LOGIC Heng Ji Feb 19, 2016

Logic

Knowledge-based agents Knowledge base (KB) = set of sentences in a formal language 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 Distinction between data and program Fullest realization of this philosophy was in the field of expert systems or knowledge-based systems in the 1970s and 1980s Inference engine Knowledge base Domain-independent algorithms Domain-specific content

What is logic? Logic is a formal system for manipulating facts so that true conclusions may be drawn “The tool for distinguishing between the true and the false” – Averroes (12 th cen.) Syntax: rules for constructing valid sentences E.g., x + 2  y is a valid arithmetic sentence,  x2y + is not Semantics: “meaning” of sentences, or relationship between logical sentences and the real world Specifically, semantics defines truth of sentences E.g., x + 2  y is true in a world where x = 5 and y = 7

Overview Propositional logic Inference rules and theorem proving First order logic

Propositional logic: Syntax Atomic sentence: A proposition symbol representing a true or false statement Negation: If P is a sentence,  P is a sentence Conjunction: If P and Q are sentences, P  Q is a sentence Disjunction: If P and Q are sentences, P  Q is a sentence Implication: If P and Q are sentences, P  Q is a sentence Biconditional: If P and Q are sentences, P  Q is a sentence , , , ,  are called logical connectives

Propositional logic: Semantics A model specifies the true/false status of each proposition symbol in the knowledge base E.g., P is true, Q is true, R is false With three symbols, there are 8 possible models, and they can be enumerated exhaustively Rules for evaluating truth with respect to a model:  Pis true iff P is false P  Q is true iff P is true and Q is true P  Q is true iff P is true or Q is true P  Q is true iff P is false or Q is true P  Qis true iff P  Q is true and Q  P is true

Truth tables A truth table specifies the truth value of a composite sentence for each possible assignments of truth values to its atoms The truth value of a more complex sentence can be evaluated recursively or compositionally

Logical equivalence Two sentences are logically equivalent iff they are true in same models

Validity, satisfiability A sentence is valid if it is true in all models, e.g., True, A  A, A  A, (A  (A  B))  B A sentence is satisfiable if it is true in some model e.g., A  B, C A sentence is unsatisfiable if it is true in no models e.g., A  A

Entailment Entailment means that a sentence follows from the premises contained in the knowledge base: KB ╞ α Knowledge base KB entails sentence α if and only if α is true in all models where KB is true E.g., x = 0 entails x * y = 0 Can α be true when KB is false? KB ╞ α iff (KB  α) is valid KB ╞ α iff (KB  α) is unsatisfiable

Inference Logical inference: a procedure for generating sentences that follow from a knowledge base KB An inference procedure is sound if whenever it derives a sentence α, KB╞ α A sound inference procedure can derive only true sentences An inference procedure is complete if whenever KB╞ α, α can be derived by the procedure A complete inference procedure can derive every entailed sentence

Inference How can we check whether a sentence α is entailed by KB? How about we enumerate all possible models of the KB (truth assignments of all its symbols), and check that α is true in every model in which KB is true? Is this sound? Is this complete? Problem: if KB contains n symbols, the truth table will be of size 2 n Better idea: use inference rules, or sound procedures to generate new sentences or conclusions given the premises in the KB

Inference rules Modus Ponens And-elimination premises conclusion

Inference rules And-introduction Or-introduction

Inference rules Double negative elimination Unit resolution

Resolution Example:  : “The weather is dry”  : “The weather is rainy” γ : “I carry an umbrella” or

Resolution is complete To prove KB╞ α, assume KB   α and derive a contradiction Rewrite KB   α as a conjunction of clauses, or disjunctions of literals Conjunctive normal form (CNF) Keep applying resolution to clauses that contain complementary literals and adding resulting clauses to the list If there are no new clauses to be added, then KB does not entail α If two clauses resolve to form an empty clause, we have a contradiction and KB╞ α

Complexity of inference Propositional inference is co-NP-complete Complement of the SAT problem: α ╞ β if and only if the sentence α   β is unsatisfiable Every known inference algorithm has worst-case exponential running time Efficient inference possible for restricted cases

Definite clauses A definite clause is a disjunction with exactly one positive literal Equivalent to (P 1  …  P n )  Q Basis of logic programming (Prolog) Efficient (linear-time) complete inference through forward chaining and backward chaining premise or body conclusion or head

Forward chaining Idea: find any rule whose premises are satisfied in the KB, add its conclusion to the KB, and keep going until query is found

Forward chaining example

Backward chaining Idea: work backwards from the query q: to prove q by BC, check if q is known already, or prove by BC all premises of some rule concluding q

Backward chaining example

Forward vs. backward chaining Forward chaining is data-driven, automatic processing May do lots of work that is irrelevant to the goal Backward chaining is goal-driven, appropriate for problem-solving Complexity can be much less than linear in size of KB

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 Resolution is complete for propositional logic Forward, backward chaining are linear-time, complete for definite clauses

FIRST-ORDER LOGIC Chapter 8

Outline Why FOL? Syntax and semantics of FOL Using FOL Wumpus world in FOL Knowledge engineering in FOL

Pros and cons of propositional logic Propositional logic is declarative Propositional logic allows partial/disjunctive/negated information (unlike most data structures and databases) Propositional logic is compositional: meaning of B 1,1  P 1,2 is derived from meaning of B 1,1 and of P 1,2 Meaning in propositional logic is context-independent (unlike natural language, where meaning depends on context)  Propositional logic has very limited expressive power (unlike natural language) E.g., cannot say "pits cause breezes in adjacent squares“ except by writing one sentence for each square

First-order logic Whereas propositional logic assumes the world contains facts, first-order logic (like natural language) assumes the world contains Objects: people, houses, numbers, colors, baseball games, wars, … Relations: red, round, prime, brother of, bigger than, part of, comes between, … Functions: father of, best friend, one more than, plus, …

Syntax of FOL: Basic elements ConstantsKingJohn, 2, NUS,... PredicatesBrother, >,... FunctionsSqrt, LeftLegOf,... Variablesx, y, a, b,... Connectives , , , ,  Equality= Quantifiers , 

Atomic sentences Atomic sentence =predicate (term 1,...,term n ) or term 1 = term 2 Term =function (term 1,...,term n ) or constant or variable E.g., Brother(KingJohn,RichardTheLionheart) > (Length(LeftLegOf(Richard)), Length(LeftLegOf(KingJohn)))

Complex sentences Complex sentences are made from atomic sentences using connectives  S, S 1  S 2, S 1  S 2, S 1  S 2, S 1  S 2, E.g. Sibling(KingJohn,Richard)  Sibling(Richard,KingJohn) >(1,2)  ≤ (1,2) >(1,2)   >(1,2)

Truth in first-order logic Sentences are true with respect to a model and an interpretation Model contains objects (domain elements) and relations among them Interpretation specifies referents for constant symbols → objects predicate symbols → relations function symbols →functional relations An atomic sentence predicate(term 1,...,term n ) is true iff the objects referred to by term 1,...,term n are in the relation referred to by predicate

Models for FOL: Example

Universal quantification  Everyone at NUS is smart:  x At(x,NUS)  Smart(x)  x P is true in a model m iff P is true with x being each possible object in the model Roughly speaking, equivalent to the conjunction of instantiations of P At(KingJohn,NUS)  Smart(KingJohn)  At(Richard,NUS)  Smart(Richard)  At(NUS,NUS)  Smart(NUS) ...

A common mistake to avoid Typically,  is the main connective with  Common mistake: using  as the main connective with  :  x At(x,NUS)  Smart(x) means “Everyone is at NUS and everyone is smart”

Existential quantification  Someone at NUS is smart:  x At(x,NUS)  Smart(x)$  x P is true in a model m iff P is true with x being some possible object in the model Roughly speaking, equivalent to the disjunction of instantiations of P At(KingJohn,NUS)  Smart(KingJohn)  At(Richard,NUS)  Smart(Richard)  At(NUS,NUS)  Smart(NUS) ...

Another common mistake to avoid Typically,  is the main connective with  Common mistake: using  as the main connective with  :  x At(x,NUS)  Smart(x) is true if there is anyone who is not at NUS!

Properties of quantifiers  x  y is the same as  y  x  x  y is the same as  y  x  x  y is not the same as  y  x  x  y Loves(x,y) “There is a person who loves everyone in the world”  y  x Loves(x,y) “Everyone in the world is loved by at least one person” Quantifier duality: each can be expressed using the other  x Likes(x,IceCream)  x  Likes(x,IceCream)  x Likes(x,Broccoli)  x  Likes(x,Broccoli)

Equality term 1 = term 2 is true under a given interpretation if and only if term 1 and term 2 refer to the same object E.g., definition of Sibling in terms of Parent:  x,y Sibling(x,y)  [  (x = y)   m,f  (m = f)  Parent(m,x)  Parent(f,x)  Parent(m,y)  Parent(f,y)]

Using FOL The kinship domain: Brothers are siblings  x,y Brother(x,y)  Sibling(x,y) One's mother is one's female parent  m,c Mother(c) = m  (Female(m)  Parent(m,c)) “Sibling” is symmetric  x,y Sibling(x,y)  Sibling(y,x)

Using FOL The set domain:  s Set(s)  (s = {} )  (  x,s 2 Set(s 2 )  s = {x|s 2 })  x,s {x|s} = {}  x,s x  s  s = {x|s}  x,s x  s  [  y,s 2 } (s = {y|s 2 }  (x = y  x  s 2 ))]  s 1,s 2 s 1  s 2  (  x x  s 1  x  s 2 )  s 1,s 2 (s 1 = s 2 )  (s 1  s 2  s 2  s 1 )  x,s 1,s 2 x  (s 1  s 2 )  (x  s 1  x  s 2 )  x,s 1,s 2 x  (s 1  s 2 )  (x  s 1  x  s 2 )

Interacting with FOL KBs Suppose a wumpus-world agent is using an FOL KB and perceives a smell and a breeze (but no glitter) at t=5: Tell (KB,Percept([Smell,Breeze,None],5)) Ask (KB,  a BestAction(a,5)) I.e., does the KB entail some best action at t=5? Answer: Yes, {a/Shoot} ← substitution (binding list) Given a sentence S and a substitution σ, Sσ denotes the result of plugging σ into S; e.g., S = Smarter(x,y) σ = {x/Hillary,y/Bill} Sσ = Smarter(Hillary,Bill) Ask (KB,S) returns some/all σ such that KB╞ σ

Knowledge base for the wumpus world Perception  t,s,b Percept([s,b,Glitter],t)  Glitter(t) Reflex  t Glitter(t)  BestAction(Grab,t)

Deducing hidden properties  x,y,a,b Adjacent([x,y],[a,b])  [a,b]  {[x+1,y], [x-1,y],[x,y+1],[x,y-1]} Properties of squares:  s,t At(Agent,s,t)  Breeze(t)  Breezy(s) Squares are breezy near a pit: Diagnostic rule---infer cause from effect  s Breezy(s)  \Exi{r} Adjacent(r,s)  Pit(r)$ Causal rule---infer effect from cause  r Pit(r)  [  s Adjacent(r,s)  Breezy(s)$ ]

Knowledge engineering in FOL 1. Identify the task 2. Assemble the relevant knowledge 3. Decide on a vocabulary of predicates, functions, and constants 4. Encode general knowledge about the domain 5. Encode a description of the specific problem instance 6. Pose queries to the inference procedure and get answers 7. Debug the knowledge base

The electronic circuits domain One-bit full adder

The electronic circuits domain 1. Identify the task Does the circuit actually add properly? (circuit verification) 2. Assemble the relevant knowledge Composed of wires and gates; Types of gates (AND, OR, XOR, NOT) Irrelevant: size, shape, color, cost of gates 3. Decide on a vocabulary Alternatives: Type(X 1 ) = XOR Type(X 1, XOR) XOR(X 1 )

The electronic circuits domain 4. Encode general knowledge of the domain  t 1,t 2 Connected(t 1, t 2 )  Signal(t 1 ) = Signal(t 2 )  t Signal(t) = 1  Signal(t) = 0 1 ≠ 0  t 1,t 2 Connected(t 1, t 2 )  Connected(t 2, t 1 )  g Type(g) = OR  Signal(Out(1,g)) = 1   n Signal(In(n,g)) = 1  g Type(g) = AND  Signal(Out(1,g)) = 0   n Signal(In(n,g)) = 0  g Type(g) = XOR  Signal(Out(1,g)) = 1  Signal(In(1,g)) ≠ Signal(In(2,g))  g Type(g) = NOT  Signal(Out(1,g)) ≠ Signal(In(1,g))

The electronic circuits domain 5. Encode the specific problem instance Type(X 1 ) = XOR Type(X 2 ) = XOR Type(A 1 ) = AND Type(A 2 ) = AND Type(O 1 ) = OR Connected(Out(1,X 1 ),In(1,X 2 ))Connected(In(1,C 1 ),In(1,X 1 )) Connected(Out(1,X 1 ),In(2,A 2 ))Connected(In(1,C 1 ),In(1,A 1 )) Connected(Out(1,A 2 ),In(1,O 1 )) Connected(In(2,C 1 ),In(2,X 1 )) Connected(Out(1,A 1 ),In(2,O 1 )) Connected(In(2,C 1 ),In(2,A 1 )) Connected(Out(1,X 2 ),Out(1,C 1 )) Connected(In(3,C 1 ),In(2,X 2 )) Connected(Out(1,O 1 ),Out(2,C 1 )) Connected(In(3,C 1 ),In(1,A 2 ))

The electronic circuits domain 6. Pose queries to the inference procedure What are the possible sets of values of all the terminals for the adder circuit?  i 1,i 2,i 3,o 1,o 2 Signal(In(1,C_1)) = i 1  Signal(In(2,C 1 )) = i 2  Signal(In(3,C 1 )) = i 3  Signal(Out(1,C 1 )) = o 1  Signal(Out(2,C 1 )) = o 2 7. Debug the knowledge base May have omitted assertions like 1 ≠ 0

Summary First-order logic: objects and relations are semantic primitives syntax: constants, functions, predicates, equality, quantifiers Increased expressive power: sufficient to define wumpus world