Knowledge Representation & Reasoning AI UNIT 3
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The document outlines key concepts in artificial intelligence, focusing on propositional logic, first-order logic, and various inference methods such as forward and backward chaining. It also discusses probabilistic reasoning and Bayesian networks, providing a framework for knowledge representation and uncertainty handling in AI. Additionally, it covers fundamental inference rules and resolution techniques used to prove logical statements.
![Inference in First order logic
Inference in First-Order Logic is used to deduce new facts
or sentences from existing sentences.
Before understanding the FOL inference rule, let's
understand some basic terminologies used in FOL.
Substitution:
Substitution is a fundamental operation performed on
terms and formulas.
It occurs in all inference systems in first-order logic.
The substitution is complex in the presence of quantifiers
in FOL.
If we write F[a/x], so it refers to substitute a constant "a" in
place of variable "x".
9SURBHI SAROHA](https://image.slidesharecdn.com/knowledgerepresentationreasoningai3-201022053041/85/Knowledge-Representation-Reasoning-AI-UNIT-3-9-320.jpg)
Knowledge Representation & Reasoning AI UNIT 3
- 1. UNIT 3 AI BY:SURBHI SAROHA 1SURBHI SAROHA
- 2. SYLLABUS Propositional logic Theory of first order logic Inference in First order logic Forward & Backward chaining Resolution Probabilistic reasoning Utility theory Hidden Markov Models(HMM) Bayesian Networks. 2SURBHI SAROHA
- 3. Propositional logic This logic is synonymous with sentential logic, provides ways and means of combining or changing the propositions to create a complicated structure and to build a new logical relationship and properties. It combines logical connections of all the constituent statements and the true value of the complex statement is derived. While doing, it considers other factors like reasoning, relationship, and interconnection between the constituent statements. 3SURBHI SAROHA
- 4. Cont…. This logic provides better clarity on data and information in an incomplete environment by deeper analysis and inference of the limited information presented to it. Due to this unique feature and powerful algorithms deployed in logical reasoning, it is widely adopted in advanced search in AI in solving complex problems. 4SURBHI SAROHA
- 5. What is propositional logic? The natural language words may have slightly different meanings. Example: A ∧ B and B ∧ A should always have the same meaning. But the sentences She became sick and she went to the doctor. and She went to the doctor and she became sick. have different meanings. 5SURBHI SAROHA
- 6. Theory of first order logic In the topic of Propositional logic, we have seen that how to represent statements using propositional logic. But unfortunately, in propositional logic, we can only represent the facts, which are either true or false. PL is not sufficient to represent the complex sentences or natural language statements. The propositional logic has very limited expressive power. Consider the following sentence, which we cannot represent using PL logic. "Some humans are intelligent", or "Sachin likes cricket." To represent the above statements, PL logic is not sufficient, so we required some more powerful logic, such as first-order logic. 6SURBHI SAROHA
- 7. First-Order logic: First-order logic is another way of knowledge representation in artificial intelligence. It is an extension to propositional logic. FOL is sufficiently expressive to represent the natural language statements in a concise way. First-order logic is also known as Predicate logic or First- order predicate logic. First-order logic is a powerful language that develops information about the objects in a more easy way and can also express the relationship between those objects. First-order logic (like natural language) does not only assume that the world contains facts like propositional logic but also assumes the following things in the world: 7SURBHI SAROHA
- 8. Cont…. Objects: A, B, people, numbers, colors, wars, theories, squares, pits, … Relations: It can be unary relation such as: red, round, is adjacent, or n-any relation such as: the sister of, brother of, has color, comes between Function: Father of, best friend, third inning of, end of, ...... As a natural language, first-order logic also has two main parts: Syntax Semantics 8SURBHI SAROHA
- 9. Inference in First order logic Inference in First-Order Logic is used to deduce new facts or sentences from existing sentences. Before understanding the FOL inference rule, let's understand some basic terminologies used in FOL. Substitution: Substitution is a fundamental operation performed on terms and formulas. It occurs in all inference systems in first-order logic. The substitution is complex in the presence of quantifiers in FOL. If we write F[a/x], so it refers to substitute a constant "a" in place of variable "x". 9SURBHI SAROHA
- 10. Cont…. Equality: First-Order logic does not only use predicate and terms for making atomic sentences but also uses another way, which is equality in FOL. For this, we can use equality symbols which specify that the two terms refer to the same object. Example: Brother (John) = Smith. As in the above example, the object referred by the Brother (John) is similar to the object referred by Smith. The equality symbol can also be used with negation to represent that two terms are not the same objects. Example: ￢(x=y) which is equivalent to x ≠y. 10SURBHI SAROHA
- 11. FOL inference rules for quantifier: As propositional logic we also have inference rules in first-order logic, so following are some basic inference rules in FOL: Universal Generalization Universal Instantiation Existential Instantiation Existential introduction 11SURBHI SAROHA
- 12. 1. Universal Generalization Universal generalization is a valid inference rule which states that if premise P(c) is true for any arbitrary element c in the universe of discourse, then we can have a conclusion as ∀ x P(x). It can be represented as: . This rule can be used if we want to show that every element has a similar property. In this rule, x must not appear as a free variable. Example: Let's represent, P(c): "A byte contains 8 bits", so for ∀ x P(x) "All bytes contain 8 bits.", it will also be true. 12SURBHI SAROHA
- 13. 2.Universal Instantiation: Universal instantiation is also called as universal elimination or UI is a valid inference rule. It can be applied multiple times to add new sentences. As per UI, we can infer any sentence obtained by substituting a ground term for the variable. The UI rule state that we can infer any sentence P(c) by substituting a ground term c (a constant within domain x) from ∀ x P(x) for any object in the universe of discourse. It can be represented as:. Example:1. IF "Every person like ice-cream"=> ∀x P(x) so we can infer that "John likes ice-cream" => P(c) 13SURBHI SAROHA
- 14. Cont… Example: 2. Let's take a famous example, "All kings who are greedy are Evil." So let our knowledge base contains this detail as in the form of FOL: ∀x king(x) ∧ greedy (x) → Evil (x), So from this information, we can infer any of the following statements using Universal Instantiation: King(John) ∧ Greedy (John) → Evil (John), King(Richard) ∧ Greedy (Richard) → Evil (Richard), King(Father(John)) ∧ Greedy (Father(John)) → Evil (Father(John)), 14SURBHI SAROHA
- 15. 3.Existential Instantiation: Existential instantiation is also called as Existential Elimination, which is a valid inference rule in first-order logic. It can be applied only once to replace the existential sentence. The new KB is not logically equivalent to old KB, but it will be satisfiable if old KB was satisfiable. This rule states that one can infer P(c) from the formula given in the form of ∃x P(x) for a new constant symbol c. The restriction with this rule is that c used in the rule must be a new term for which P(c ) is true. It can be represented as: 15SURBHI SAROHA
- 16. Cont…. Example: From the given sentence: ∃x Crown(x) ∧ OnHead(x, John), So we can infer: Crown(K) ∧ OnHead( K, John), as long as K does not appear in the knowledge base. The above used K is a constant symbol, which is called Skolem constant. The Existential instantiation is a special case of Skolemization process. 16SURBHI SAROHA
- 17. 4.Existential introduction An existential introduction is also known as an existential generalization, which is a valid inference rule in first-order logic. This rule states that if there is some element c in the universe of discourse which has a property P, then we can infer that there exists something in the universe which has the property P. It can be represented as: Example: Let's say that, "Priyanka got good marks in English." "Therefore, someone got good marks in English." 17SURBHI SAROHA
- 18. Forward & Backward chaining Forward chaining as the name suggests, start from the known facts and move forward by applying inference rules to extract more data, and it continues until it reaches to the goal, whereas backward chaining starts from the goal, move backward by using inference rules to determine the facts that satisfy the goal. 18SURBHI SAROHA
- 19. Forward Chaining Forward chaining is also known as a forward deduction or forward reasoning method when using an inference engine. Forward chaining is a form of reasoning which start with atomic sentences in the knowledge base and applies inference rules (Modus Ponens) in the forward direction to extract more data until a goal is reached. The Forward-chaining algorithm starts from known facts, triggers all rules whose premises are satisfied, and add their conclusion to the known facts. This process repeats until the problem is solved. 19SURBHI SAROHA
- 20. Properties of Forward-Chaining: It is a down-up approach, as it moves from bottom to top. It is a process of making a conclusion based on known facts or data, by starting from the initial state and reaches the goal state. Forward-chaining approach is also called as data- driven as we reach to the goal using available data. Forward -chaining approach is commonly used in the expert system, such as CLIPS, business, and production rule systems. 20SURBHI SAROHA
- 21. Example: "As per the law, it is a crime for an American to sell weapons to hostile nations. Country A, an enemy of America, has some missiles, and all the missiles were sold to it by Robert, who is an American citizen." Prove that "Robert is criminal." To solve the above problem, first, we will convert all the above facts into first-order definite clauses, and then we will use a forward-chaining algorithm to reach the goal. 21SURBHI SAROHA
- 22. Facts Conversion into FOL: It is a crime for an American to sell weapons to hostile nations. (Let's say p, q, and r are variables) American (p) ∧ weapon(q) ∧ sells (p, q, r) ∧ hostile(r) → Criminal(p) ...(1) Country A has some missiles. ?p Owns(A, p) ∧ Missile(p). It can be written in two definite clauses by using Existential Instantiation, introducing new Constant T1. Owns(A, T1) ......(2) Missile(T1) .......(3) All of the missiles were sold to country A by Robert. ?p Missiles(p) ∧ Owns (A, p) → Sells (Robert, p, A) ......(4) Missiles are weapons. Missile(p) → Weapons (p) .......(5) Enemy of America is known as hostile. Enemy(p, America) →Hostile(p) ........(6) 22SURBHI SAROHA
- 23. Cont…. Country A is an enemy of America. Enemy (A, America) .........(7) Robert is American American(Robert). ..........(8) 23SURBHI SAROHA
- 24. Forward chaining proof: Step-1: In the first step we will start with the known facts and will choose the sentences which do not have implications, such as: American(Robert), Enemy(A, America), Owns(A, T1), and Missile(T1). All these facts will be represented as below. 24SURBHI SAROHA
- 25. Cont… Step-2: At the second step, we will see those facts which infer from available facts and with satisfied premises. Rule-(1) does not satisfy premises, so it will not be added in the first iteration. Rule-(2) and (3) are already added. Rule-(4) satisfy with the substitution {p/T1}, so Sells (Robert, T1, A) is added, which infers from the conjunction of Rule (2) and (3). Rule-(6) is satisfied with the substitution(p/A), so Hostile(A) is added and which infers from Rule-(7). 25SURBHI SAROHA
- 26. 26SURBHI SAROHA
- 27. Cont…. Step-3: At step-3, as we can check Rule-(1) is satisfied with the substitution {p/Robert, q/T1, r/A}, so we can add Criminal(Robert) which infers all the available facts. And hence we reached our goal statement. 27SURBHI SAROHA
- 28. 28SURBHI SAROHA
- 29. Cont…. Hence it is proved that Robert is Criminal using forward chaining approach. 29SURBHI SAROHA
- 30. Backward chaining A backward chaining algorithm is a form of reasoning, which starts with the goal and works backward, chaining through rules to find known facts that support the goal. Properties of backward chaining: It is known as a top-down approach. Backward-chaining is based on modus ponens inference rule. 30SURBHI SAROHA
- 31. Cont… The Forward-thinking approach is used in AI to help an AI agent solve logical problems by inspecting the data from the previous learnings and then coming to a conclusion full of solutions. That’s not all, Forward Chaining might as well be used to explore the available information or answer a question or solve a problem. Forward chaining is extensively used to break down a long and complex logical approach by attaching each step once the previous one is completed. This way, it goes from beginning to the end with relative ease. 31SURBHI SAROHA
- 32. Steps for working of Forwarding Chaining Step 1: We start from the already stated facts, and then, we’ll subsequently choose the facts that do not have any implications at all. Step 2: Now, we will state those facts that can be inferred from available facts with satisfied premises. Step 3: In step 3 we can check the given statement that needs to be checked and check whether it is satisfied with the substitution which infers all the previously stated facts. Thus we reach our goal. 32SURBHI SAROHA
- 33. Resolution Resolution is a theorem proving technique that proceeds by building refutation proofs, i.e., proofs by contradictions. It was invented by a Mathematician John Alan Robinson in the year 1965. Resolution is used, if there are various statements are given, and we need to prove a conclusion of those statements. Unification is a key concept in proofs by resolutions. Resolution is a single inference rule which can efficiently operate on the conjunctive normal form or clausal form. Clause: Disjunction of literals (an atomic sentence) is called a clause. It is also known as a unit clause. 33SURBHI SAROHA
- 34. Cont… Conjunctive Normal Form: A sentence represented as a conjunction of clauses is said to be conjunctive normal form or CNF. Steps for Resolution: Conversion of facts into first-order logic. Convert FOL statements into CNF Negate the statement which needs to prove (proof by contradiction) Draw resolution graph (unification). To better understand all the above steps, we will take an example in which we will apply resolution. 34SURBHI SAROHA
- 35. Cont…. Example: John likes all kind of food. Apple and vegetable are food Anything anyone eats and not killed is food. Anil eats peanuts and still alive Harry eats everything that Anil eats. Prove by resolution that: John likes peanuts. 35SURBHI SAROHA
- 36. Probabilistic reasoning Probabilistic reasoning is a way of knowledge representation where we apply the concept of probability to indicate the uncertainty in knowledge. In probabilistic reasoning, we combine probability theory with logic to handle the uncertainty. 36SURBHI SAROHA
- 37. Cont…. Probabilistic reasoning is a method of representation of knowledge where the concept of probability is applied to indicate the uncertainty in knowledge. Probabilistic reasoning is used in AI: When we are unsure of the predicates When the possibilities of predicates become too large to list down When it is known that an error occurs during an experiment 37SURBHI SAROHA
- 38. Bayesian Networks Bayesian network is a directed acyclic graph model which helps us represent probabilistic data. A Bayesian network contains two basic components: Nodes: These represent attributes/functions or data. Arcs: These represent the dependencies between the nodes. If an arc is present between the two nodes, then those two nodes have some sort of relation between them. An absence of an arc denotes the absence of any relation between the nodes. When designing a Bayesian network, we keep the local probability table at each node. 38SURBHI SAROHA
- 39. Utility theory Utility Theory is the discipline that lays out the foundation to create and evaluate Utility Functions. Typically, Utility Theory uses the notion of Expected Utility (EU) as a value that represents the average utility of all possible outcomes of a state, weighted by the probability that the outcome occurs. The agents use the utility theory for making decisions. It is the mapping from lotteries to the real numbers. An agent is supposed to have various preferences and can choose the one which best fits his necessity. 39SURBHI SAROHA
- 40. Hidden Markov Models(HMM) Hidden Markov Model is a statistical Markov model in which the system being modeled is assumed to be a Markov process – call it – with unobservable states. HMM assumes that there is another process whose behavior "depends" on. The goal is to learn about by observing. Hidden Markov Model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process – call it {displaystyle X} – with unobservable ("hidden") states. HMM assumes that there is another process {displaystyle Y} whose behavior "depends" on {displaystyle X}. The goal is to learn about {displaystyle X} by observing {displaystyle Y} 40SURBHI SAROHA
- 41. Thank you 41SURBHI SAROHA