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Logical agents
(Logické agenty)
Lubica Benuskova
Reading: AIMA 2nd ed., Chapter 7.1 – 7.5
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Intro 2 Artificial Intelligence: Lecture 02
Logic
• Logic (from the Ancient Greek: λογική, logike) consists of the systematic study of the forms of arguments.
• A valid argument is one where there is a logical support between
the assumptions of the argument and its conclusion.
• Logic has traditionally included the classification of arguments, the systematic exposition of the 'logical form' common to all valid
arguments, the study of inference, and the study of semantics.
• Logic has been studied in philosophy (Aristotle, 384-322 BC) and mathematics (since the mid-1800s), and recently logic has been
studied in computer science, linguistics, psychology, and other fields.
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Logical form
• The logical form of a an argument is the form obtained by abstracting from the subject matter of its content.
• Original argument
• All humans are mortal.
• Socrates is human.
• Therefore, Socrates is mortal.
• All w e have done in the Argument form is to put 'H' for 'human' and
'humans', 'M' for 'mortal', and 'S' for 'Socrates'; w hat results is the logical form of the original argument.
• The concept of logical form is central to logic. The validity of an
argument is determined by its logical form, not by its content.
• Argument logical form
• All H are M.
• S is H.
• Therefore, S is M.
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Subdivisions of logic
• Informal logic is the study of natural language arguments.
• Formal logic is the study of inference with purely formal content . No formal logic captures all of the nuances of natural language.
– Propositional logic (Výroková logika)
– Predicate logic (Predikátová logika)
– And others (e.g. Fuzzy logic)
• Inference possesses a purely formal content if it can be expressed as a
particular application of an abstract rule, that is, a rule that is not about any particular (concrete) thing or property.
• Inference is the act or process of deriving logical conclusions from premises known or assumed to be true, i.e. from the observed facts holding
in the world. 4
Znalostný agent: knowledge-based agent
Čo obsahuje znalostný agent?
1. Databázu poznatkov o svete : knowledge base (KB).
2. Mechanizmus usudzovania: the (empty) inference mechanism (IM).
KB, which is
the model of
the world
IM
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Working of the knowledge-based agent
Dodá do KB
aktuálny
vnem
Opýta sa KB
na možnú
akciu
Záznam
vybranej akcie
The function KB-AGENT(percept)
1) Tells KB, which fact/s is/are true through the function TELL.
2) Asks KB, which action to choose through the function ASK.
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2
Knowledge-Based Agents
• KB = know ledge base
– A set of facts (e.g. A, B, C) and – A set of statements in a logic language, e.g.: If A then (B OR C)
• IM = Inference module
– uses a set of logical statements to infer new ones
• Example of a simple model of reasoning:
– Agent is told or perceives new evidence • E.g., A is true and C is false
– Agent then infers new facts to be added to the KB
• E.g., KB = {if A then (B OR C) }, then given A and not C w e can infer that B is true
• B is now added to the KB even though it w as not explicitly observed, i.e., the agent inferred B is true.
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The Wumpus World: PEAS
• Performance measures:
• +1000 for picking up gold
• -1000 got falling into pit
• -1 for each move
• -10 for using arrow
• Environment
– Cave of 4×4 rooms
• Wumpus: A deadly beast who kills anyone entering his room.
• Pits: Bottomless pits that will
trap you forever.
• Gold
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„Wumpus world“ charakteristiky
• Plne pozorovateľné? Nie – vnem je len lokálny
• Deterministické? Áno
• Epizodické? Nie
• Statické Áno – obluda a priepasti sa nehýbu, sú to
charakteristiky prostredia, nie agenti
• Diskrétne? Áno
• Jednoagentové? Áno, obluda nie je agent, ale črta
prostredia
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The Wumpus World: PEAS
• Agents Sensors:
– Stench next to Wumpus – Breeze next to pit
– Glitter in square w ith gold – Bump w hen agent moves into
a w all – Scream from w umpus when
killed
• Agents actions – Agent can move forw ard, turn
left or turn right – Shoot, one shot
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• The goal is to explore environment, make inferences (reasoning) to try to find the gold.
Exploring the Wumpus World
The KB initially contains the rules of the environment; i.e. rooms next to the pit are breezy, the rooms next to the wumpus are smelly. There is only one wumpus, agent has only 1 arrow, etc.
The first percept is [1,1] = [stench=0; breeze=0, glitter=0; bump=0; scream=0],
THUS it is safe to move to cell [2,1] = OK or [1,2]=OK. Let’s try [2,1].
Exploring the Wumpus World
[2,1] = [stench=0; breeze=1, glitter=0; bump=0; scream=0],
Percept indicates that there is a pit in [2,2] = P? or [3,1] = P?,
Action: return to [1,1] to try the next safe cell [1,2]
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Exploring the Wumpus World
[1,2] Stench in the cell, which means that wumpus is either in [1,3] or [2,2] AND not in [2,2] = OK, because there was no stench in [2,1]
THUS Wumpus is not in [2,2] or stench would have been detected in [2,1] (this is relatively sophisticated reasoning!)
Action: move to cell [2,2] = OK
Exploring the Wumpus World
Percept [2,2] = [stench=0; breeze=0, glitter=0; bump=0; scream=0],
Thus it is safe to move either to [2,3] or [3,2]. Let’s move to [2,3]
Percept [2,3] = [stench=1; breeze=1, glitter=1; bump=0; scream=0],
THUS gold in [2,3] and pit in [3,3] or [2,4] and Wumpus in [1,3]
Exploring the Wumpus World
Percept [2,3] = [stench=1; breeze=1, glitter=1; bump=0; scream=0], THUS gold in [2,3] and pit in [3,3] or [2,4] and Wumpus in [1,3]
Action: pick up gold in [2,3] and then move to [1,3] to kill the Wumpus Percept [1,3] (after killing Wumpus) = [0,0,0,0,0]
THUS it is safe to move to [1,4] and from there continue to explore the
cave to find more gold, while avoiding the pits…
What the Wumpus example has shown us
• Logical agent can represent general know ledge about the environment by a set of rules and facts.
• Logical agent can gather evidence and then infer new facts by
combining evidence w ith the rules.
• The conclusions are guaranteed to be correct if
– The evidence and rules are correct and
– The inference procedure is correct (thanks to logic)
• The inference may be quite complex
– E.g., evidence at different times, combined w ith different rules, etc.
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Logical language
• The rules in the KB are expressed as propositions (sentences) according to the syntax of the representation language.
– SYNTAX specifies all the propositions that are w ell formed.
• in arithmetic: “x+y=4” is w ell formed w hile “+x4y=“ not
• A logic must also define the semantics of the language. Loosely speaking, SEMANTICS means the “meaning” of propositions.
• In logic, the definition is more precise. The TRUTH semantics of the language defines the truth of each sentence w ith respect to each possible world.
– For example, the usual semantics adopted for arithmetic specifies that the sentence “x + y =4” is true in a w orld w here x=2 and y=2, but false in a w orld w here x=1 and y=1.
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Possible worlds – models
• Ludw ig Wittgenstein (1922, Tractatus):
– The w orld is everything that is the case:
• The w orld is the complete collection of facts, not things.
• The w orld is determined by the facts, and by being the complete collection of facts.
• To be precise, w e w ill use the term model in place of “possible w orld.”
• Models are mathematical abstractions, each of which fixes the truth or falsehood of every relevant proposition/sentence.
– In standard logics, every sentence/proposition must be either true or false in each possible w orld—there is no “in betw een.
– In fuzzy logic, degrees of truth are allow ed.
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Propositional logic: Syntax
• Atomic sentences
– Are single proposition symbols, e.g., P, Q, R, and so on
– Plus tw o special symbols with fixed meaning: T = true, F = false • Complex sentences: If S, S1 and S2 are sentences, then
– S is a sentence negation
– S1 S2 is a sentence conjunction (its parts are called conjuncts)
– S1 S2 is a sentence disjunction (its parts are called disjuncts)
– S1 S2 is a sentence implication (also known as if-then statement) – S1 is a premise/antecedent; S2 is called conclusion / consequent
– S1 S2 is a sentence biconditional
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Backus-Naur Form (BNF)
• A BNF (Backus–Naur Form) of sentences in propositional logic.
• The syntax must be completely unambiguous (jednoznačná).
• Hence w e use the order of precedence (priority): , , , ,
• If not enough then use parentheses to disambiguate. 20
Propositional logic: Semantics
• The semantics defines the rules for determining the truth of a sentence w ith respect to a particular model (possible w orld).
• First, w e fix the truth value for every symbol (atomic sentence), e.g. P
= T, Q = F and R = T (there are 23=8 possible w orlds for 3 symbols).
• Next, the semantics for propositional logic must specify how to compute the truth value of any sentence, given a model.
• All sentences are constructed from atomic sentences and the five
connectives; therefore, w e need to specify how to compute the truth of sentences formed w ith each of the five connectives.
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Truth tables for connectives
• Implication is alw ays true w hen the premise is false. Why also w hen P is false and Q is true?
• Consider an example: “It rains, therefore I am a w oman.”
• Propositional logic does not require any relation of causation or relevance betw een P and Q. Thus, if I am a w oman (Q=T), then
(P => Q) = T, too. • There is an additional connective called “exclusive or” (“xor”
for short) that yields true only w hen a single disjunct is true. 22
An example of a simple knowledge base
• A simple KB for the cave that contains only pits. For each [i, j]:
– Let P[i, j ] = T if there is a pit in [i, j ]
– Let B[i,j ] = T if there is a breeze in [i, j]
• The initial KB includes the sentences (R1 R2 R3) :
– R1 : P[1, 1] // There is no pit in [1,1]
– R2 : B[1, 1] ( P[1, 2] P[2, 1] )
– R3 : B[2, 1] ( P[1, 1] P[2, 2] P[3, 1] )
• IMPORTANT: After the agent moves to [2,1], the KB expands to include additional sentences (KB = R1R2 R3 R4 R5) , where
– R4 : B[1, 1] // There is no breeze in [1,1]
– R5 : B[2, 1] // There is a breeze in [2,1]
• After each agent action, the KB expands to include new sentences!
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Inference: logické vyvodzovanie (entailment)
• Logical inference involves the relation of logical entailment between sentences—the idea that a sentence follows logically from another sentence. In mathematical notation, we write as
– a ╞ b
– To mean that sentence a entails the sentence b.
• The formal definition of entailment is this:
– a ╞ b if and only if, in every model, in which the sentence a is true, the sentence b is also true.
• Another way to say this is that if a is true then b must also be true, or the truth of b is “contained” in the truth of a.
• If an inference algorithm I can derive a from KB, we write: KB - Ia, which reads “ a is derived from KB by I”.
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Logical inference – example: cave with pits
• Consider the situation the agent has detected nothing in [1,1] and a
breeze in [2,1], and the KB = R1R2 R3 R4 R5.
• The agent w ants to infer w hether there is a pit in [1,2], [2,2] and [3,1]
• First, w e generate all the possible w orlds, i.e. models.
• There are 23 = 8 possible models.
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Model checking: 1st part
• The KB is false in those models that contradict w hat the agent knows, e.g. the KB is false in any model in w hich [1,2] contains a pit, because
there is no breeze in [1,1].
• Thus out of total 8 possible models, the KB is true only in 3 of them.
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Logical inference from KB
• Let’s consider the first conclusion, i.e. that α1 = "[1,2] is safe"
• We can see, in every model, in which KB is true, a1 is also true.
• Thus KB ╞ α1, i.e. a1 is derived from the KB. KB entails a1.
[1,2]
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Logical inference from KB
[2,2]
• Let’s consider the 2nd conclusion, i.e. that α2 = "[2,2] is safe"
• We can see, in some models, in which KB is true, a2 is false.
• Thus KB ╞ α1, i.e. a2 is not derived from the KB.
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Logical inference by model checking
• The definition of entailment can be applied to derive conclusions
from the KB, i.e. by entailment w e carry out the logical inference.
• The w hole procedure is called model checking:
– First w e generate all possible models and check, in w hich of them
the KB is true.
– Next, for each conclusion a, w e check w hether a is true in all
models, in w hich the KB is true.
• In deriving the conclusions, w e have watch out for logical
equivalence, validity and satisfiability of derived sentences
(ekvivalencia, platnosť a splniteľnosť).
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Equivalence: standard logical equivalencies
• Logical equivalence: tw o sentences a and b are logically equivalent if
they are true in the same set of models: a b.
• Alternative definition: a b, if and only if a ╞ b and b╞ a.
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Validity (platnosť) and tautology
• A sentence (proposition) is logically valid if it is true in all models.
• We call sentences / propositions that are valid, TAUTOLOGIES.
• The concept of validity is crucial for the deduction theorem:
For any sentences a and b; a entails b, i.e. a ╞ b , if and only if the implication (a b) is valid.
• Thus, w e can think of the model checking inference as checking the validity of KB a, i.e KB entails a.
• In turn, every valid implication describes a legitimate inference.
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Tautológia: formula pravdivá v ľubovoľnej interpretácii
Príklad:
PQP
P Q PQP PQ
True
True
True
True
True
True
True
True
True
True
True
False
False False
False
False
Tautology: an example
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Satisfiability (splniteľnosť)
• A sentence is satisfiable if it is true in some model.
– For example, the KB given earlier, (R1 R2 R3 R4 R5), is satisfiable because there are 3 models, in w hich it is true.
• If a sentence a is true in a model m, then w e say that m satisfies a, or
that m is a model of a.
• Satisfiability can be checked by enumerating the possible models until one is found that satisfies the sentence.
• Validity and satisfiability are related:
– a is valid if a is unsatisfiable
– a is satisfiable if a is not valid 33
Proof by contradiction (refutation): dôkaz sporom
• Theorem:
– a ╞ b if and only if the sentence (a b) is unsatisfiable.
• Proving b from a by checking the unsatisfiability of (a b) corresponds to the standard mathematical proof technique of
reductio ad absurdum (i.e. reduction to an absurd thing).
• This kind of proof is also called refutation or proof by contradiction.
• One assumes a proposition b to be false and show s that this leads to a contradiction w ith the know n axiom(s) a. This contradiction is exactly
w hat is meant by saying that (a b) is unsatisfiable.
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Rules of logical inference (logical reasoning)
• At last, w e w ill cover patterns of inference that the inference mechanism (IM) applies to derive conclusions that lead to the desired goal.
• These patterns of inference are called INFERENCE RULES.
• An IM that derives only sentences entailed by the KB is called sound or
truth-preserving (korektný algoritmus).
• An IM is complete (kompletný algoritmus) if it can derive any sentence that is entailed by the KB.
Ak je KB pravdivá v reálnom svete, potom každá veta
odvodená z KB korektnou vyvodzovacou procedúrou je tiež
pravdivá v reálnom svete. 35
Modus Ponens
• This means that w henever a sentence of the form a and a b are
given, then the sentence b can be inferred.
• For example if Wumpus and Wumpus Shoot are given, then Shoot
can be inferred.
b
baa ,
Shoot
ShootWumpusWumpus ,
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Modus Tollens
• This means that w henever a sentence of the form b and a b are
given, then the sentence a can be inferred.
• For example if Glitter = False and Gold Glitter are given, then
Gold = F can be inferred.
a
bab
,
FGold
GoldGlitterFGlitter
,
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And-Elimination
• Simplifikácia (pravidlo odstránenia konjunkcie): Z konjunkcie sa dá odvodiť ľubovoľný z jej konjunktov. Ak celá konjunkcia je pravdivá,
všetky konjunkty musia byť pravdivé, i.e.
• Pravidlo odstránenia dvojitej negácie: Keď nejaký výrok dvakrát znegujeme, dostaneme ten samý výrok, i.e.
i
n
a
aaa ...21
a
a
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• Pravidlo vovedenia konjunkcie: Umožní z viacerých formúl odvodiť ich konjunkciu. Ak sú pravdivé, aj ich
konjunkcia je pravdivá.
• Adícia (pravidlo vovedenia dizjunkcie): Z jednej formuly odvodí jej dizjunkciu s hocijakými formulami. Ak je formula
pravdivá, jej dizjunkcia s čímkoľvek je pravdivá.
n
n
aaa
aaa
...
,....,,
21
21
ni
i
aaaaa
a
.....321
naaa ,....,, 21
Vovedenie konjunkcie a dizjunkcie
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Example of inference in the cave with pits
• The preceding derivation—a sequence of applications of inference rules—is called a PROOF.
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Vyvodzovanie vo výrokovej logike – iný príklad
4
3
2
1
predpokladTS
predpokladSR
predpokladPR
predpokladQP
T
S
R
Q
P
1
2
3 4
5
6
7 8
9
simplifikácia predpokladu 1
simplifikácia predpokladu 1
medzivýsledok 5 a modus tollens na predpoklad 2
medzivýsledok 7 a modus ponens na predpoklad 3
medzivýsledok 8 a modus ponens na predpoklad 4
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Resolution (rezolvencia)
• RESOLUTION is a single inference rule, that yields a complete inference algorithm w hen coupled w ith any complete search
algorithm.
• REZOLVENCIA je špeciálny prípad modus ponens:
• In order to generalize, let’s introduce this notation:
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b
baa
b
baa
,,
mlwherel
llm
l
llm
1
2
21
2
21 ,,
8
kii
k
llll
lllm
- ......
....,
111
21
njjkii
kn
mmmmllll
lllmm
-- ............
....,...
111111
211
Resolution inference rule
• Unit resolution (jednotková rezolvencia), li = m
• Full resolution (plná rezolvencia ), li = mj
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• Resolution for 3 propositions: takes tw o clauses and produces a new clause containing all the literals of the tw o original clauses except the
tw o complementary literals, i.e.:
• Note: a literal is the atomic proposition and clause (klauzula) is a disjunction of literals.
• What about conjunctions? Every sentence of propositional logic is
logically equivalent to a conjunction of disjunctions of literals.
• A sentence expressed as a conjunction of disjunctions of literals is said to be in conjunctive normal form or CNF.
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3221 ,
ll
llll
Conjunctive normal form (CNF)
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vstup : formula A , výstup CNF A
algoritmus:
1. Podformule nahradiť
2. nahradiť
3. Urobiť úpravy negácií:
Algoritmus CNF
YX XYYX
YX YX
YX
YX
X
YX
YX
X
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4. Uplatniť distributivitu voči :
5. Odstrániť redundancie napr. nahradiť X.
ZYX
ZYX
ZYZX
ZXYX
)(
XX
A resolution algorithm
• Inference by resolution w orks by using the proof by contradiction.
• That is, to show that KB a, i.e KB entails a, w e show that (KB a) is not satisfiable (i.e. not true in any model).
• First, (KB a) is converted into CNF.
• Then the resolution is applied to the resulting clauses.
• Each pair that contains complementary literals is resolved to produce a
new clause, w hich is added to the set if it is not already present. The process continues until one of tw o things happens:
– there are no new clauses that can be added
– resolution derives an empty clause, i.e. a disjunction of no disjuncts.
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A resolution algorithm
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Conclusion
• In propositional logic, any complete search algorithm, applying only the resolution rule, can derive any conclusion entailed by any KB.
• Vo výrokovej logike, ktorýkoľvek kompletný algoritmus prehľadávania,
ktorý používa len rezolvenčné pravidlo, dokáže odvodiť ktorúkoľvek
vetu vyplývajúcu z ktorejkoľvek bázy znalostí.
• Logical reasoning thus ensures that the new clauses represent aspects of the w orld that actually follow from the aspects that the old clauses
represent.
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