This presentation briefly introduces Logic Based System in Artificial Intelligence

1. Formal Logic
Dr. C.V. Suresh Babu
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Demo
https://youtu.be/IY1sPV9e9H0

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Why?
• Logic can help AI and ML in complex domains or in domains with very
little data.
https://www.youtube.com/watch?v=K4ChzesrWKI

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What?
• By logic we mean symbolic, knowledge-based, reasoning and other
similar approaches to AI that differ, at least on the surface, from
existing forms of classical machine learning and deep learning.
• It is crucial to keep in mind just as there are many forms of machine
learning; there are many different forms of logic-based approaches to
AI with their own sets of tradeoffs.
• Very briefly, logic-based AI systems can be thought of as high-level
programming systems that can easily encode human knowledge in a
compact and usable manner.
https://www.youtube.com/watch?v=QQwBLv9LarI

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Introduction
• Starting with the simplest of logic systems, we have propositional
logic (sometimes called zero-order logic).
• Within propositional logic, we have objects known as sentences or
formulae that encode information.
• They denote some statement about the world.
• For building blocks, we have basic sentences known as atoms, usually
denoted by “P, Q, R, S, …..”
• For example, P might stand for “It is raining” and is an atom.
https://www.youtube.com/watch?v=_MhgsoPHvYo

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Building blocks of logic: atoms
Building blocks of logic: more complex formulae
Atoms can combine with logical connectives such as “and, or, if
then” to form more sentences called compound formulae.

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• Formulae give us a way to represent information.
• Then, how do we go from information we already have to new
information?
• We do this through reasoning or inference.
• Propositional logic comes equipped with a set of schemes called
inference methods.
• Inference methods can be thought of as little programs that take a set
of sentences as input and spit out one or more sentences.
https://www.youtube.com/watch?v=ywKZgjpMBUU

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• The figure below shows a very straightforward inference method
that takes in a formula that represents “Q and R” and produces as
output a formula R.
For example, if the input is “It
is sunny, and It is warm,” the
output will be “It is sunny.”
Building blocks of logic: simple inferences

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• All logics come equipped with a set of
inference methods like the above.
• These primitive inbuilt methods of a logic play
a role similar to that of the standard library in
any programming language.
• Given these methods, we can combine them
to form even more sophisticated methods
such as the one shown, just like you can create
large programs from a few simpler primitives.

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Propositional logic
• Propositional logic is a good starting place for pedagogical reasons
but is unwieldy for modeling domains with a large number of objects.
• For example, let us say we want to write down the constraints that a
Sudoku puzzle should satisfy.
• Let us say we have for each number k and each row i and column j an
atom A_ijk that stands for “row i and column k contain number k”.
• With this, we can quickly write down constraints for Sudoku in
propositional logic.

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For example, the sentence in the figure below states that the
number 5 should appear in the first row.
• We can similarly write down
other constraints for other
numbers.
• We will have a total of 9³ =
729 atoms in our constraints.
• Each atom can be true or
false, giving us a total of
possible 2⁷²⁹ states (much
much greater than the
number of physical atoms in
our universe).

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First-order logic
• First-order logic improves upon propositional logic by introducing
atoms that can take in arguments that stand in for objects in a
domain.
For example, A(1,3,5) says that row 1 and column 3 has a 5 in it.
The figure below says that one column exists in row 1 that has
the number 5 in it.
As you can see, the representation for “Row 1 has a 5 in some
column” is much more compact in first-order logic.
• In first-order logic, instead of
having an atom A_ijk for every
combination of i, j and k, we will
have a single atom that takes in
those variables as arguments.

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Other logics
• First-order logic forms the basis of many modern logic systems used
in research and industry.
• Many other logic systems build upon and extend first-order logic (e.g.,
second-order logic, third-order logic, higher-order logic, and modal
logic).
• Each logic adds new a dimension or feature that makes it easy to
model some aspect of the world.
• For example, logics that are known as temporal logics are used to
model time and change.

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Applications of Logic in AI and ML
• Automated Discovery in Science
• Inductive Programming
• Automation of Mathematical Reasoning
• Verification of Computer Systems (including ML)
• Logic-like Systems along with Machine Learning Models