field of artificial intelligence
• Knowledge and reasoning are important to artificial agents because they enable successful behaviors that would be very hard to achieve otherwise (no piece in chess can be on two different squares at the same time)
• Knowledge and reasoning also play a crucial role in dealing with partially observable environments (inferring hidden states in diagnosing diseases, natural language understanding)
• Knowledge also allows flexibility.
![Page 4: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/4.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
4
Knowledge bases
• Knowledge base = set of sentences in a formal language• Each sentence is expressed in a knowledge representation language and represents some
assertions about the world• There should be a way to add new sentences to KB, and to query what is known• 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
• Both tasks may involve inference – deriving new sentences from old• In logical agents – when one ASKs a question to KB, the answer should follow from what
has been TELLed• 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
![Page 5: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/5.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
5
A simple knowledge-based agent
• KB : maintain the background knowledge• Each time the agent program is called it does three things
– TELLs the KB what it perceives– ASK the KB what action it should perform– TELL the KB that the action is executed
• 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
• Declarative versus procedural approaches
![Page 6: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/6.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
6
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 (stench)– Squares adjacent to pit are breezy– Glitter iff 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
![Page 7: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/7.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
7
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
![Page 8: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/8.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
8
Exploring a wumpus world
![Page 9: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/9.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
9
Exploring a wumpus world
![Page 10: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/10.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
10
Exploring a wumpus world
![Page 11: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/11.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
11
Exploring a wumpus world
![Page 12: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/12.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
12
Exploring a wumpus world
![Page 13: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/13.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
13
Exploring a wumpus world
![Page 14: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/14.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
14
Exploring a wumpus world
![Page 15: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/15.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
15
Exploring a wumpus world
![Page 16: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/16.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
16
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 iff 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
• Possible world – model • m is a model of α – the sentence α is true in model m
![Page 17: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/17.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
17
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
– If α true then KB must also be 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
![Page 18: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/18.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
18
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
![Page 19: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/19.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
19
Entailment in the wumpus world
• Situation after detecting nothing in [1,1], moving right, breeze in [2,1]
![Page 20: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/20.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
20
Wumpus models
3 Boolean choices ⇒ 8 possible models for the adjacent squares [1,2], [2,2] and [3,1] to contain pits
![Page 21: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/21.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
21
Wumpus models
• KB = wumpus-world rules + observations• KB is false in any model in which [1,2] contains a
pit, because there is no breeze in [1,1]
Consider possible models for KB assuming only pits
![Page 22: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/22.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
22
Wumpus models
• Consider α1 = “[1,2] is safe” = “There is no pit in [1,2]”• In every model KB is true α1 is also true• KB ╞ α1, proved by model checking• We can conclude that there is no pit in [1,2]
![Page 23: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/23.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
23
Wumpus models
• Consider α2 = “[2,2] is safe” = “There is no pit in [2,2]”• In some models in which KB is true α2 is false• KB ╞/ α2
• We cannot conclude that there is no pit in [2,2]
![Page 24: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/24.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
24
Inference• KB ├i α = sentence α can be derived from KB by a procedure i (an inference algorithm)
• Soundness: i is sound if whenever KB ├i α, it is also true that KB╞ α• An inference algorithm that derives only entailed sentences is sound or truth preserving
(model checking is a sound procedure)• Completeness: i is complete if whenever KB╞ α, it is also true that KB ├i α • An inference algorithm is complete if it can derive any sentence that is entailed
• Think set of all consequences of KB as a haystack and α as a needle. Entailment is like the needle being in the haystack, and inference is like finding it
• An unsound inference procedure essentially makes things up as it goes along – it announces the discovery of nonexistent needles
• For completeness, a systematic examination can always decide whether the needle is in the haystack which is finite
• If KB is true in the real world then any sentence α derived from KB by a sound inference procedure is also true in real world
– The conclusions of the reasoning process are guaranteed to be true in any world in which the premises are true
![Page 25: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/25.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
25
Propositional logic: Syntax
• Propositional logic is the simplest logic – illustrates basic ideas
• Atomic sentences : consists of proposition symbols P1, P2
• Complex sentences : constructed from atomic sentences using logical connectives
– 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)
![Page 26: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/26.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
26
Precedence– Use parentheses to specify the precedence– Otherwise the precedence from highest to lowest is: ¬ , ∧, ∨, ⇒ , ⇔ – A ⇒ Β ⇒ C is not allowed
![Page 27: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/27.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
27
Propositional logic: SemanticsSemantics defines the rules for determining the truth of a sentence with respect to a particular modelEach model specifies true/false for each proposition symbol
E.g. P1,2 P2,2 P3,1
false true false
True is true in every model, False is false in every modelThe truth value of every other proposition symbol must be specified directly in the modelFor the complex sentencesRules 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 trueS1 ∨ S2 is true iff S1is true or S2 is trueS1 ⇒ S2 is true iff S1 is false orS2 is true i.e., is false iff S1 is true and S2 is falseS1 ⇔ S2 is true iff S1⇒S2 is true andS2⇒S1 is true
Important shorthandS1 ⇒ S2 ≡ ¬S1 ∨ S2
S1 ⇔ S2 ≡ S1 ⇒ S2 ∧ S2 ⇒ S1
![Page 28: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/28.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
28
Truth tables for connectives
Simple recursive process evaluates an arbitrary sentence, e.g.,¬P1,2 ∧ (P2,2 ∨ P3,1) = true ∧ (true ∨ false) = true ∧ true = true
Implication: if P is true then I am claming that Q is true, otherwise I am making no claimThe sentence is false, if P is true but Q is false
Biconditional: True whenever both P->Q and Q->P is true(e.g. a square is breezy if and only if adjacent square has a pit: implication requires the presence of pit if there is a breeze, biconditional also requires the absence of pit if there is no breeze)
![Page 29: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/29.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
29
Wumpus world sentences
Let Pi,j be true if there is a pit in [i, j].Let Bi,j be true if there is a breeze in [i, j].Knowledge base includes:
R1: ¬ P1,1 No pit in [1,1]R2: ¬B1,1 No breeze in [1.1]R3: B2,1 Breeze in [2,1]
• "Pits cause breezes in adjacent squares"R4: B1,1 ⇔ (P1,2 ∨ P2,1)R5: B2,1 ⇔ (P1,1 ∨ P2,2 ∨ P3,1)
KB = R1 ∧ R2 ∧ R3 ∧ R4 ∧ R5
![Page 30: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/30.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
30
Inference
• Decide whether KB╞ α• First method: enumerate the models and check that α is true in every
model in which KB is true• B1,1 B2,1, P1,1 , P1,2, P2,1, P3,1
• 7 symbols : 27 = 128 possible models
![Page 31: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/31.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
31
Truth tables for inferenceR1: ¬ P1,1
R2: ¬B1,1
R3: B2,1
R4: B1,1 ⇔ (P1,2 ∨ P2,1)
R5: B2,1 ⇔(P1,1 ∨ P2,2 ∨ P3,1)
KB = R1 ∧ R2 ∧ R3 ∧ R4 ∧ R5
α1 = ¬ P1,2
α2 = P2,2
α1 is true in all models that KB is true
α2 is true only in two models that KB is true, but false in the other one
![Page 32: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/32.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
32
Inference by enumeration
• Depth-first enumeration of all models is sound and complete
• For n symbols, time complexity is O(2n), space complexity is O(n)
![Page 33: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/33.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
33
Logical equivalence
• Two sentences are logically equivalent iff they are true in same models: • α ≡ ß iff α╞ β and β╞ α
![Page 34: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/34.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
34
Validity and satisfiabilityA sentence is valid if it is true in all models,
e.g., True, A ∨¬A, A ⇒ A, (A ∧ (A ⇒ B)) ⇒ B
Valid sentences are tautologiesEvery valid sentence is equivalent to True
Validity is connected to inference via the Deduction Theorem:KB ╞ α if and only if (KB ⇒ α) is validEvery valid implication sentence describes a legitimate inference
A sentence is satisfiable if it is true in some model e.g., A∨ B, C
If a sentence is true in a model m, then we say m satisfies the sentence, or a model of the sentence
A sentence is unsatisfiable if it is true in no modelse.g., A∧¬Aα is valid iff ¬ α is unsatisfiable, α is satisfiable iff ¬ α is not valid
Satisfiability is connected to inference via the following:KB ╞ α if and only if (KB ∧¬α) is unsatisfiable
![Page 35: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/35.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
35
Examples
![Page 36: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/36.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
36
Satisfiability
• Related to constraint satisfaction• Given a sentence S, try to find an interpretation i
where S is true• Analogous to finding an assignment of values to
variables such that the constraint hold• Example problem: scheduling nurses in a hospital
– Propositional variables represent for example that Nurse1 is working on Tuesday at 2
– Constraints on the schedule are represented using logical expressions over the variables
• Brute force method: enumerate all interpretations and check
![Page 37: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/37.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
37
Example Problem
![Page 38: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/38.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
38
Checking Interpretations
• Start by figuring out what set of interpretations make our original sentences true.
• Then, if G is true in all those interpretations, it must be OK to conclude it from the sentences we started out with (our knowledge base).
• In a universe with only three variables, there are 8 possible interpretations in total.
![Page 39: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/39.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
39
Checking Interpretations
• Only one of these interpretations makes all the sentences in our knowledge base true:
• S = true, H = true, G = true.
![Page 40: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/40.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
40
Checking Interpretations
• it's easy enough to check that G is true in that interpretation, so it seems like it must be reasonable to draw the conclusion that the lecture will be good.
![Page 41: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/41.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
41
Computing entailment
![Page 42: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/42.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
42
Entailment and Proof
![Page 43: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/43.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
43
Proof
![Page 44: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/44.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
44
Logical equivalence
![Page 45: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/45.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
45
Natural deduction
α⇔β--------------------(α⇒β) ∧ (β⇒α)(α⇒β) ∧ (β⇒α)-------------------- α⇔β
Biconditional Elimination
![Page 46: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/46.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
46
Example
![Page 47: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/47.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
47
Example
![Page 48: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/48.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
48
Example
![Page 49: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/49.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
49
Example
![Page 50: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/50.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
50
Example
![Page 51: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/51.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
51
Example
![Page 52: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/52.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
52
Example from Wumpus World
R1: ¬ P1,1
R2: ¬B1,1
R3: B2,1
R4: B1,1 ⇔ (P1,2 ∨ P2,1)
R5: B2,1 ⇔(P1,1 ∨ P2,2 ∨ P3,1)
KB = R1 ∧ R2 ∧ R3 ∧ R4 ∧ R5
Prove α1 = ¬ P1,2
![Page 53: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/53.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
53
Example from Wumpus World
R1: ¬ P1,1
R2: ¬B1,1
R3: B2,1
R4: B1,1 ⇔ (P1,2 ∨ P2,1)
R5: B2,1 ⇔(P1,1 ∨ P2,2 ∨ P3,1)
R6 : B1,1 ⇔ (B1,1 ⇒ (P1,2 ∨ P2,1)) ∧((P1,2 ∨ P2,1) ⇒ B1,1) Biconditional elimination
![Page 54: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/54.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
54
Example from Wumpus World
R1: ¬ P1,1
R2: ¬B1,1
R3: B2,1
R4: B1,1 ⇔ (P1,2 ∨ P2,1)
R5: B2,1 ⇔(P1,1 ∨ P2,2 ∨ P3,1)
R6 : B1,1 ⇔ (B1,1 ⇒ (P1,2 ∨ P2,1)) ∧((P1,2 ∨ P2,1) ⇒ B1,1) Biconditional elimination
R7 : ((P1,2 ∨ P2,1) ⇒ B1,1) And Elimination
![Page 55: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/55.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
55
Example from Wumpus World
R1: ¬ P1,1
R2: ¬B1,1
R3: B2,1
R4: B1,1 ⇔ (P1,2 ∨ P2,1)
R5: B2,1 ⇔(P1,1 ∨ P2,2 ∨ P3,1)
R6 : B1,1 ⇔ (B1,1 ⇒ (P1,2 ∨ P2,1)) ∧((P1,2 ∨ P2,1) ⇒ B1,1) Biconditional elimination
R7 : ((P1,2 ∨ P2,1) ⇒ B1,1) And Elimination
R8: (¬ B1,1 ⇒ ¬ (P1,2 ∨ P2,1)) Equivalence for contrapositives
![Page 56: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/56.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
56
Example from Wumpus World
R1: ¬ P1,1
R2: ¬B1,1
R3: B2,1
R4: B1,1 ⇔ (P1,2 ∨ P2,1)
R5: B2,1 ⇔(P1,1 ∨ P2,2 ∨ P3,1)
R6 : B1,1 ⇔ (B1,1 ⇒ (P1,2 ∨ P2,1)) ∧((P1,2 ∨ P2,1) ⇒ B1,1) Biconditional elimination
R7 : ((P1,2 ∨ P2,1) ⇒ B1,1) And Elimination
R8: (¬ B1,1 ⇒ ¬ (P1,2 ∨ P2,1)) Equivalence for contrapositives
R9: ¬ (P1,2 ∨ P2,1) Modus Ponens with R2 and R8
![Page 57: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/57.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
57
Example from Wumpus World
R1: ¬ P1,1
R2: ¬B1,1
R3: B2,1
R4: B1,1 ⇔ (P1,2 ∨ P2,1)
R5: B2,1 ⇔(P1,1 ∨ P2,2 ∨ P3,1)
R6 : B1,1 ⇔ (B1,1 ⇒ (P1,2 ∨ P2,1)) ∧((P1,2 ∨ P2,1) ⇒ B1,1) Biconditional elimination
R7 : ((P1,2 ∨ P2,1) ⇒ B1,1) And Elimination
R8: (¬ B1,1 ⇒ ¬ (P1,2 ∨ P2,1)) Equivalence for contrapositives
R9: ¬ (P1,2 ∨ P2,1) Modus Ponens with R2 and R8
R10: ¬ P1,2 ∧ ¬ P2,1 De Morgan’s Rule
![Page 58: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/58.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
58
Example from Wumpus World
R1: ¬ P1,1
R2: ¬B1,1
R3: B2,1
R4: B1,1 ⇔ (P1,2 ∨ P2,1)
R5: B2,1 ⇔(P1,1 ∨ P2,2 ∨ P3,1)
R6 : B1,1 ⇔ (B1,1 ⇒ (P1,2 ∨ P2,1)) ∧((P1,2 ∨ P2,1) ⇒ B1,1) Biconditional elimination
R7 : ((P1,2 ∨ P2,1) ⇒ B1,1) And Elimination
R8: (¬ B1,1 ⇒ ¬ (P1,2 ∨ P2,1)) Equivalence for contrapositives
R9: ¬ (P1,2 ∨ P2,1) Modus Ponens with R2 and R8
R10: ¬ P1,2 ∧ ¬ P2,1 De Morgan’s Rule
R11: ¬ P1,2 And Elimination
![Page 59: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/59.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
59
Monotonicity
• The set of entailed sentences can only increase as information is added to the knowledge base
• If • KB╞ α • Then • KB ∧ β╞ α
![Page 60: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/60.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
60
Proof systems
![Page 61: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/61.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
61
ResolutionR1: ¬ P1,1
R2: ¬B1,1
R3: B2,1
R4: B1,1 ⇔ (P1,2 ∨ P2,1)
R5: B2,1 ⇔ (P1,1 ∨ P2,2 ∨ P3,1)
….
R11: ¬B1,2
R12: B1,2 ⇔ (P1,1 ∨ P2,2 ∨ P1,3)
R13: ¬ P2,2
R14: ¬ P1,3
R15: (P1,1 ∨ P2,2 ∨ P3,1) biconditional elimination on R3, followed by a Modus Ponens with R5
R16: (P1,1 ∨ P3,1) Resolution with ¬ P2,2 in R13
If there is a pit in one of [1,1], [2,2] and [3,1] and it is not in [2,2] then it is in [1,1] or [3,1]
R17: P3,1 Resolve with ¬ P1,1 in R1
![Page 62: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/62.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
62
Resolution
![Page 63: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/63.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
63
Conjunctive Normal form
![Page 64: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/64.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
64
Converting to CNF
![Page 65: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/65.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
65
CNF Conversion Example
![Page 66: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/66.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
66
Resolution
![Page 67: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/67.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
67
Example
![Page 68: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/68.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
68
Example
![Page 69: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/69.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
69
Example
![Page 70: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/70.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
70
The power of false
![Page 71: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/71.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
71
Example
![Page 72: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/72.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
72
Example
![Page 73: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/73.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
73
Resolution
Conjunctive Normal Form (CNF) conjunction of disjunctions of literals
clausesE.g., (A ∨ ¬B) ∧ (B ∨ ¬C ∨ ¬D)
• Resolution inference rule (for CNF):li ∨… ∨ lk, m1 ∨ … ∨ mn
li ∨ … ∨ li-1 ∨ li+1 ∨ … ∨ lk ∨ m1 ∨ … ∨ mj-1 ∨ mj+1 ∨... ∨ mn
where li and mj are complementary literals. E.g., P1,3 ∨ P2,2, ¬P2,2
P1,3
• Resolution is sound and complete for propositional logic
![Page 74: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/74.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
74
Conversion to CNF
B1,1 ⇔ (P1,2 ∨ P2,1)
3. Eliminate ⇔, replacing α ⇔ β with (α ⇒ β)∧(β ⇒ α).(B1,1 ⇒ (P1,2 ∨ P2,1)) ∧ ((P1,2 ∨ P2,1) ⇒ B1,1)
2. Eliminate ⇒, replacing α ⇒ β with ¬α∨ β.(¬B1,1 ∨ P1,2 ∨ P2,1) ∧ (¬(P1,2 ∨ P2,1) ∨ B1,1)
3. Move ¬ inwards using de Morgan's rules and double-negation:(¬B1,1 ∨ P1,2 ∨ P2,1) ∧ ((¬P1,2 ∧ ¬P2,1) ∨ B1,1)
4. Apply distributivity law (∧ over ∨) and flatten:(¬B1,1 ∨ P1,2 ∨ P2,1) ∧ (¬P1,2 ∨ B1,1) ∧ (¬P2,1 ∨ B1,1)
![Page 75: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/75.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
75
Resolution algorithm
• Proof by contradiction, i.e., show KB∧¬α unsatisfiable
![Page 76: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/76.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
76
Resolution example
• KB = (B1,1 ⇔ (P1,2∨ P2,1)) ∧¬ B1,1 α = ¬P1,2
![Page 77: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/77.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
77
Forward and backward chaining
• Horn Form (restricted)KB = conjunction of Horn clauses
– Horn clause = • proposition symbol; or• (conjunction of symbols) ⇒ symbol
– E.g., C ∧ (B ⇒ A) ∧ (C ∧ D ⇒ B)• Modus Ponens (for Horn Form): complete for Horn KBs
α1, … ,αn, α1 ∧ … ∧ αn ⇒ ββ
• Can be used with forward chaining or backward chaining.• These algorithms are very natural and run in linear time
![Page 78: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/78.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
78
Forward chaining
• Idea: fire any rule whose premises are satisfied in the KB,– add its conclusion to the KB, until query is found
![Page 79: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/79.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
79
Forward chaining algorithm
• Forward chaining is sound and complete for Horn KB
![Page 80: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/80.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
80
Forward chaining example
![Page 81: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/81.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
81
Forward chaining example
![Page 82: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/82.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
82
Forward chaining example
![Page 83: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/83.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
83
Forward chaining example
![Page 84: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/84.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
84
Forward chaining example
![Page 85: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/85.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
85
Forward chaining example
![Page 86: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/86.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
86
Forward chaining example
![Page 87: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/87.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
87
Forward chaining example
![Page 88: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/88.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
88
Proof of completeness
• FC derives every atomic sentence that is entailed by KB1. FC reaches a fixed point where no new atomic sentences are
derived2. Consider the final state as a model m, assigning true/false to
symbols3. Every clause in the original KB is true in m
a1 ∧ … ∧ ak ⇒ b
4. Hence m is a model of KB5. If KB╞ q, q is true in every model of KB, including m
![Page 89: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/89.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
89
Backward chaining
Idea: work backwards from the query q:to prove q by BC,
check if q is known already, orprove by BC all premises of some rule concluding q
Avoid loops: check if new subgoal is already on the goal stack
Avoid repeated work: check if new subgoal1. has already been proved true, or2. has already failed
![Page 90: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/90.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
90
Backward chaining example
![Page 91: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/91.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
91
Backward chaining example
![Page 92: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/92.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
92
Backward chaining example
![Page 93: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/93.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
93
Backward chaining example
![Page 94: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/94.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
94
Backward chaining example
![Page 95: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/95.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
95
Backward chaining example
![Page 96: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/96.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
96
Backward chaining example
![Page 97: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/97.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
97
Backward chaining example
![Page 98: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/98.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
98
Backward chaining example
![Page 99: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/99.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
99
Backward chaining example
![Page 100: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/100.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
100
Forward vs. backward chaining
• FC is data-driven, automatic, unconscious processing,– e.g., object recognition, routine decisions
• May do lots of work that is irrelevant to the goal
• BC is goal-driven, appropriate for problem-solving,– e.g., Where are my keys? How do I get into a PhD program?
• Complexity of BC can be much less than linear in size of KB
![Page 101: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/101.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
101
Proof methods
• Proof methods divide into (roughly) two kinds:
– Application of inference rules• Legitimate (sound) generation of new sentences from old• Proof = a sequence of inference rule applications
Can use inference rules as operators in a standard search algorithm• Typically require transformation of sentences into a normal form
– Model checking• truth table enumeration (always exponential in n)• improved backtracking, e.g., Davis--Putnam-Logemann-Loveland (DPLL)• heuristic search in model space (sound but incomplete)
e.g., min-conflicts-like hill-climbing algorithms
![Page 102: Chapter 7 Logical Agents - Bilkent Universityduygulu/Courses/CS461/Notes/LogicalAgents.pdf · – Squares adjacent to wumpus are smelly (stench) – Squares adjacent to pit are breezy](https://reader031.fdocuments.us/reader031/viewer/2022020114/5be3a37809d3f281048c063e/html5/thumbnails/102.jpg)
CS461 Artificial Intelligence © Pinar Duygulu Spring 2008
102
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 wrt 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
• Wumpus world requires the ability to represent partial and negated information, reason by cases, etc.
• Resolution is complete for propositional logicForward, backward chaining are linear-time, complete for Horn clauses
• Propositional logic lacks expressive power