UNIT 3 - Artificial Intelligence - PDF

Artificial Intelligence (AI) is about building machines and systems that can reason, learn, and solve problems, similar to how humans do. In many AI applications, logical reasoning plays a crucial role, allowing systems to make decisions based on given conditions and facts. Propositional Logic is a...

Artificial Intelligence (AI) is about building machines and systems that can reason, learn, and solve problems, similar to how humans do. In many AI applications, logical reasoning plays a crucial role, allowing systems to make decisions based on given conditions and facts. Propositional Logic is a fundamental part of AI that deals with reasoning using declarative statements. These statements are either true or false, helping AI systems process facts and make decisions logically. By breaking down complex reasoning into simple propositions, AI systems can solve puzzles, represent knowledge, and make automated decisions efficiently. What is Propositional Logic in Artificial Intelligence? Propositional Logic (PL) is a branch of logic that focuses on statements (propositions) that can be either true or false. It is also known as Boolean logic since the truth values are binary—either True (1) or False (0). In AI, propositional logic forms the foundation for logical reasoning, allowing systems to represent facts and rules about a problem domain. These rules help the system infer new information or make decisions based on the given inputs. Propositional logic simplifies knowledge representation by breaking down reasoning into atomic statements or propositions. For example, an AI system used in home automation might have propositions such as: P: “The light is on.” Q: “The window is open.” Using logical connectives, the system can combine these propositions to represent more complex statements like: “If the light is on and the window is open, turn off the light.” By using propositional logic, AI systems can reason effectively and perform tasks like automated decision-making, knowledge representation, and game playing. Basic Facts About Propositional Logic 1. Propositions are Declarative Statements: In propositional logic, each statement, known as a proposition, is either True or False. Example: P: “It is raining.” (True or False) Q: “The heater is on.” (True or False) 2. Atomic Propositions: These are simple, indivisible statements that cannot be broken down further. Each atomic proposition represents a basic fact or condition. Example: “The door is closed.” 3. Compound Propositions:  Multiple atomic propositions can be combined using logical connectives (like AND, OR, NOT) to create compound propositions. Example: “The door is closed AND the heater is on.” 4. Binary Truth Values: Every proposition has a binary truth value: it can only be True (1) or False (0). There are no intermediate states. This simplicity makes propositional logic ideal for clear-cut decisions. 5. Logical Connectives Combine Propositions: Logical connectives such as AND, OR, NOT, IF-THEN, and IF AND ONLY IF allow us to create more complex propositions from simple ones. Syntax of Propositional Logic The syntax of propositional logic defines the rules for creating valid propositions. In propositional logic, we combine atomic propositions using logical connectives to form more complex statements, known as compound propositions. Building Blocks of Propositional Logic Syntax 1. Atomic Propositions: These are basic statements that represent individual facts or conditions. Example: P: “It is raining.” Q: “The heater is on.” 2. Logical Connectives: Connectives are used to combine atomic propositions to form compound propositions. AND ( ∧ ): True if both propositions are true. OR ( ∨ ): True if at least one proposition is true. NOT ( ¬ ): Negates the truth value of a proposition. IF-THEN ( → ): True unless the first proposition is true and the second is false. IF AND ONLY IF ( ↔ ): True if both propositions have the same truth value. 3. Compound Propositions: These are more complex statements formed by connecting atomic propositions using logical connectives. Example: “If it is raining and the heater is on, then the room will be warm.” This can be written in propositional logic syntax as: (P∧Q)→R Where: P: “It is raining.” Q: “The heater is on.” R: “The room is warm.” Example of Propositional Logic  Let’s explore a real-world scenario where propositional logic is applied in AI. Consider a home automation system that needs to decide whether to turn on the air conditioner based on the weather conditions and indoor temperature. Scenario: P: “It is hot outside.” Q: “The windows are open.” R: “Turn on the air conditioner.” Using propositional logic, we can represent the system’s decision-making with the following compound proposition: (P∧¬Q)→R This logic reads as: “If it is hot outside AND the windows are not open, then turn on the air conditioner.” Explanation of the Logic: AND ( ∧ ) ensures that both conditions must be true (hot outside and windows closed) for the air conditioner to turn on. NOT ( ¬ ) negates the condition, meaning the windows must be closed. IF-THEN ( → ) states that if the first part is true, the second part (turning on the AC) will follow. Logical Connectives in Propositional Logic Logical connectives are essential operators that combine atomic propositions to form compound propositions. These connectives allow AI systems to build more complex rules and perform logical reasoning. Below are the most common connectives used in propositional logic: Common Logical Connectives Connective Symbol Meaning Example AND ∧ True if both propositions are true. (P ∧ Q): “It is raining AND cold.” OR ∨ True if at least one proposition is true. (P ∨ Q): “It is raining OR cold.” NOT ¬ Negates the truth value of a proposition. ¬P: “It is not raining.”  True unless the first is true and second is IF-THEN → (P → Q): “If it rains, then it will flood.” false. True if both propositions are either true or IF AND ONLY IF ↔ (P ↔ Q): “It rains if and only if it is cloudy.” false. These connectives allow us to create logical rules that AI systems can use to make decisions. Let’s take a quick look at how each works: 1. AND ( ∧ ): The result is True only if both propositions are true. Example: If P is “It is hot” and Q is “The fan is on”, then (P ∧ Q) means both conditions are satisfied. 2. OR ( ∨ ): The result is True if at least one of the propositions is true. Example: (P ∨ Q) will be true if either it is hot or the fan is on. 3. NOT ( ¬ ): This inverts the truth value of the proposition. Example: If P is true, ¬P will be false. 4. IF-THEN ( → ): This implies that if the first proposition is true, the second must also be true for the compound statement to be true. Example: “If it rains, then the ground will be wet” (P → Q). 5. IF AND ONLY IF ( ↔ ): This is true only when both propositions have the same truth value (either both true or both false). Example: “It is cloudy if and only if it will rain” (P ↔ Q). Truth Table A truth table is a useful tool for determining the truth value of a compound proposition based on the truth values of its atomic propositions. It systematically lists all possible combinations of truth values and the corresponding output for a given logical expression. How Truth Tables Work Let’s consider two propositions: P: “It is raining.” Q: “The ground is wet.” We can build a truth table to evaluate the compound proposition P ∧ Q (It is raining AND the ground is wet).  P Q P∧Q TRUE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE The result of P ∧ Q is True only when both P and Q are True. Truth Table with Three Propositions Let’s extend the concept to three propositions: P: “It is hot.” Q: “The air conditioner is on.” R: “The windows are closed.” We can create a truth table for the compound proposition (P ∨ Q) ∧ R (It is hot OR the air conditioner is on, AND the windows are closed). P Q R (P ∨ Q) ∧ R TRUE TRUE TRUE TRUE TRUE TRUE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE TRUE FALSE FALSE  FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE Purpose of Truth Tables Truth tables help in evaluating the outcomes of complex logical expressions. They ensure correct reasoning by listing all possibilities, making them a vital tool for AI systems that rely on logical reasoning. Precedence of Connectives in Propositional Logic When evaluating compound propositions with multiple logical connectives, it’s important to follow a specific order of precedence to ensure accurate results. Similar to arithmetic operations, logical operators are evaluated in a defined sequence, from highest to lowest precedence. Order of Precedence 1. NOT ( ¬ ) – Negation has the highest precedence and is evaluated first. 2. AND ( ∧ ) – Conjunction is evaluated next, after negations are resolved. 3. OR ( ∨ ) – Disjunction comes after AND operations. 4. IF-THEN ( → ) – Implication is evaluated after OR. 5. IF AND ONLY IF ( ↔ ) – Biconditional has the lowest precedence. Example: Precedence in Action Consider the following logical expression: ¬P∨(Q∧R) Step 1: Evaluate ¬P (Negation). Step 2: Evaluate Q ∧ R (AND). Step 3: Evaluate ¬P ∨ (Q ∧ R) (OR). The final result depends on the proper evaluation order, ensuring the correct outcome. Using Parentheses for Clarity To avoid ambiguity, it’s good practice to use parentheses in complex expressions. For example: (P∨Q)→R In this case, (P ∨ Q) is evaluated first, followed by the implication → Logical Equivalence in Propositional Logic Logical equivalence occurs when two or more logical expressions produce the same truth values for all possible combinations of their propositions. In other words, two statements are logically equivalent if they always have the same result, regardless of the truth values of the individual propositions. Definition of Logical Equivalence Two propositions P and Q are logically equivalent if: P≡Q This means that both P and Q yield identical truth values for all possible cases. Logical equivalence allows AI systems to simplify complex expressions without changing their meaning. Example of Logical Equivalence 1. De Morgan’s Laws: These laws show how negations of conjunctions and disjunctions behave: ¬(P∧Q)≡(¬P∨¬Q) ¬(P∨Q)≡(¬P∧¬Q) 2. Double Negation: Negating a negation gives the original proposition: ¬(¬P)≡P 3. Implication and Disjunction: An implication can be rewritten as: P→Q≡¬P∨Q Tautologies and Contradictions Tautology:A tautology is a statement that is always true, no matter the truth values of its individual propositions. Example: P∨¬P≡True Contradiction: A contradiction is a statement that is always false. Example: P∧¬P≡False Properties of Operators in Propositional Logic In propositional logic, logical operators follow specific properties that allow us to manipulate and simplify logical expressions. Understanding these properties is essential for building efficient AI systems that rely on logical reasoning. 1. De Morgan’s Laws These laws describe how negations distribute over AND ( ∧ ) and OR ( ∨ ) operations: First Law: ¬(P∧Q)≡(¬P∨¬Q) This means that the negation of a conjunction is equivalent to the disjunction of the negated propositions. Second Law: ¬(P∨Q)≡(¬P∧¬Q) This means that the negation of a disjunction is equivalent to the conjunction of the negated propositions. 2. Commutative Property This property states that the order of the propositions does not affect the result of AND ( ∧ ) and OR ( ∨ ) operations: AND: P∧Q≡Q∧P OR: P∨Q≡Q∨P 3. Associative Property This property allows us to group propositions in any order when using AND or OR operations: AND: (P∧Q)∧R≡P∧(Q∧R) OR: (P∨Q)∨R≡P∨(Q∨R) 4. Distributive Property This property states that AND distributes over OR, and vice versa: AND over OR: P∧(Q∨R)≡(P∧Q)∨(P∧R)P OR over AND: P∨(Q∧R)≡(P∨Q)∧(P∨R) Applications of Propositional Logic in AI 1. Knowledge Representation in Expert Systems: Represents rules and facts to solve domain-specific problems (e.g., medical diagnosis systems). 2. Reasoning and Decision-Making: AI agents use logical rules to make decisions (e.g., robot vacuum cleaners deciding when to start cleaning). 3. Natural Language Processing (NLP): Helps analyze text and respond logically (e.g., chatbots understanding weather- related queries). 4. Game-Playing AI:  Uses logic to make strategic moves (e.g., deciding checkmate in chess). Limitations of Propositional Logic 1. Inability to Handle Complex Relationships Propositional logic cannot represent relationships between multiple objects or deal with hierarchies of information. 2. No Handling of Uncertainty It works only with true or false values and cannot deal with probabilities or uncertain outcomes, limiting its use in real-world applications involving incomplete data. 3. Limited Expressiveness It cannot represent time-based sequences or dynamic events, which are crucial in some AI systems like speech recognition and robotics. 4. Scalability Issues As the number of propositions grows, the complexity of expressions increases, making reasoning slower and harder to manage. Conclusion Propositional logic is a foundational tool in artificial intelligence, offering a simple and effective way to represent facts and perform logical reasoning. Its binary structure and use of logical connectives enable AI systems to make clear-cut decisions, making it ideal for applications like expert systems, decision-making agents, NLP, and game-playing AI. However, propositional logic also has its limitations, such as the inability to handle complex relationships and uncertainty, which restricts its application in more dynamic or uncertain environments. To overcome these limitations, advanced systems often use first-order logic or probabilistic approaches. Despite its simplicity, propositional logic remains an essential concept for beginners in AI, providing a solid foundation for more advanced topics in logical reasoning. In Artificial Intelligence (AI), reasoning plays a crucial role in building systems that can make decisions and infer knowledge based on facts and conditions. However, propositional logic is limited in its ability to represent complex relationships or detailed information. Predicate logic, also known as first-order logic (FOL), extends propositional logic by allowing AI systems to represent relationships between objects and their properties. This makes predicate logic a powerful tool for knowledge representation and reasoning. It enables AI systems to understand relationships like “John is the father of Mary” or “All humans are mortal.”  Predicate Logic in AI What is Predicate Logic in Artificial Intelligence? Predicate logic, also known as first-order logic (FOL), is an extension of propositional logic that allows us to express relationships between objects and their properties. In AI, predicate logic is widely used to represent knowledge and perform reasoning in more complex scenarios where relationships matter. Unlike propositional logic, which deals with simple true/false statements, predicate logic introduces predicates, variables, constants, and quantifiers. These elements help in modeling real- world problems that involve multiple objects and their interactions. Role of Predicate Logic in AI Knowledge Representation: It provides a structure for representing complex facts about objects and their relationships in a system. Reasoning: AI systems use predicate logic to infer new information from existing facts, making it suitable for decision-making tasks. Example: “John is the father of Mary” can be represented as: Father(John,Mary)Father(John, Mary)Father(John,Mary) This expression tells us that there is a relationship (Father) between two objects (John and Mary). Predicate logic enables us to model and reason about such relationships effectively. Components of Predicate Logic: Predicate logic involves several key components that allow it to represent relationships and properties of objects in a structured way. 1. Predicates: A predicate is a function that returns either true or false based on the relationship between its arguments. Example: IsHungry(John)IsHungry(John)IsHungry(John) This predicate represents whether John is hungry, returning true if he is and false if not. 2. Variables: Variables are placeholders for objects within a domain. They allow us to represent general statements that apply to multiple objects. Example: In IsHungry(x), the variable x can represent any person. 3. Constants: Constants represent specific objects or entities in the domain. Example: John is a constant in the predicate IsHungry(John). Structure of Predicates Predicate consists of two key elements: the predicate symbol and arguments. Predicates are enhanced with quantifiers to specify the scope of variables involved. 1. Predicate Symbol The predicate symbol defines the property or relationship being described. Example: IsHungry(x) represents whether a person (x) is hungry. Married(x, y) denotes that person x is married to person y. Predicates are named based on the relationship or property they represent. The symbol is followed by arguments enclosed in parentheses. 2. Arguments and Arity Arguments refer to the specific objects that the predicate is applied to. The arity of a predicate refers to the number of arguments it takes. Examples: IsHungry(x): A predicate with 1 argument (arity = 1). Married(x, y): A predicate with 2 arguments (arity = 2). X(a, b, c): A predicate with 3 arguments (arity = 3), representing something like “a + b + c = 0.” 3. Quantifiers in Predicate Logic Quantifiers allow us to specify the scope of variables. There are two main types: 1. Existential Quantifier ( ∃ ) Meaning: There exists at least one object that satisfies the given condition. Example: ∃x IsHungry(x) This statement means that at least one person is hungry. Negation: The negation of the existential quantifier means that no such object exists. ¬∃x IsHungry(x) This means that no one is hungry. 2. Universal Quantifier ( ∀ ) Meaning: The given condition holds for all objects in the domain. Example: ∀x (IsHuman(x)→IsMortal(x)) This means that all humans are mortal.  Negation: The negation of the universal quantifier means there is at least one exception. ¬∀x IsHuman(x)→IsMortal(x) This implies that at least one human is not mortal. Examples of Predicate Logic 1. Simple Predicate Example: Predicate: IsHungry(John) Meaning: This predicate represents the state of whether John is hungry. It takes one argument (John) and returns true if John is hungry, otherwise false. Application in AI: In NLP-based chatbots, predicates like this could help infer the user’s intent. For example, if a chatbot detects that the user is hungry, it could suggest nearby restaurants. 2. Equality Predicate Example: Predicate: E(x,y)≡(x=y) Meaning: This predicate denotes that x is equal to y. It returns true if the two objects are identical. Application in AI: AI-based reasoning systems use equality predicates to match objects. For example, in a robot warehouse, a robot may use this predicate to determine if an object picked matches the one requested (e.g., E(Package1, RequestedItem)). 3. Mathematical Predicate Example: Predicate: X(a,b,c)≡(a+b+c=0) Meaning: This predicate checks whether the sum of a, b, and c equals zero. It returns true if the equation holds, otherwise false. Application in AI: In optimization problems, AI models might use mathematical predicates to check if constraints are satisfied. For example, in scheduling systems, such predicates can validate if certain conditions are met (e.g., X(shiftA, shiftB, totalTime) checks if the total shift hours are balanced). 4. Relationship Predicate Example: Predicate: M(x,y)≡x is married to y Meaning: This predicate expresses a relationship between two objects, indicating that x is married to y. Application in AI: In family tree AI systems, relationship predicates are used to infer relationships among family members. For instance, if M(John, Mary) is true, the system can infer that John is Mary’s spouse. 5. Universal Quantification Example: Expression: ∀x (IsHuman(x)→IsMortal(x))  Meaning: This statement reads as “For all x, if x is human, then x is mortal.” It applies to every object in the domain of humans. If an object is found to be human, it must also be mortal for the statement to hold true. Application in AI: Knowledge-based systems use such rules to infer properties about objects. For example, in medical diagnosis systems, rules like “All viruses can spread infections” (∀x IsVirus(x) → CanSpreadInfection(x)) help the system reason about diseases. 6. Existential Quantification Example: Expression: ∃x IsHungry(x) Meaning: This reads as “There exists at least one x such that x is hungry.” It indicates that at least one object in the domain satisfies the condition of being hungry. Application in AI: In robot planning, a robot could use existential quantifiers to plan actions. For example, “There exists a task that requires charging” (∃x TaskRequires(x, Charging)) might guide the robot to prioritize charging tasks. 7. Compound Example with Multiple Quantifiers: Expression: ∀x∃y (Parent(x,y)) Meaning: This statement means “For every person x, there exists a person y such that x is the parent of y.” It shows how multiple quantifiers can be used together to represent complex relationships. Application in AI: This logic is often used in social network AI models to analyze relationships. In a family tree system, the model could use such logic to infer relationships between family members. Propositions with Multiple Quantifiers In predicate logic, multiple quantifiers can be used within a single proposition to express more complex ideas. The order of quantifiers is crucial, as it can change the meaning of the statement. Example 1: Order of Quantifiers Matters ∀x∃y (Parent(x,y)) Meaning: For every person x, there exists at least one person y such that x is the parent of y. Example in AI: In a family tree system, this could represent the rule that every parent must have at least one child. Now, let’s reverse the quantifiers: ∃y∀x (Parent(x,y)) Meaning: There exists a person y such that every person x is the parent of y. Interpretation: This is logically impossible under normal circumstances, as a single person cannot have all people as parents. Example 2: Nested Quantifiers in AI ∀x∃y (RobotCanPerform(x,y)) Meaning: For every task x, there exists a robot y that can perform the task. AI Application: This could represent a rule in a robot planning system, where every task must have at least one robot capable of completing it. Now, consider the reversed version: ∃y∀x (RobotCanPerform(x,y)) Meaning: There exists a robot y that can perform every task x. AI Application: This would imply that a single robot can perform all tasks, which may not always be practical. Impact of Quantifier Order on Meaning As seen in the examples above, changing the order of quantifiers can completely change the meaning of a proposition. In AI systems, it’s essential to use the correct order of quantifiers to ensure that logical statements align with the desired behavior or knowledge representation. Knowledge Representation using Predicate Logic in AI Knowledge representation is a critical part of AI, enabling systems to store, reason, and derive new facts from existing information. Predicate logic offers a structured way to represent complex relationships, facts, and rules in a logical and understandable format. How Predicate Logic Helps in Knowledge Representation 1. Representing Relationships Between Objects: Example: Married(John,Mary) This expression indicates that John is married to Mary. Predicate logic captures such binary relationships between objects, helping AI systems understand connections. 2. Storing Rules in Knowledge Bases: Rules can be defined using predicates and quantifiers. Example: ∀x (IsHuman(x)→IsMortal(x)) This rule states that all humans are mortal. AI systems can store these rules and apply them during reasoning. 3. Handling Multiple Entities and Properties: AI systems can store complex information by using predicates with multiple arguments. Example: Parent(John,Mary)∧Sibling(Mary,Kate) This expresses that John is Mary’s parent and Mary is Kate’s sibling. 4. Reasoning and Inference: AI systems can use inference rules to derive new facts from stored knowledge. Example: If the system knows that: ∀x (Parent(John,x)→Loves(John,x)) It can infer that John loves all his children. Examples of AI Applications Using Predicate Logic  Natural Language Processing (NLP): NLP systems use predicate logic to understand relationships in text and answer complex queries. For example, in a chatbot, predicate logic helps infer the user’s intent. Robot Planning Systems: In a warehouse, robots use predicate logic to plan tasks based on conditions like: ∃x TaskRequires(x,Charging) This ensures robots prioritize tasks that require charging. Expert Systems: Predicate logic is used in expert systems (like medical diagnosis tools) to store if-then rules and infer new facts about diseases based on symptoms. Aspect Predicate Logic Propositional Logic Extends propositional logic by Definition allowing variables and relationships Deals with simple statements that are either true or false. between objects. Uses predicates, variables, Elements Uses atomic propositions (simple statements). constants, and quantifiers. More expressive; can represent Expressiveness Less expressive; limited to simple true/false statements. relationships and properties. Example ∀x (IsHuman(x) → IsMortal(x)) It is raining (P). Supports existential (∃) and Quantifiers No quantifiers are used. universal (∀) quantifiers. Can represent complex Handling of relationships, like Parent(John, Cannot represent relationships between objects. Relationships Mary). Used in NLP, expert systems, and Applications Used in decision-making systems with simpler conditions. robot planning. More complex due to the use of Complexity Simpler with fewer components. variables and quantifiers. Reasoning Enables inference through rules Limited to evaluating truth values of propositions. Capability and relationships. Difference Between Predicate Logic and Propositional Logic Conclusion Predicate logic is an essential tool in AI, providing a powerful framework for representing and reasoning about complex relationships and properties of objects. It builds on propositional logic by introducing predicates, variables, and quantifiers, making it possible to express more intricate facts and rules. Through examples like robot planning, NLP, and expert systems, we see how predicate logic allows AI systems to store knowledge logically, infer new information, and make decisions. However, while predicate logic is more expressive, it also introduces greater complexity compared to propositional logic. In summary, predicate logic plays a critical role in enabling AI systems to handle advanced reasoning tasks, forming the foundation for building intelligent, knowledge-based systems.  Resolution Method in AI Resolution method is an inference rule which is used in both Propositional as well as First-order Predicate Logic in different ways. This method is basically used for proving the satisfiability of a sentence. In resolution method, we use Proof by Refutation technique to prove the given statement. The key idea for the resolution method is to use the knowledge base and negated goal to obtain null clause(which indicates contradiction). Resolution method is also called Proof by Refutation. Since the knowledge base itself is consistent, the contradiction must be introduced by a negated goal. As a result, we have to conclude that the original goal is true. Applying resolution method Convert the given axiom into clausal form, i.e., disjunction form. Apply and proof the given goal using negation rule. Use those literals which are needed to prove. Solve the clauses together and achieve the goal. In (2), Q → R will be converted as (QV R) In (3), P → S will be converted as (-P V S) Negation of Goal (-R): It will not rain. Resolution Method in FOPI/Predicate Logic: Resolution method in FOPL is an uplifted version of propositional resolution method. In FOPL, the process to apply the resolution method is as follows: 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). Example 1. John likes all kind of food. 2. Apple and vegetable are food 3. Anything anyone eats and not killed is food. 4. Anil eats peanuts and still alive 5. Harry eats everything that Anil eats. Prove by resolution that: John likes peanuts. Solution: Step-1: Conversion of Facts into FOL a) John likes all kind of food. a. Ɐx: food(x) → likes(John, x) b) Apple and vegetable are food b. food(Apple) A food(vegetables) c) Anything anyone eats and not killed is food. c. Ɐx Ɐy: eats(x, y) A - killed(x) → food(y) d) Anil eats peanuts and still alive d. eats (Anil, Peanuts) ^ alive(Anil). e) Harry eats everything that Anil eats. e. Ɐx: eats(Anil, x) → eats(Harry, x) John likes peanuts. likes(John, Peanuts) Added predicates Ɐx: alive(x) →- killed(x) Added predicates Ɐx: - killed(x) → alive(x) Step-2: Conversion of FOL into CNF 1. Eliminate all implication (→) and rewrite. 2. Move negation (~) inwards and rewrite. 3. Rename variables or standardize variables to unique variable. 4. Move all quantifiers to the left without changing their relative order. 5. Eliminate existential instantiation quantifier by elimination ∃ (there exist). 6. Drop Universal quantifiers Ɐ(for all). 7. Convert the formula into a conjunction of disjuncts. Draw the resolution graph Unification is a fundamental concept in artificial intelligence (AI), particularly within the realms of logic programming and automated reasoning. It is the process of making two or more logical expressions identical by finding a suitable substitution for their variables. This technique is widely used in AI for tasks such as theorem proving, natural language processing, and the implementation of logic programming languages like Prolog. Definition - Unification is a process in which two terms, which may include variables, constants, and functions, are made identical by finding a substitution for the variables. A substitution is a mapping of variables to terms. For instance, if we have a variable (X) and we substitute it with a constant (a), the variable (X) is unified with (a). Basic Terminology - Term: A term can be a constant, a variable, or a compound term (a function with arguments). Substitution: A mapping from variables to terms. Unifier: A substitution that, when applied to two terms, makes them identical. Most General Unifier (MGU): The simplest unifier that can be used to unify two terms, meaning that any other unifier can be derived from the MGU by further substitution Unification Algorithm - The unification algorithm attempts to find the MGU for two terms. The algorithm involves recursively applying substitutions until the terms become identical or a conflict is found that prevents unification. Steps in the Unification Algorithm: 1. Initialization: Start with the two terms you want to unify. 2. Decompose Compound Terms: If the terms are compound (i.e., functions with arguments), break them down into their constituent parts. 3. Check for Conflicts: If a variable is being unified with a term that contains that variable (occurs check), unification fails. 4. Apply Substitutions: Continuously apply the substitutions and simplify the terms until they are identical or no further simplification is possible. Example 1: Suppose we have the following expressions: Expression 1: f(a, X, g(Y)) Expression 2: f(Z, b, g(h)) To unify these expressions, we need to find a substitution that makes them equal. The unification process involves matching corresponding parts and finding a set of variable assignments that satisfy both expressions: Match f(a, X, g(Y)) with f(Z, b, g(h)): X unifies with b (X/b) Y unifies with h (Y/h) Z unifies with a (Z/a) The resulting substitution is: {X/b, Y/h, Z/a}. Applying this substitution to both expressions gives us: f(a, b, g(h)) = f(a, b, g(h)) The expressions are now unified. Example 2: Let's consider a more complex example involving predicates and quantifiers in first-order logic: Expression 1: ∀x P(x, f(Y)) Expression 2: P(Z, f(a)) To unify these expressions, we need to find a substitution that makes them equal. The unification process involves matching the quantifiers and predicates and finding a set of variable assignments: Match ∀x P(x, f(Y)) with P(Z, f(a)): x unifies with Z (x/Z) Y unifies with a (Y/a) The resulting substitution is: {x/Z, Y/a}. Applying this substitution to both expressions gives us: ∀Z P(Z, f(a)) = P(Z, f(a)) The expressions are now unified. Applications in AI: 1. Logic Programming (Prolog): - Unification is the core mechanism for pattern matching in Prolog. When a query is made, Prolog uses unification to match the query with facts and rules in the database to infer new information or find solutions. 2. Theorem Proving: - Automated theorem provers use unification to match premises with conclusions of rules, thereby deriving new statements and ultimately proving or disproving theorems. 3. Natural Language Processing (NLP): - In NLP, unification is used for parsing and understanding sentences. Feature structures representing grammatical properties are unified to check for agreement and syntactic correctness. 4. Type Inference in Programming Languages: - Unification is used in type inference algorithms to determine the type of expressions in statically typed languages like Haskell and ML. Unification is a fundamental operation used in various AI applications, such as logic programming (e.g., Prolog), natural language processing, and automated reasoning. It enables AI systems to find common ground between different expressions, allowing for efficient manipulation, inference, and deduction.