Introduction to
Knowledge Representation
and Logic
Motivation
All our search programs seem quite special
purpose.
The search algorithm may be general, but we have
to code up the problem representation and
heuristic functions ourselves.
Wouldn’t it be nice if we could build a more
“general purpose” program.
E.g., suppose we could just
– tell our program about the rules of chess or checkers,
and it would be able to play
– give it some ideas about how to play well, and it would use
them.
Knowledge Representation
Such a general program would have to be
capable of “knowing” lots of things:
– E.g., the rules of chess
– Good strategies
– Lots of things needed to make sense out of
these rules, etc. (“Background knowledge”)
We would need to be able to express an
open-ended variety of things, in some form
that they can be properly manipulated.
The means of expressing general “facts”,
or beliefs, etc., is called a knowledge
representation language.
KR: The Basic Idea
The
World
Mental
Representations
“A man is on the
girder.”
“The girder is
on the mountain.”
mental processes
“The man is above
the mountain.”
Some Things to Do with
Representations
Given some beliefs,
– determine what else we should believe.
– agree that some other things are true
(likely?).
– predict that something is going to
happen.
– predict what happens conditionally.
Note that the last two complicate the
idea of correspondence with “the
world”.
Knowledge Representation
Languages
A KRL is a sme way of writing down
beliefs/facts/etc.
Not really a language, any more than a
programming language is.
Needs to be
– Very expressive: In it, we need to be
able to express anything we want.
– What might some possibilities be?
KRL Candidates: LISP?
I.e., suppose everything we know is a program.
– After all, we only care about knowledge that we can use.
– E.g., knowing someone’s telephone number might be the program
you run to dial that number.
Pretty expressive.
Good for certain kinds of things.
– Certainly, some of the things we know (how to play tennis) might
best we thought of this way.
But:
– We miss certain generalizations if everything is a program.
– E.g.: If our knowledge of a phone number were the procedure
that dials the number, then that procedure doesn’t work
» when we move from a rotary to a touch-tone phone
» for any other number
The Procedural/Declarative
Distinction
“Know-how” versus “know-that”.
In general, procedural knowledge is
– Hard to use generally, reason with, introspect on,
articulate.
– Enable fast behavior
Declarative knowledge is
– More flexible (multi-use), amenable to introspection,
articulation
– This is the focus of most KR work.
– (Note, though, that there must always be some sort of
(procedural) mechanism around that is interprets
declarative knowledge.)
KRL Candidates: English?
I.e., suppose the KRL is some natural language.
Expressive!
Suitably declarative.
But:
– Ambiguous
» “I saw her duck.”
– Context-dependent meanings
» Pronouns, deixis
– Non-compositional
» Idioms, collocations, etc.
In other words, a KR language should represent
facts in form that expresses what they mean
after they have been understood.
KRL Desiderata
Expressive
Unambiguous (wrt relevant
distinctions)
Context-independent
Compositional
Artificial Languages
We’ll have to make up an artificial
“language” that has all these nice
properties.
If we are being psychological, we might
hope to uncover the language of thought.
There is at least some evidence that people
think in something other than language per
se
– Consider “false-alarming”; multi-lingual recall.
– We are hard-pressed to know exactly what this
language is.
Actual KRLs
There have been various candidates
proposed for KRLs over the years.
One set of proposals is that formal logic be
used as a basic framework for such
languages.
Logic consists of
– A language
» which tells us how to build up sentences in the
language (i.e., syntax)
» and what those sentences mean (i.e, semantics)
– An inference procedure
» Which tells us which sentences are valid inferences
from other sentences
Logic as a KRL
There are really many logics.
– differing mostly in terms of how much they can express,
– but also, in a lot of details.
Using logic seems reasonable in that the language was
designed not to have the “flaws” of natural languages.
– I.e., we can engineer our way around ambiguity, etc.
Using logic seems questionable, in that:
– The inference procedure is designed for “valid reasoning”, i.e, it
is a proof theory. Human reason seems rife with useful nonvalid processed, e.g., assumption, likelihood, etc.
– Designing a complete valid inference procedure is not possible
for all but the simplest logics.
– The semantics is “model theory”, which some find problematic
for real domains.
Alternatives?
There have been various representational alternatives to
logic proposed over the years:
– Frames, semantic networks, Conceptual Dependency, Conceptual
Graphs
However, all the other formalisms people have proposed over
the years seem to have not worked out very well.
– Most could be viewed as special cases of logical frameworks, or
organizations layered on top of them, etc.
– The shortcomings of logic don’t disappear no matter what we
do.
It was also claimed by some that neural networks, etc., would
eliminate the need for representations, somehow.
– But, in fact, the fundamental problems don’t go away.
– And things are are very easy with logical representations can be
very hard to do with neural network formalisms.
There’s Lots of Room to
Maneuver
Bear in mind, if we agree on a logic,
that is just the beginning.
We still have to figure out what the
vocabulary is.
– This is at least as much of a problem.
In addition, there is no reason why
one has to buy into both the language
and the inference procedure.
– We might find that the general
inference procedure isn’t what we want,
but still use the language, etc.
Knowledge Bases and
Beliefs, etc.
We'll assume that we can describe the world in
terms of sentences in our KRL.
A knowledge base is a set of sentences.
– I.e., it is partial description of the world (we hope).
– (“Knowledge” is a bad word for this….)
Sentences can be completely disembodied, but
they are most interesting when used to describe
mental states of agents.
For example, sentences in a knowledge based can
be said to represent an agent’s beliefs.
– Actually, there are lots of relations an agent can have
toward sentences: belief, doubt, thinking about. The
philosophers call these propositional attitudes.
What Does a Sentence
Mean?
We said that sentences are supposed to
correspond to “the world”.
Exactly how a sentence corresponds to the
world is its interpretation.
– We'll have to specify this explicitly.
– More on this latter.
The “world” as we will talk about it will
generally be fairly small part of the whole
world.
– We call this the universe (or domain) of
discourse.
Truth and All That
A sentence is true if its interpretation corresponds to
what’s actually out there.
– E.g., “Bush likes baseball.” is true given the normal
interpretation of things.
– (Obviously, we’re using English here, but we really mean “the
content of this sentence, represented in our formal language”,
which we haven’t seen yet.)
Some sentences are true no matter how we interpret them.
These are said to be valid.
– E.g., “Everyone who likes baseball likes baseball.”
– What about “Everyone who hates baseball doesn't like it.”?
More often, sentences will be true or not depending on how
we interpret them.
– Such sentences are said to be satisfiable.
But some are never true, and hence are unsatisfiable.
– E.g.. “Bush likes baseball and Bush doesn't like baseball.”
(Assertions Aren’t
Everything!)
Note that all our sentences are just
assertions, i.e., declarative sentences.
In natural language, we can “do” lots more.
– E.g., we can
» ask questions
» give commands
» express incredulity
We don't (yet) have anything that
corresponds to these notions, a big
limitation.
Note in particular that “truth” per se
doesn't apply to questions or commands.
Our First Logic: Propositional
Calculus
The syntax:
– Vocabulary:
» A set of propositional symbols
◆
e.g., P, Q, …
» A set of logical connectives or operators
◆
usually ∨ (or), ∧ (and), ¬ (not), → (implication),
maybe ≡ (equivalence)
» Parenthesis (for grouping)
» The special symbols True, False (logical
constants)
Propositional Calculus (con’t)
Syntax (con’t)
– Rules for forming sentences:
» Each symbol (i.e., a constant or a propositional symbol)
is a sentence (an atomic sentence).
» A sentence in parentheses is a sentence.
» If α and β are sentences, then so are
◆
◆
◆
◆
◆
◆
α ∨ β (conjunction)
α ∧ β (disjunction)
¬α (negation)
α → β (implication)
and similarly for whatever other connectives we allow
These are complex sentences.
» (Actually, we should insist on parentheses, or specify
a precedence order of operators, but we drop these
when they are not needed.)
Sample Sentences
P
True
P∨Q
¬P
(P ∨ Q)
¬(P ∨ Q)
¬P ∨ Q
(P ∨ Q) → R
P ∧ ¬P
But What Do the
Sentences Mean?
We specify semantics by stating:
– The interpretation of each propositional symbol.
– The interpretation of each complex sentences as a
function of its parts.
We interpret each propositional symbol as
referring to some proposition (whichever one you
like).
– E.g., P might be “It is raining in Oakland”, and Q, “Paris is
the capital of Germany.”
We interpret logical connectives in the obvious
way.
– E.g., ¬P means that P is not the case; P ∨ Q means that at
least one of P or Q is true.
Truth
For sentences, we also get to say
whether they are true or false.
– True is always true; False always false.
– P, Q, etc., are true or false depending on
their interpretation.
» So these are satisfiable, but not valid.
– Complex sentences are true or false as a
function of their connective.
» Usually specified as a truth table.
Truth Tables
Conjunction (∧)
α
β
Disjunction (∨)
α∧β
α
β
α∨β
false
false
false
false
false
false
false
true
false
false
true
true
true
false
false
true
false
true
true
true
true
true
true
true
Implication (→)
α
β
Negation (¬)
α→β
false
false
true
false
true
true
true
false
false
true
true
true
α
¬α
false
true
true
false
Comments
Note that connectives are truth
conditional, i.e., their truth is
completely determined by the truth
of their subparts.
This doesn’t seem true even for the
corresponding named terms of
ordinary language.
– “Touch that and you die!”
– “Bush having used heroin implies that he
will legalize drugs.”
A Proof Theory for
Propositional Logic
It is easy to devise a procedure to determine the
truth of an arbitrary sentence in propositional
logic.
Just write down a big truth table, and see if the
sentence is always true.
E.g., suppose we want to know if ¬(P∧Q)→ ¬P∨¬Q.
Here is a truth table:
P
Q
¬(P∧Q)
¬P∨¬Q
¬(P∧Q)→ ¬P∨¬Q
false
false
true
true
true
false
true
true
true
true
true
false
true
true
true
true
true
false
false
true
So we have proved this sentence.
Reasoning in Propositional
Logic
Similarly, if we assume a few things,
we can determine if something
follows.
E.g., if we assume P, then P∨Q, say,
degenerates into True∨Q, which a
truth table will tell us is always true.
So, we can always draw valid
conclusions from premises, regardless
of that any of this means.
More Convenient Reasoning
While it is not strictly necessary—we can
always use truth-tables—it is convenient to
capture common patterns of inference,
using inference rules.
We often write these as
α, β, …
γ
or, sometimes, as
α, β, …|− γ
meaning, if we know α, β, …, it is okay to
conclude γ.
Common Inference Rules
Modus Ponens:
α→β, α |− β
And-Elimination:
α1∧α2 |− αi
And-Introduction:
α1, α2 |− α1∧α2
Or-Introduction:
α |− α∨β
Double-Negation Elimination:
¬¬α |− α
Resolution:
α∨β, ¬β∨γ |− α∨γ
The Complexity of Propositional
Inference
While propositional logic inference procedures are
complete, the problem of finding a proof is
generally regarded to be exponential.
– In general, determining whether a set of sentences is
satisfiable is NP-complete.
Of course, any particular proof might be easy.
And there are useful special cases which are
better.
E.g., if we can express all our knowledge in form
P1 ∧ P2 ∧ P3 … ∧ Pn → Q
then there is a polynomial-time inference
procedure.
– These are called Horn clauses.
– Not all knowledge bases can be expressed this way.
But!
Propositional calculus is not expressive enough to
much interesting reasoning.
For example, if
P = “It is raining”
Q = “Jan is outside”
R = “Jan is wet”
we can say P ∧ Q → R. Then, by MP, if P ∧ Q, we
could deduce R.
But we would need an entirely different rule for
Pat.
What we would like instead is to be able to say
that this is true in general for things of people.
Example: Using Propositional
Logic for a Blocks World Problem
Suppose we have tiny world, and try to solve
problems in it with propositional logic.
E.g.: Suppose our world is this:
A
B
C
Table
with rules like
– If you move a block x from y to z, it is now on z.
– To move a block, nothing can be on top of it.
Our goal might be building a tower with A on top
of B on top of C.
A
B
C
Table
With Propositional Logic
All we have to work with is propositions, so
suppose we have a set of propositions like
CA, OAB , OA, meaning things like “Nothing is
on A”, “A is on B”, and “A is on table”, resp.
Then initial state of world might be:
CA: Nothing is on A.
CC: Nothing is on C.
OAB: A is on B.
OB: B is on the table.
OC: C is on the table.
And our goal:
OAB ∧ OBC
And some facts
We might add that
OAB ∨ OCB → ¬CB
OAB → ¬OAC ∧ ¬OA
etc.
Note that we can indeed use logic for something:
– We can ask if our goal state is true.
But how do we encode actions, like “moving A to
the table?” Several subproblems:
– How do we say “A must be clear to move it”?
– How do we say “After moving A to the table, it is on the
table”?
I.e., we want to say things like
CA → MA (“If A is clear, then you can move it to the table.”)
MA → OA (“If you move A to the table, it is on the table.”)
but MA is used inconsistently here.
Representing Actions
One trick helps somewhat: Can say
(CA ∧ MA) → OA
i.e., if A is clear and you move it to the table, then
it is on the table.
Of course, we will need a lot of sentences like this
one.
Then we might try different moves by assuming
MA, say, and seeing if we can find a sequence of
such assumptions that will let us achieve our goal.
But this isn’t quite right.
– If we assume MA, then since, CA is true, we could prove
OA, which isn’t true.
– I.e., we have no notion of time at all.
A Quick Fix: Introduce
States of the World
Instead of
(CA ∧ MA) → OA
we will say
(CA0 ∧ MA0) → OA1
i.e., CA0 means A is clear in state 0, etc.
We will also need
(CA1 ∧ MA1) → OA2
up to some finite number.
And we will have to upgrade all our formulas as
well, so, e.g., instead of
OAB ∨ OCB → ¬CB
we would have
OAB0 ∨ OCB0 → ¬CB0
Our New Goal
Instead of
OAB ∧ OBC
something like this:
OAB0 ∧ OBC0 ∨
OAB1 ∧ OBC1 ∨
OAB2 ∧ OBC2 ∨ …
to some finite number of steps.
Now we could try to prove this goal.
(Actually, if we try, we’ll find there are quite a few more
rules we have to add to make this work. We’ll explore the
difficulties later.)
Then we could use the formulas in the proof that have MXYs
in them to as the sequence of steps to take.
Bottom Line
The big problem with propositional
logic is
–
–
–
–
we would lots and lots of symbols, and
lots and lots of formulas,
even for simple problems,
and even then, we couldn’t really express
what we want (i.e., find a solution in any
number of steps).
Let’s see if another representation
language will help us.