Logical Agents Philipp Koehn 5 March 2020 Philipp Koehn - PowerPoint PPT Presentation

Logical Agents Philipp Koehn 5 March 2020 Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

1 The world is everything that is the case. Wittgenstein, Tractatus Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Outline 2 ● Knowledge-based agents ● Logic in general—models and entailment ● Propositional (Boolean) logic ● Equivalence, validity, satisfiability ● Inference rules and theorem proving – forward chaining – backward chaining – resolution Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

3 knowledge-based agents Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Knowledge-Based Agent 4 ● Knowledge base = set of sentences in a formal language ● Declarative approach to building an agent (or other system): T ELL it what it needs to know ● Then it can A SK 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 Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

A Simple Knowledge-Based Agent 5 function KB-A GENT ( percept ) returns an action static : KB , a knowledge base t , a counter, initially 0, indicating time T ELL ( KB , M AKE -P ERCEPT -S ENTENCE ( percept , t )) action ← A SK ( KB , M AKE -A CTION -Q UERY ( t )) T ELL ( KB , M AKE -A CTION -S ENTENCE ( action , t )) t ← t + 1 return action ● 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 Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

6 example Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Hunt the Wumpus 7 Computer game from 1972 Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Wumpus World PEAS Description 8 ● 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 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 ● Actuators Left turn, Right turn, Forward, Grab, Release, Shoot ● Sensors Breeze, Glitter, Smell Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Wumpus World Characterization 9 ● 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 Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Exploring a Wumpus World 10 Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Exploring a Wumpus World 11 Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Exploring a Wumpus World 12 Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Exploring a Wumpus World 13 Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Exploring a Wumpus World 14 Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Exploring a Wumpus World 15 Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Exploring a Wumpus World 16 Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Exploring a Wumpus World 17 Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Tight Spot 18 ● Breeze in (1,2) and (2,1) � ⇒ no safe actions ● Assuming pits uniformly distributed, (2,2) has pit w/ prob 0.86, vs. 0.31 Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Tight Spot 19 ● Smell in (1,1) � ⇒ cannot move ● Can use a strategy of coercion: shoot straight ahead – wumpus was there � ⇒ dead � ⇒ safe – wumpus wasn’t there � ⇒ safe Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

20 logic in general Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Logic in General 21 ● 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; x 2 + 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 Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Entailment 22 ● 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 Ravens won” and “the Jays won” entails “the Ravens won or the Jays won” ● E.g., x + y = 4 entails 4 = x + y ● Entailment is a relationship between sentences (i.e., syntax ) that is based on semantics ● Note: brains process syntax (of some sort) Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Models 23 ● 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 ⇒ KB ⊧ α if and only if M ( KB ) ⊆ M ( α ) ● E.g. KB = Ravens won and Jays won α = Ravens won Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Entailment in the Wumpus World 24 ● Situation after detecting nothing in [1,1], moving right, breeze in [2,1] ● Consider possible models for all ? , assuming only pits ● 3 Boolean choices � ⇒ 8 possible models Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Possible Wumpus Models 25 Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Valid Wumpus Models 26 KB = wumpus-world rules + observations Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Entailment 27 KB = wumpus-world rules + observations α 1 = “[1,2] is safe”, KB ⊧ α 1 , proved by model checking Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Valid Wumpus Models 28 KB = wumpus-world rules + observations Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Not Entailed 29 KB = wumpus-world rules + observations α 2 = “[2,2] is safe”, KB / ⊧ α 2 Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Inference 30 ● KB ⊢ i α = sentence α can be derived from KB by procedure i ● Consequences of KB are a haystack; α is a needle. Entailment = needle in haystack; inference = finding it ● 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 . Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

31 propositional logic Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Propositional Logic: Syntax 32 ● Propositional logic is the simplest logic—illustrates basic ideas ● The proposition symbols P 1 , P 2 etc are sentences ● If P is a sentence, ¬ P is a sentence (negation) ● If P 1 and P 2 are sentences, P 1 ∧ P 2 is a sentence (conjunction) ● If P 1 and P 2 are sentences, P 1 ∨ P 2 is a sentence (disjunction) ● If P 1 and P 2 are sentences, P 1 � ⇒ P 2 is a sentence (implication) ● If P 1 and P 2 are sentences, P 1 ⇔ P 2 is a sentence (biconditional) Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020

Propositional Logic: Semantics 33 ● Each model specifies true/false for each proposition symbol E.g. P 1 , 2 P 2 , 2 P 3 , 1 false true false (with these symbols, 8 possible models, can be enumerated automatically) ● Rules for evaluating truth with respect to a model m : ¬ P is true iff P is false P 1 ∧ P 2 is true iff is true and is true P 1 P 2 P 1 ∨ P 2 is true iff P 1 is true or P 2 is true P 1 � ⇒ P 2 is true iff P 1 is false or P 2 is true is true and i.e., is false iff P 1 P 2 is false P 1 ⇔ P 2 P 1 � ⇒ P 2 is true and P 2 � ⇒ P 1 is true iff is true ● Simple recursive process evaluates an arbitrary sentence, e.g., ¬ P 1 , 2 ∧ ( P 2 , 2 ∨ P 3 , 1 ) = true ∧ ( false ∨ true )= true ∧ true = true Philipp Koehn Artificial Intelligence: Logical Agents 5 March 2020