Transcript Lecture 8 - Computer Science at CCSU

Pattern-directed inference systems
We can describe any problem domain in terms of 2 types of knowledge:
1.
Declarative knowledge: facts about the domain, which can be
expressed as assertions (statements in some language).
Examples:
(Today is a beautiful day)
Assertion in natural language
(= (Today) (beautiful-day)
Assertion in FOL language
(Today beautiful-day)
Assertion in OPS language
2.
Procedural knowledge: represents dependencies among facts,
and can be expressed as “if – then” rules. Example:
(> temperature 650F) and (no-rain)  beautiful-day
In an AI program, the declarative component is called a "knowledge”
base", and the procedural component is called a "rule base".
Pattern-directed inference systems: basic
architecture
Adding new assertions
Fact 1
Fact 2
....
Fact n
represented
as sentences
in some KR
language
Assertion/fact base
Adding new rules
Rule 1
Rule 2
.....
Rule k
Rule base
Data base
define what
follows from
the facts in
the KB
Inference
Engine
Knowledge representation: expressing knowledge
in a form understandable by a computer.
Choosing an appropriate language to represent knowledge is the first and the
most important step in building an intelligent system. Each language has 2 sides:
– Syntax: defines how to build sentences (formulas).
– Semantics: defines the meaning and the truth value of sentences by
connecting them to the facts in the outside world.
If the syntax and the semantics of a language are precisely defined, we call that
language a logic.
Logic = Syntax + Semantics
Connection between sentences in a KR
language and facts in the outside world
Internal
Representation
Entails
Sentences
Sentences
Outside world
Facts
Facts
Follows
There must exist an exact correspondence between the sentences entailed by
the logic and the facts that follow from other facts in the outside world. If this
requirement does not hold, the logic will be unpredictable and irrational.
Entailment and inference
Entailment defines if sentence A is true with respect to a given KB (we denote it as
KB |== A). Inference defines if sentence A can be derived from the KB (we denote it
as KB |-- A).
Let the KB contains only true sentences representing explicit knowledge about some
domain. To find all consequences (or derived knowledge) that follow from that KB,
the system must “run” an inference procedure. If this inference procedure generates
only entailed sentences, then it is called sound, and if the inference procedure
generates all entailed sentences, then it is called complete. Ideally, we want an
inference procedure to be both, sound and complete. In many cases, however, this
may not be possible (for example, if the KB is infinite) and we are willing to drop the
requirement for completeness.
If A is derivable from a KB by a sound inference procedure, i, i.e. KB |--i A, then the
derivation process is called a proof of A.
Knowledge Representation Languages
To formally express knowledge we need a language which is expressive
and concise, unambiguous and context independent, and computationally
efficient. Among the languages that fulfill at least partially these requirements are:
– Propositional Logic (PL). It can represent only facts, which are true or
false.
– First-Order Logic (FOL). It can represent objects, facts and relations
between objects and facts, which are true or false.
– Temporal Logic. This is an extension of FOL which takes the time into
account.
– Probabilistic Logic. Limits the representation to facts only, but can these
facts can be uncertain, true or false. To express uncertainty, it attaches a
degree of belief (0..1) to each fact.
– Truth Maintenance Logic. Represents facts only, but these can be unknown
and uncertain as well as true and false.
– Fuzzy Logic. Represents facts which degree of truth can be explicitly
defined.
Interpretation and model of a representation
Interpretation establishes a connection between sentences of a selected KR
language and facts from the outside world.
Example: Assume that A, B and C are sentences of our logic. If we refer to
the “Moon world”, A may have the following interpretation “The moon is
green”, B -- “There are people on the moon”, and C -- “It is sunny and nice
on the moon, and people there eat a lot of green cheese".
Given an interpretation, a sentence can be assigned a truth value. In PL, for
example, it can be true or false, where true sentences represent facts that hold
in the outside world, and false sentences represent facts that do not hold.
Any world in which a sentence is true under a particular interpretation is called a
model of that sentence under that interpretation.
Sentences may have different interpretations depending on the
meaning given to them.
Example: Consider English language. The word “Pope” is to be understood as
a “microfilm”, and the word “Denver” is to be understood as “pumpkin on the
left side of the porch”. In this interpretation, sentence “The Pope is in
Denver” means “the microfilm is in the pumpkin”.
Assume that we can enumerate all possible interpretations in all possible worlds
that can be given to the sentences from our representation. Then:
– A sentence is called valid (or tautology) if it is true in all these
interpretations. Example: (A v not A) is always true even if we refer to
the “Moon world” (“There are people on the moon or there are no people
on the moon”).
– A sentence is called satisfiable if it is true in some interpretation.
Example: “The snow is red and the day is hot” is satisfiable if this is the
case on Mars.
– A sentence is called unsatisfiable if it is not true in any interpretation.
Propositional logic
To define any logic, we must address the following three questions:
1. How to make sentences (i.e. define the syntax).
2. How to relate sentences to facts (i.e. define the semantics).
3. How to generate implicit consequences (i.e. define the proof theory).
From the syntactic point of view, sentences are finite sequences of primitive
symbols. Therefore, we must first define the alphabet of PL. It consists of the
following classes of symbols:
– propositional variables A, B, C ...
– logical constants true and false
– parentheses (, )
– logical connectives &, v, <=>, =>, not
Well-formed formulas (wff)
Given the alphabet of PL, a wff (or sentence, or proposition) is inductively
defined as:
– a propositional variable;
– A v B, where A, B are sentences;
– A & B, where A, B are sentences;
– A => B, where A, B are sentences;
– A <=> B, where A, B are sentences;
– not A, where A is a sentence;
– true is a sentence;
– false is a sentence.
The following hierarchy is imposed on logical operators: not, &, v, =>, <=>.
Composite statements are evaluated with respect to this hierarchy, unless
parentheses are used to alter it.
Example: ((A & B) => C) is equivalent to A & B => C
(A & (B => C)) is a different sentence.
The semantics of PL is defined by specifying
the interpretation of wwf and the meaning of
logical connectives.



If a sentence is composed by only one propositional symbol, then it
may have any possible interpretation. Depending on the interpretation,
the sentence can be either true or false (i.e. satisfiable).
If a sentence is composed by a logical constant (true or false), then its
interpretation is fixed:
– true has as its interpretation a true fact;
– false has as its interpretation a false fact.
If a sentence is composite (complex), then its meaning is derived from
the meaning of its parts as follows (such semantics is called
compositional, and this is known as a truth table method):
P
Q
not P
P&Q
PvQ
P => Q
P <=> Q
F
F
T
F
F
T
T
F
T
T
F
T
T
F
T
F
F
F
T
F
F
T
T
F
T
T
T
T
Example: using a truth table, define the validity of
P & (Q & R) <=> (P & Q) & R
P
Q
R
Q&R
P & (Q & R)
(P & Q) & R
F
T
F
T
F
T
F
T
F
F
T
T
F
F
T
T
F
F
F
F
T
T
T
T
F
F
F
F
F
F
T
T
F
F
F
F
F
F
F
T
F
F
F
F
F
F
F
T
P & (Q & R)<=>(P & Q) & R
T
T
T
T
T
T
T
T
This formula is valid, because it is true in all possible interpretations of its
propositional variables. It is known as the “associativity of conjunction” law.
Inference: this is a process of building a proof of a
sentence.
Inference is carried out by inference rules, which allow one formula to be
inferred from a set of other formulas. For example, A |-- B meaning that
B can be derived from A.
An inference procedure is sound if and only if (iff) its inference
rules are sound. In turn, an inference rule is sound iff its
conclusion is true whenever the rule premises are true.
PL inference rules are sound (this can be proven by means of truth tables);
therefore they can be used for building proofs of other formulas.
PL inference rules





Modus ponens: if sentence A and implication A => B hold, then B also
holds, i.e. (A, A => B) |-- B.
Example: Let A means “lights are off”,
A => B means “if lights are off, then there is no one
in the office”
B means “there is no one in the office”
AND-elimination: if conjunction A1 & A2 & ... & An holds, then any of
its conjuncts also holds, i.e. A1 & A2 & ... & An |-- Ai.
AND-introduction: if a list of sentences holds, then their conjunction
also holds, i.e. A1, A2,...,An |-- (A1 & A2 & ... & An).
OR-introduction: If Ai holds, then any disjunction containing Ai also
holds, i.e. Ai |-- (A1 v ... v Ai v ... v An).
Double-negation elimination: states that a formula can be either true
or false, i.e. (not (not A)) |-- A
PL inference rules (cont.)


Unit resolution: (A v B), not B |-- A. Note that (A v B) is equivalent to
(not B => A), i.e. unit resolution is a modification of modus ponens.
Resolution: (A v B), (not B v C) |-- (A v C). Note that
(A v B) is equivalent to
(not A => B),
(not B v C) is equivalent to
(B => C)
By eliminating the intermediate
conclusion, we get
(not A
=> C).
The soundness of each one of these rules can be checked by means of the
truth table method. Once the soundness of a rule has been established, it
can be used for building proofs. Proofs are sequences of applications of
inference rules, starting with sentences initially contained in the KB
Complexity of propositional inference
1. Propositional inference is complete, i.e. any valid formula can be proved
by means of the truth tables method.
2. However, the truth table may have as much as 2^N rows, where N is the
number of propositional variables in the KB. To build such a table, takes
time proportional to N, i.e. the problem of proving the validity of a PL
formulais NP-complete.
3. Inference rules in PL are monotonic. That is,
if KB1 |-- A, then KB1 U KB2 |-- A
Here, KB2 can be contradictory to KB1, which makes monotonicity of
PL rule a major representation problem.
4. Inference rules in PL are local, i.e. they depend only on their premises
(this is a consequent of the monotonicity property). This, in turn, makes
the inference procedure much better than exponential, because only a
small number of propositions are involved in each inference.
Horn formulas
Although inference in PL in the worst case takes an exponential time, for one
special class of PL formulas, there exists a polynomial time inference
procedure. These formulas are called Horn formulas, and they have the
following form:
A1 & A2 & ... & An  B,
where A1, A2, ..., An, B are positive literals.
A literal is a formula or its negation.
If the KB can be represented as a collection of Horn formulas, then by just
applying modus ponens, we can infer all conclusions.
Pattern-matching and unification
Assume that the KB consists of Horn formulas, whose literals are not simple
propositional variables or constants, but assertions such as:
(Robot Robbie)
(Robot ?x)  (Can_reason ?x), where ?x is called a pattern variable.
We can apply MP in this case, if our inference procedure is augmented with a
pattern-matching facility. It allows pattern (Robot ?x) to be “matched” to
data (Robot Robbie). Note that the match must be propagated to the right hand
side of the implication making (Can_reason Robbie) the conclusion of this
inference step.
In some cases, there is a need to match two patterns (rather than a pattern and a
data). This requires a special procedure, called unification, which finds all values
of pattern variables that make the two patterns identical.