Logical Agents CE417: Introduction to Artificial Intelligence - PowerPoint PPT Presentation

Logical Agents CE417: Introduction to Artificial Intelligence Sharif University of Technology Spring 2017 Soleymani “ Artificial Intelligence: A Modern Approach ” , 3 rd Edition, Chapter 7

Knowledge-based agents  Knowledge-based agents  Reasoning operates on internal representation of knowledge  Logic as a general class of representation  Propositional logic  First-order logic 2

A generic knowledge-based agent function KB_AGENT(𝑞𝑓𝑠𝑑𝑓𝑞𝑢) returns an 𝑏𝑑𝑢𝑗𝑝𝑜 persistent : 𝐿𝐶 , a knowledge base 𝑢, 𝑏 counter for time, initially 0 TELL(𝐿𝐶, MAKE_PERCEPT_SENTENCE(𝑞𝑓𝑠𝑑𝑓𝑞𝑢, 𝑢)) 𝑏𝑑𝑢𝑗𝑝𝑜 ← ASK(𝐿𝐶, MAKE_ACTION_QUERY(𝑢)) TELL(𝐿𝐶, MAKE_ACTION_SENTENCE(𝑏𝑑𝑢𝑗𝑝𝑜, 𝑢)) 𝑢 ← 𝑢 + 1 • Makes a sentence asserting that the agent return 𝑏𝑑𝑢𝑗𝑝𝑜 perceived the given percept at the given time. • Makes a sentence that asks what action should be done at the current time.  The agent must be able to: • Makes a sentence asserting that the chosen  Represent states, actions, percepts, … . action was executed.  Incorporate new percepts  Update internal representations of the world  Deduce hidden properties of the world  Deduce appropriate actions 3

Knowledge Base (KB)  KB = a set of sentences expressed in a knowledge representation language  TELL : adds new sentences to the knowledge base  ASK : asks a question of KB  the answer follows from previously TELL ed sentences to the KB  Inference: derives new sentences from old ones  Basis of TELL and ASK operations 4

Wumpus world  Wumpus  Pitts  Gold Wumpus  Agent 5

Wumpus world PEAS description  Performance measure  +1000 for garbing gold  -1000 for death  -1 for each action  -10 for using up the arrow  Game ends when the agent dies or climbs out of the cave  Environment  4×4 grid  Agent starts in [1,1] while facing to the right  Gold and wumpus are located randomly in the squares except to [1,1]  Each square other than [1,1] can be a pit with probability 0.2 6

Wumpus world PEAS description  Sensors: Stench, Breeze, Glitter, Bump, Scream  In the squares adjacent to wumpus, agent perceives a Stench  It the squares adjacent to a pit, agent perceives a Breeze  In the gold square, agent perceives a Glitter  When walking into a wall, agent perceives a Bump  When Wumpus is killed, agent perceives a Scream  Actuators: Forward, TurnLeft, TurnRight, Shoot, Grab, Climb  Forward,TurnLeft,TurnRight: moving and rotating face actions.  Moving to a square containing a pit or a live wumpus causes death.  If an agent tries to move forward and bumps into a wall then it does not move.  Shoot: to fire an arrow in a straight line in the facing direction of the agent  Shooting kills wumpus if the agent is facing it (o.w. the arrow hits a wall) The first shoot action has any effect (the agent has only one arrow)   Grab: to pick up the gold if it is in the same square as the agent .  Climb: climb out of the cave but only from [1,1] 7

Wumpus world characterization  Fully Observable? No – many aspects are not directly perceivable  Episodic? No – sequential at the level of actions  Static? Yes  Discrete?Yes  Single-agent?Yes 8

Wumpus world example A: agent OK: safe square 9

Wumpus world example A: agent A: agent OK: safe square OK: safe square B: Breeze 10

Wumpus world example A: agent OK: safe square B: Breeze P: Pit 11

Wumpus world example A: agent OK: safe square B: Breeze P: Pit S: Stench 12

Wumpus world example A: agent OK: safe square B: Breeze P: Pit S: Stench 13

Wumpus world example A: agent OK: safe square B: Breeze P: Pit S: Stench 14

Wumpus world example A: agent OK: safe square B: Breeze P: Pit S: Stench 15

Wumpus world example A: agent OK: safe square B: Breeze P: Pit S: Stench 16

Logic & Language  Logic includes formal languages for representing information such that conclusions can be drawn  Syntax defines the well-formed sentences in a language  Semantics define the "meaning" of sentences  i.e., define truth of a sentence with respect to each possible world  We will introduce Propositional Logic in this lecture  It talks about facts  Propositions can be true, false, or unknown 17

Propositional logic: Syntax  Propositional logic is the simplest logic  To illustrate basic ideas about logic & reasoning  The proposition symbols 𝑄 , 𝑅 , … are sentences.  If 𝑄 is a sentence,  𝑄 is a sentence (negation)  If 𝑄 and 𝑅 are sentences, 𝑄  𝑅 is a sentence (conjunction)  If 𝑄 and 𝑅 are sentences, 𝑄  𝑅 is a sentence (disjunction)  If 𝑄 and 𝑅 are sentences, 𝑄  𝑅 is a sentence (implication)  If 𝑄 and 𝑅 are sentences, 𝑄  𝑅 is a sentence (biconditional) 18

Propositional logic (Grammar) 𝑇𝑓𝑜𝑢𝑓𝑜𝑑𝑓 → 𝐵𝑢𝑝𝑛𝑗𝑑𝑇𝑓𝑜𝑢𝑓𝑜𝑑𝑓 | 𝐷𝑝𝑛𝑞𝑚𝑓𝑦𝑇𝑓𝑜𝑢𝑓𝑜𝑑𝑓 𝐵𝑢𝑝𝑛𝑗𝑑𝑇𝑓𝑜𝑢𝑓𝑜𝑑𝑓 → 𝑈𝑠𝑣𝑓 𝐺𝑏𝑚𝑡𝑓 𝑄 𝑅 𝑆 | … 𝐷𝑝𝑛𝑞𝑚𝑓𝑦𝑇𝑓𝑜𝑢𝑓𝑜𝑑𝑓 → (𝑇𝑓𝑜𝑢𝑓𝑜𝑑𝑓) | ¬𝑇𝑓𝑜𝑢𝑓𝑜𝑑𝑓 | 𝑇𝑓𝑜𝑢𝑓𝑜𝑑𝑓 ∧ 𝑇𝑓𝑜𝑢𝑓𝑜𝑑𝑓 | 𝑇𝑓𝑜𝑢𝑓𝑜𝑑𝑓 ∨ 𝑇𝑓𝑜𝑢𝑓𝑜𝑑𝑓 | 𝑇𝑓𝑜𝑢𝑓𝑜𝑑𝑓 ⟹ 𝑇𝑓𝑜𝑢𝑓𝑜𝑑𝑓 | 𝑇𝑓𝑜𝑢𝑓𝑜𝑑𝑓 ⇔ 𝑇𝑓𝑜𝑢𝑓𝑜𝑑𝑓 𝑄𝑠𝑓𝑑𝑓𝑒𝑓𝑜𝑑𝑓: ¬ , ∧ , ∨ , ⟹ , ⇔ 19

Truth tables for connectives 20

Propositional logic: Semantics  Each model specifies true/false of each proposition symbol  e .g., Propistion symbols: 𝑄 1,2 , 𝑄 2,2 , 𝑄 3,1  8 possible models  𝑛 1 = {𝑄 1,2 = 𝑔𝑏𝑚𝑡𝑓, 𝑄 2,2 = 𝑔𝑏𝑚𝑡𝑓, 𝑄 3,1 = 𝑢𝑠𝑣𝑓}  Semantics of a complex sentence in any model m :  𝑄 is true iff 𝑄 is false in 𝑛 .  𝑄  𝑅 is true iff 𝑄 is true and 𝑅 is true in 𝑛 .  𝑄 ∨ 𝑅 is true iff 𝑄 is true or 𝑅 is true in 𝑛 .  𝑄 ⟹ 𝑅 is true unless 𝑄 is true and 𝑅 is false in 𝑛 .  𝑄 ⟺ 𝑅 is true iff 𝑄 and 𝑅 are both true or both false in 𝑛 .  21

Wumpus world sentences  𝑄 𝑗,𝑘 is true if there is a pit in [𝑗, 𝑘] . 𝑗,𝑘 is true if there is a wumpus in [𝑗, 𝑘] , dead or alive.  𝑋  𝐶 𝑗,𝑘 is true if the agent perceives a breeze in [𝑗, 𝑘] .  𝑇 𝑗,𝑘 is true if the agent perceives a stench in [𝑗, 𝑘] .  General rules (only related ones to the current agent position)  𝑆 1 :  𝑄 (no pit in [1,1]) 1,1 (Pits cause breezes in adjacent squares)  𝑆 2 : 𝐶 1,1 ⇔ 𝑄 1,2 ∨ 𝑄 2,1 (Pits cause breezes in adjacent squares)  𝑆 3 : 𝐶 2,1 ⇔ 𝑄 1,1 ∨ 𝑄 2,2 ∨ 𝑄 3,1  Perception  𝑆 4 :  𝐶 1,1  𝑆 5 : 𝐶 2,1 22

Models  Models are mathematical abstraction of possible worlds  For each logical sentence, we can consider a set that specifies all possible worlds in which 𝛽 is true  𝑁(𝛽) : set of all models of the sentence 𝛽 (all satisfying 𝛽 )  𝑛 ∈ 𝑁(𝛽) if 𝛽 is true in model 𝑛  E.g., For 𝛽 : 𝑦 + 𝑧 = 4 , all possible assignments of values to 𝑦 , 𝑧 are models. 𝑁(𝛽) contains a subset of models satisfying 𝑦 + 𝑧 = 4. 23

Wumpus models  Agents starts at [1,1] with no active sensor, then moves to [2,1] and senses Breeze in it  Possible models for pits in [1,2], [2,2], [3,1]: 24

Logical reasoning: entailment  𝛽 ⊨ 𝛾 (entailment): 𝛽 entails 𝛾  𝛽 ⇒ 𝛾 is a tautology or valid  A sentence is valid or tautology if it is T𝑠𝑣𝑓 in all models ( e.g., 𝑄  𝑄 , (𝑄  (𝑄  𝑅))  𝑅 )  𝛾 logically follows from 𝛽 or 𝛽 is stronger than 𝛾  𝛽 ⊨ 𝛾 iff 𝑁 𝛽  𝑁 𝛾  𝛽 entails 𝛾 iff 𝛾 is true in all worlds where 𝛽 is true 25

Entailment iff  𝐿𝐶 ⊨ 𝛽 𝑁(𝐿𝐶) ⊆ 𝑁(𝛽)  Example:  KB = “ A is red ” & “ B is blue ”  α = “ A is red ” 26

Entailment in the wumpus world (before perception): rules of the  𝐿𝐶 wumpus world  Perception: After detecting nothing in [1,1], moving right, a breeze in [1,2] 27

Wumpus models  Possible models for pits in [1,2], [2,2], [3,1] Consider possible models for only pits in neighboring squares 2 3 = 8 possible models 28

Wumpus models  𝐿𝐶 = wumpus-world rules + perceptions 29

Wumpus models  𝐿𝐶 = wumpus-world rules + perceptions  𝛽 1 = "[1,2] is safe"  𝑁(𝐿𝐶) ⊆ 𝑁(𝛽 1 ) ⇒ 𝐿𝐶 ⊨ 𝛽 1 30

Wumpus models  KB = wumpus-world rules + perceptions 31

Wumpus models  𝐿𝐶 = wumpus-world rules + perceptions  𝛽 2 = " [2,2] is safe", 𝐿𝐶 ⊭ 𝛽 2 32