Artificial Intelligence Core Concepts: Logic, Planning, Uncertainty
Artificial Intelligence: Logic and Reasoning Fundamentals
First-Order Logic (FOL)
First-Order Logic (FOL) is a formal system used in AI to express knowledge about objects, their properties, and relations. Unlike propositional logic, which only handles true or false values, FOL allows the use of quantifiers like “for all” (∀) and “there exists” (∃), making it more expressive. FOL uses predicates to describe properties of objects and relationships between them. For example, “All humans are mortal” can be written as ∀x(Human(x) → Mortal(x)). FOL is widely used in AI for knowledge representation, reasoning, and natural language processing. It enables systems to handle complex reasoning with multiple variables and domains.
Unification
Unification is a key process in AI and logic, especially in automated reasoning and first-order logic. It finds a substitution that makes different logical expressions identical. This allows inference algorithms like resolution to combine facts and rules effectively. For example, unifying P(x, Cat) with P(John, Cat) results in {x/John}. Unification checks variable and constant compatibility, producing a most general unifier (MGU) if possible. It is widely used in logic programming (e.g., Prolog), natural language processing, and expert systems. Efficient unification ensures that AI systems can match patterns, deduce facts, and apply rules accurately during reasoning.
Resolution
Resolution is an inference technique used to derive conclusions by combining pairs of logical clauses containing complementary literals. It is a complete rule of inference for first-order logic and propositional logic. The resolution process works by eliminating contradictory terms and producing new clauses until either the desired conclusion is reached or a contradiction is found. For example, resolving A ∨ B and ¬B ∨ C gives A ∨ C. In AI, resolution is the basis for automated theorem proving, logic programming, and expert systems. Its completeness ensures that if a conclusion logically follows from premises, resolution can prove it.
Ontological Engineering
Ontological Engineering is the practice of designing formal representations of knowledge domains, known as ontologies. It defines concepts, relationships, and rules that describe the structure of knowledge in a domain. Ontologies consist of classes, objects, properties, and constraints, enabling AI systems to share, reuse, and reason about knowledge effectively. For example, in a medical ontology, concepts like “Patient,” “Disease,” and “Treatment” are defined with their relationships. Ontologies support interoperability between systems, semantic web applications, and knowledge-based systems. They help AI systems understand and manipulate domain knowledge systematically, promoting consistency, reusability, and efficient reasoning.
Forward Chaining
Forward Chaining is a data-driven reasoning technique that starts from known facts and applies inference rules to derive new facts. The process continues until a goal is reached or no new facts can be generated. It is widely used in expert systems, rule-based engines, and production systems. For example, if “Socrates is a man” and “All men are mortal” are known, forward chaining can infer “Socrates is mortal.” It is efficient for scenarios where many rules can apply to an evolving knowledge base. Forward chaining works well when the goal state is unknown and all consequences of facts need exploration.
Backward Chaining
Backward Chaining is a goal-driven reasoning method that starts with a hypothesis or goal and works backward to verify whether the known facts support the goal. It attempts to break down the goal into sub-goals recursively until known facts are reached. For example, to prove “Socrates is mortal,” it checks whether “Socrates is a man” and whether “All men are mortal.” Backward chaining is commonly used in diagnostic systems, expert systems, and logic programming (e.g., Prolog). It is efficient when the goal is specific, as it focuses on proving particular queries rather than exploring all possible facts.
AI Planning: Strategies and Techniques
Classical Planning
Classical Planning is a method in AI where an agent creates a sequence of actions to transform an initial state into a goal state. It assumes a fully observable, deterministic, and static environment where the effects of actions are known and predictable. Each plan consists of actions with preconditions and effects, ensuring that the goal can be reached step-by-step. Classical planning uses techniques like state-space search, STRIPS representation, and heuristic search to generate valid plans. It’s widely applied in robotics, logistics, and automated systems where tasks can be broken into smaller, clearly defined steps.
State Space Search
State Space Search is a problem-solving method where all possible states of a system are represented as nodes in a graph. The search begins at the initial state and explores neighboring states by applying available actions, aiming to reach the goal state. Algorithms like Breadth-First Search (BFS), Depth-First Search (DFS), and A* are commonly used for state space search. This method is widely used in AI planning, pathfinding, game playing, and problem-solving applications. Each state transition corresponds to an action that changes the system’s condition, enabling systematic exploration of possible solutions.
Planning Graph
A Planning Graph is a layered structure used to efficiently solve planning problems by representing actions and states over time. Each level alternates between proposition levels (states) and action levels (possible actions). It helps identify reachable goals, detect conflicts (mutex relations), and estimate plan lengths. Planning graphs simplify complex planning by quickly identifying irrelevant actions and infeasible states, improving search efficiency. They are central to algorithms like Graphplan. Planning graphs are particularly useful in classical planning where deterministic and fully observable environments allow prediction of action effects.
STRIPS
STRIPS (Stanford Research Institute Problem Solver) is a formal language used to represent actions in classical planning. Each action is described by its preconditions (what must be true to apply the action) and effects (how the world changes after the action). STRIPS simplifies planning by structuring actions into clear, logical steps. For example, the action “boil water” may have the precondition “kettle is filled” and effect “water is boiled.” STRIPS-based planners search for sequences of actions that lead from the initial state to the goal state by satisfying these logical conditions.
Hierarchical Planning
Hierarchical Planning decomposes complex tasks into simpler sub-tasks, making planning more manageable. Instead of creating a plan from scratch, it breaks goals into hierarchies of subtasks, each of which can be solved separately. High-level actions (abstract actions) are refined into detailed sequences of low-level actions. This approach reflects how humans plan: for example, “prepare tea” decomposes into “boil water,” “add tea leaves,” and “pour tea.” Hierarchical planning reduces complexity, improves efficiency, and allows reuse of sub-plans for different goals. It’s widely used in robotics, scheduling, and multi-agent systems.
Forward Search
Forward Search starts from the initial state and explores possible actions step-by-step toward the goal. It applies all applicable actions at each step, generating successor states until the goal is found. Forward search is data-driven and works well when the initial state is fully known. Algorithms like Breadth-First Search (BFS), Depth-First Search (DFS), and heuristic-based A* are often used. This approach is effective when the state space is not too large or when heuristic information guides the search efficiently. It’s commonly used in planning, robotics, and navigation problems.
Backward Search
Backward Search begins from the goal state and works backward to determine the sequence of actions required to reach it from the initial state. At each step, it identifies actions that can achieve the current goal and adds their preconditions as sub-goals. This method is goal-driven and efficient when the number of possible goal states is small. Backward search is often used in theorem proving, diagnostic systems, and certain planning problems where the desired outcome is well defined. It reduces unnecessary exploration of irrelevant states, focusing directly on achieving the goal.
Mutex Relation
A Mutex (Mutual Exclusion) Relation in planning graphs indicates that two actions or two propositions cannot be true simultaneously. These relations help prune infeasible paths during the planning process by identifying conflicting actions or states. For example, “Boil water” and “Turn off kettle” might be mutex if they interfere with each other. Mutex relations are computed at each level of the planning graph, simplifying the search by eliminating incompatible action combinations. They play a crucial role in ensuring consistency and efficiency in planning algorithms like Graphplan.
Uncertainty and Probability in AI Systems
Uncertainty in AI
Uncertainty in AI arises when the system lacks complete, reliable, or precise information to make decisions. Real-world data often contains noise, incomplete facts, or unpredictable changes, making outcomes uncertain. AI systems must handle such uncertainty to function effectively, using probabilistic models, fuzzy logic, or statistical approaches. For example, predicting weather, diagnosing diseases, or recognizing speech involves uncertainty due to variability in input. Techniques like Bayesian inference, Markov models, and probabilistic reasoning help AI systems deal with uncertain information, updating beliefs and making informed decisions even with partial or imperfect data.
Probability
Probability is a mathematical measure that quantifies the likelihood of an event occurring, ranging between 0 (impossible) and 1 (certain). In AI, probability provides a formal way to handle uncertain knowledge and reason under incomplete information. Probabilistic models represent beliefs about unknown outcomes and guide decision-making processes. For example, if there is a 70% chance of rain, AI systems can plan accordingly. Probabilities can be assigned based on historical data, expert knowledge, or learning algorithms. Probability theory forms the foundation of many AI techniques such as Bayesian networks, Markov models, and decision theory.
Conditional Probability
Conditional Probability is the probability of one event occurring given that another event has already occurred. It is denoted as P(A|B), meaning the probability of event A given event B. Conditional probability allows AI systems to update beliefs based on new evidence. For example, the probability of having a disease given a positive test result is a conditional probability. It plays a central role in probabilistic reasoning, Bayesian networks, and diagnostic systems. Conditional probabilities are calculated using the formula P(A|B) = P(A ∩ B) / P(B), where P(A ∩ B) is the joint probability.
Bayes’ Rule
Bayes’ Rule is a fundamental theorem in probability that allows updating of probabilities as new evidence becomes available. It combines prior knowledge with new data to calculate posterior probabilities. The formula is P(A|B) = [P(B|A) × P(A)] / P(B). In AI, Bayes’ Rule is used extensively in applications like medical diagnosis, spam detection, and machine learning. It helps refine predictions by incorporating evidence, making AI systems adaptive and evidence-based. Bayesian inference relies on this rule to continuously revise beliefs as more information is gathered.
Probabilistic Reasoning
Probabilistic Reasoning enables AI systems to make decisions and draw conclusions under uncertainty by using probability theory. Instead of relying on absolute true/false values, it assigns likelihoods to events and outcomes. Probabilistic reasoning handles incomplete, noisy, or ambiguous data effectively. Techniques such as Bayesian inference, Hidden Markov Models (HMMs), and probabilistic graphical models are used in various AI domains like speech recognition, robotics, and medical diagnosis. By quantifying uncertainty, AI systems can prioritize decisions with the highest expected success, improving reliability and robustness in real-world scenarios.
Bayesian Network
A Bayesian Network is a graphical model that represents probabilistic relationships among variables using nodes and directed edges. Each node represents a random variable, and edges show dependencies. The network encodes conditional probabilities for each node based on its parents, simplifying complex joint probability calculations. Bayesian networks are widely used in medical diagnosis, risk assessment, and decision support systems. They efficiently handle uncertainty, missing data, and causal relationships, enabling AI to make informed predictions and inferences based on available evidence.
Naive Bayes Classifier
The Naive Bayes Classifier is a simple probabilistic classifier based on Bayes’ Rule, assuming that features are independent given the class label. Despite this strong independence assumption, it performs well in many real-world applications such as text classification, spam detection, and sentiment analysis. The classifier calculates the probability of each class given the input features and selects the class with the highest probability. Its simplicity, fast training, and effectiveness with large datasets make it a popular choice for supervised learning tasks in AI.
Markov Chain
A Markov Chain is a mathematical model describing a sequence of possible events where the probability of each event depends only on the current state, not previous states. This memoryless property is called the Markov property. Markov Chains are used in AI for modeling sequential processes like weather prediction, stock market analysis, and robot navigation. The model consists of states, transition probabilities between states, and often a starting distribution. Markov Chains form the basis for more advanced models like Hidden Markov Models (HMMs) used in speech recognition and time-series analysis.
MCMC (Markov Chain Monte Carlo)
Markov Chain Monte Carlo (MCMC) methods are algorithms used to sample from complex probability distributions where direct sampling is difficult. MCMC constructs a Markov Chain whose stationary distribution matches the target distribution, and then draws samples from this chain. Common MCMC methods include Gibbs sampling and Metropolis-Hastings algorithm. In AI, MCMC is applied in Bayesian inference, probabilistic graphical models, and machine learning for estimating posterior distributions and handling high-dimensional problems. It enables approximate inference when exact computation is computationally infeasible.