Logic: The Big Picture Propositional logic: atomic statements are - - PowerPoint PPT Presentation

Logic: The Big Picture

• Propositional logic: atomic statements are facts

– Inference via resolution is sound and complete Inference via resolution is sound and complete (though likely computationally intractable)

• First-order logic: adds variables, relations, and

quantification

– Inference is essentially a generalization of propositional i f inference – Resolution is still sound and complete, but not guaranteed to terminate on non-entailed sentences guaranteed to terminate on non entailed sentences (semidecidable) – Simple inference procedures (forward chaining and b k d h i i ) il bl f k l d b backward chanining) available for knowledge bases consisting of definite clauses

Logic programming: Prolog

• FOL:

King(x) ∧ Greedy(x) ⇒ Evil(x) Greedy(y) King(John)

• Prolog:

g

evil(X) :- king(X), greedy(X). greedy(Y). king(john). g(j )

• Closed-world assumption:

Every constant refers to a unique object – Every constant refers to a unique object – Atomic sentences not in the database are assumed to be false

• Inference by backward chaining, clauses are tried in the
• rder in which they are listed in the program, and literals

(predicates) are tried from left to right

Prolog example Prolog example

parent(abraham,ishmael). parent(abraham,isaac). parent(isaac,esau). parent(isaac,jacob) parent(isaac,jacob). grandparent(X,Y) :- parent(X,Z), parent(Z,Y). descendant(X Y) : parent(Y X) descendant(X,Y) :- parent(Y,X). descendant(X,Y) :- parent(Z,X), descendant(Z,Y). ? parent(david,solomon). ? parent(abraham,X). ? grandparent(X,Y). g p ? descendant(X,abraham).

Prolog example Prolog example

parent(abraham,ishmael). parent(abraham,isaac). parent(isaac,esau). parent(isaac,jacob) parent(isaac,jacob).

• What if we wrote the definition of descendant like this:

descendant(X Y) : descendant(Z Y) parent(Z X) descendant(X,Y) :- descendant(Z,Y), parent(Z,X). descendant(X,Y) :- parent(Y,X). ? descendant(W,abraham).

• Backward chaining would go into an infinite loop!

g g p – Prolog inference is not complete, so the ordering of the clauses and the literals is really important

Backward chaining algorithm Backward chaining algorithm

Graph coloring Graph coloring

colorable(Wa,Nt,Sa,Q,Nsw,V) :- diff(Wa,Nt), diff(Wa,Sa), diff(Nt,Q), diff(Nt,Sa), diff(Q,Nsw), diff(Q,Sa), diff(Nsw,V), diff(Nsw,Sa), diff(V,Sa). diff(red,blue). diff(red,green). diff(green,red). diff(green,blue). diff(blue,red). diff(blue,green).

Prolog lists Prolog lists

• Appending two lists to produce a third:

append([],Y,Y). append([X|L],Y,[X|Z]) :- append(L,Y,Z).

• query:

append(A,B,[1,2]) [] [1 2]

• answers:

A=[] B=[1,2] A=[1] B=[2] A=[1,2] B=[]

Logic: The Big Picture

• The original goal of formal logic was to axiomatize mathematics

– Hilbert’s program (1920’s): find a formalization of mathematics that i i t t l t d d id bl is consistent, complete, and decidable

• Completeness theorem (Gödel, 1929):

– Deduction in FOL is consistent and complete – Unfortunately, FOL is not strong enough to describe infinite structures such as natural or real numbers

• Incompleteness theorem (Gödel, 1931):

p ( ) – Any consistent logic system strong enough to capture natural numbers and arithmetic will contain true sentences that cannot be proved p

• Halting problem (Turing, 1936):

– There cannot be a general algorithm for deciding whether a given statement about natural numbers is true statement about natural numbers is true

• Profound implications for foundations of mathematics

– What about implications for AI?

Applications of logic Applications of logic

• Automated theorem proving in mathematics

Automated theorem proving in mathematics

– Robbins conjecture proved in 1996

• Software verification

Software verification

• Software synthesis
• VLSI verification

VLSI verification

• VLSI design
• Planning
• Planning

http://www.cs.miami.edu/~tptp/OverviewOfATP.html

Planning Planning

• What is planning?

p g

– Finding a sequence of actions to achieve one’s goals

• How is planning different from regular search?

– States and action sequences typically have complex internal structure – State space and branching factor are huge – Multiple objectives, resource constraints

• Examples of planning applications

Scheduling of tasks in space missions – Scheduling of tasks in space missions – Logistics planning for the army – Assembly lines, industrial processes Assembly lines, industrial processes

Propositional planning

• Start state, goal state are specified as conjunctions of

predicates

Start state: At(P1 RDU) ∧ Plane(P1) ∧ Airport(RDU) ∧ – Start state: At(P1, RDU) ∧ Plane(P1) ∧ Airport(RDU) ∧ Airport(ORD) – Goal state: At(P1, ORD)

A ti d ib d i t f th i diti

• Actions are described in terms of their preconditions

and effects:

– Fly(p, source, destination) y(p, , )

• Precond: At(p, source) ∧ Plane(p) ∧ Airport(source) ∧

Airport(destination)

• Effect: ¬At(p source) ∧ At(p destination)

Effect: At(p, source) ∧ At(p, destination)

• Search problem: starting with the start state, find all

applicable actions (actions for which preconditions are ti fi d) t th t t b d th satisfied), compute the successor state based on the effects, etc.

Complexity of planning Complexity of planning

• Planning is PSPACE-complete

Planning is PSPACE complete

– Plans can be exponential in length! Example: tower of Hanoi – Example: tower of Hanoi

From propositional planning to real-world planning

• Incorporating the time dimension

Incorporating the time dimension

• Resource constraints
• Contingencies: actions failing

Contingencies: actions failing

• “Qualification problem”
• Hierarchical planning
• Hierarchical planning
• Uncertainty
• Observations
• Observations
• Multiagent planning