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The basis for intelligent mathematical software is the integration of the "power of symbolic mathematical tools" with the suitable "proof technology".

**Symbolic Reasoning**

The basis for intelligent
mathematical software is the integration of the "power of symbolic
mathematical tools" with the suitable "proof technology".

Mathematical reasoning enjoys a
property called monotonic.

"If a conclusion follows
from given premises A, B, C, …

then it also follows from any
larger set of premises, as long as the original premises are included."

Human reasoning is not monotonic.

People arrive to conclusions only
tentatively, based on partial or incomplete information, reserve the right to
retract those conclusions while they learn new facts. Such reasoning is **non-monotonic**, precisely because the
set of accepted conclusions have become smaller when the set of premises is
expanded.

**1. Non-Monotonic Reasoning**

Non-Monotonic reasoning is a
generic name to a class or a specific theory of reasoning. Non-monotonic
reasoning attempts to formalize reasoning with incomplete information by
classical logic systems.

The Non-Monotonic reasoning are of
the type

■ Default reasoning

■ Ci umscription

■ Truth Maintenance Systems

**Default Reasoning**

This is a very common from of
non-monotonic reasoning. The conclusions are drawn based on what is most likely
to be true.

There are two approaches, both are
logic type, to Default reasoning :

one is Non-monotonic logic and the other is Default logic.

**Non-monotonic logic**

It has already been defined. It
says, "the truth of a proposition may change when new information (axioms)
are added and a logic may be build to allows the statement to be
retracted."

**Non-monotonic logic is predicate logic with one extension **called** **modal** **operator **M** which means “consistent with
everything we know”. The purpose of **M** is to allow consistency.

A way to define consistency with
PROLOG notation is : To show that fact **P** is true, we attempt to prove **¬P**.

If we fail we may say that **P** is consistent since **¬P** is false.

**Example :**

**∀**** x : plays_instrument(x) ****∧**** M
manage(x) → jazz_musician(x)**

**States that for all x, the x plays
an instrument and if the
fact**

that **x** can manage is *consistent*
with all other knowledge then we can conclude that **x** is a jazz musician.

**■ ****Default Logic**

Read the above inference rule as:

" if A, and if it is
consistent with the rest of what is known to assume that B, then conclude that
C ".

The rule says that given the
prerequisite, the consequent can be inferred, provided it is consistent with
the rest of the data.

‡ Example : Rule that "birds typically fly" would be
represented as

‡ Note : Since, all we know about
Tweety is that :

Tweety is a bird, we therefore inferred that Tweety flies.

The idea behind non-monotonic
reasoning is to reason with first order logic, and if an inference can not be
obtained then use the set of default rules available within the first order
formulation.

**‡ **Applying Default Rules :

While applying default rules, it is
necessary to check their justifications for consistency, not only with initial
data, but also with the consequents of any other default rules that may be
applied. The application of one rule may thus block the application of another.
To solve this problem, the concept of default theory was extended.

**‡ **Default Theory

Example :

A Default Rule says " *Typically an American adult owns a car*
".

The rule is explained below :

The rule is only accessed if we
wish to know whether or not John owns a car then an answer can not be deduced
from our current beliefs.

This default rule is applicable if
we can prove from our beliefs that John is an American and an adult, and
believing that there is some car that is owned by John does not lead to an
inconsistency.

If these two sets of premises are
satisfied, then the rule states that we can conclude that John owns a car.

**Ci umscription**

Ci umscription
is a non-monotonic logic to formalize the common sense assumption. Ci umscription is a formalized rule of
conjecture (guess) that can be used along with the rules of inference of first
order logic.

Ci umscription involves
formulating rules of
thumb with
"abnormality"

predicates and then restricting
the extension of
these predicates,

ci umscribing
them, so that they apply to only those things to which they are currently
known.

Example : Take the case of Bird
Tweety

The
rule of thumb
is that "birds
typically fly" is
conditional. The predicate
"Abnormal" signifies abnormality with respect to flying ability.

Observe that the rule ∀ **x(Bird(x) & ¬ Abnormal(x)** → **Flies))** does not allow us to infer that *"Tweety flies"*, since we do
not know that he is abnormal with respect to flying ability.

But if we add axioms which ci
umscribe the *abnormality predicate* to
which they are currently known say *"Bird
Tweety*" then the inference can be drawn. This inference is
non-monotonic.

**Truth Maintenance Systems**

Reasoning Maintenance System (RMS)
is a critical part of a reasoning system. Its purpose is to assure that
inferences made by the reasoning system (RS) are valid.

The RS provides the RMS with
information about each inference it performs, and in return the RMS provides
the RS with information about the whole set of inferences.

Several implementations of RMS have
been proposed for non-monotonic reasoning. The important ones are the :

Truth Maintenance Systems (TMS) and
Assumption-based Truth Maintenance Systems (ATMS).

The TMS maintains the consistency of a knowledge base as soon as new
knowledge is added. It considers only one state at a time so it is not possible
to manipulate environment.

The ATMS is intended to maintain multiple environments.

The typical functions of TMS are
presented in the next slide.

Truth Maintenance Systems (TMS)

A truth maintenance system
maintains consistency in knowledge representation of a knowledge base.

The functions of TMS are to :

■ **Provide justifications for conclusions**

When a problem solving system gives
an answer to a user's query, an explanation of that answer is required;

Example : An advice to a stockbroker
is supported by an explanation of the reasons for that advice. This is
constructed by the Inference Engine (IE) by tracing the justification of the
assertion.

**Recognize inconsistencies**

The Inference Engine (IE) may tell
the TMS that some sentences are contradictory. Then, TMS may find that all
those sentences are believed true, and reports to the IE which can eliminate
the inconsistencies by determining the assumptions used and changing them
appropriately. Example : A statement that either Abbott, or Babbitt, or Cabot is
guilty together with other statements that Abbott is not guilty, Babbitt is not
guilty, and Cabot is not guilty, form a contradiction.

**Support default reasoning**

In the absence of any firm
knowledge, in many situations we want to reason from default assumptions.

Example : If "Tweety is a
bird", then until told otherwise, assume that "Tweety flies" and
for justification use the fact that "Tweety is a bird" and the
assumption that "birds fly".

**2. Implementation Issues**

The issues and weaknesses related
to implementation of non-monotonic reasoning in problem solving are :

How to derive exactly those
non-monotonic conclusion that are relevant to solving the problem at hand while
not wasting time on those that are not necessary.

How to update our knowledge
incrementally as problem solving progresses.

How to over come the problem where
more than one interpretation of the known facts is qualified or approved by the
available inference rules.

In general the theories are not
computationally effective, decidable or semi decidable.

The solutions offered, considering
the reasoning processes into two parts : one, a problem solver that uses
whatever mechanism it happens to have to draw conclusions as necessary, and
second, a truth maintenance system whose job is to maintain consistency in
knowledge representation of a knowledge base.

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