15-281 - Fall 2022

Artificial Intelligence:
Representation and Problem Solving

Key Information

Sunday and Tuesday: 10:00am - 11:15am

Wednesday: 8:30am - 9:45am

Samar Rahmouni - Office hours: Mon, Wed 6:00pm - 7:00pm


30% Final, 10% Midterm 1, 10% Midterm 2, 10% Quizzes, 40% Homework

(optional) S. Russel and P. Norvig, Artificial Intelligence: A Modern Approach (3rd edition, 2010)


This course is about the foundations of theory and practice of Artificial Intelligence. But, what AI means/is?
A number of different definitions of AI have been given so far. In general, all include the underlying idea of equipping machines (e.g., computer programs, robots) with "tools" to display behaviors that we, as humans, would be inclined to categorize as "intelligent". This is still quite vague, since it requires to clarify in turn what an intelligent behavior is.

In this course, this is issue is somehow narrowed down to the concept of rationality : the notion of Artificial Intelligence is treated as equivalent to that of Computational Rationality. Given well defined preferences, a rational agent always selects the actions whose outcomes result in the maximization of the expected utility.

The course will provide an overview of modern techniques for rational decision-making under different conditions regarding available information, certainty or uncertainty of the information, presence of one or more agents in both cooperative and adversarial scenarios.

More precisely, during the course, fundamental questions about the design of AI systems will be addressed, such as:

  • how to represent knowledge,

  • how to effectively generate provably optimal sequences of actions, (i.e., how to make optimal decision plans or define decision policies);

  • how to search among (large sets of) alternatives to find optimal or near-optimal solutions;

  • how to deal with uncertainity;

  • how to take decisions in adversarial environments;

  • how to learn to make optimal decisions.

Both deterministic and stochastic environments will be considered and treated separately.


There are three primary objectives for the course:

  • To provide a broad survey of AI and of the most popular techniques that are employed for: knowledge representation, problem solving, mathematical optimization, automated planning, probabilistic prediction and inference, sequential decision-making, supervised and reinforcement learning, deep learning, decision-making in multi-agent adversarial scenarios.

  • To develop a thorough understanding of the algorithmic foundations of AI and acquire a strong appreciation of the big-picture aspects of designing fully autonomous intelligent agents.

  • To develop operational known-how about how to build/program AI agents, and analyze and improve their performance.

Course Layout

Students are expected to attend all classes and to actively participate with questions.

Each theory lecture will feature short Quizzes at the beginning and at the end of the lecture.

For each one of the different topics, the course will present relevant techniques, discuss formal results, and show the application to problems of practical and theoretical interest.

Weekly recitations will be used to practice together with the algorithms and their application to different problems.

Homework will be assigned weekly. Each homework will include both questions to be answered and programming assignments.

  • Written questions will involve working through different algorithms, deriving and proving mathematical results, and critically analyzing the material presented in class.

  • Programming assignments will involve writing code in Python to implement and test algorithms.


Since the course will cover a number of different topics, students should have some background in programming (in Pyhton), algorithms, calculus, linear algebra, and probability theory. Please talk to the instructor if you are unsure whether your background is suitable for the course.

The formal prequisites for this course are:

  • 15-122 Principles of Imperative Computation
  • 21-241 Matrices and Linear Transformations
  • 21-127 Concepts of Mathematics or 15-151 Mathematical Foundations of Computer Science.

The corequisite for this course is:

  • 21-122 Integration and Approximation

For this corequisite, you should either have completed it prior to starting 15-281 or have it on your schedule for Spring 2023.


Grading will assigned based on the following weighting: 40% Homework, 30% Final exam, 10% Midterm exam 1, 10% Midterm exam 2, 10% Quizzes. The final exam will include questions about all topics considered in the course, with an emphasis on the topics introduced after the first midterm exam.


There is no required textbook for this class. The students should be able to learn everything from lecture handouts and homework. However, the Russel and Norvig textbook (available on Canvas) will cover good parts of the course. During the course, other books will be pointed out by the instructor to cover specific parts of the course.

Schedule (subject to changes)

Dates Topic Slides References
7/31 Introduction: Concepts, History, Definitions, Problems, Road map PDF R&N Ch. 1,2,26,27, Brief AI history, 2016 Stanford AI report
8/2 No class (professor's issue)
8/3 AI Problems and agents, Search problems: Taxonomy of problems and types of agents; planning and search problems; performance metrics; search meta-algorithms: Tree and Graph search; Informed vs. Uninformed search PDF R&N Ch. 3

8/7 Uninformed search: BFS, DFS, DL-DFS, IDS, UCS: algorithms and properties PDF R&N Ch. 3
8/9 Informed Search: Goal-drirected search using heuristic functions; Best-first strategy; Greedy; A*, admissible and consistent heuristics, properties and applications. PDF R&N Ch. 3
8/10 Recitation: Search algorithms PDF no solutions

8/14 Adversarial search 1: From goals to terminal states with a value; value / evaluation function; single player deterministic games; general AI games and adversarial scenarios; elements and types of games; zero-sum games; adversarial search trees; minimax search: algorithm, properties, role of rationality. PDF R&N Ch. 5
8/16 Adversarial search 2: More on minimax search; running examples; coordination and cooperation; analysis of properties; depth-limited approaches; evaluation functions; tree pruning and alpha-beta pruning; role of ordering and ordering heuristics. (no) pruning. PDF R&N Ch. 5
8/17 Adversarial search 3: Search under uncertainty; expectimax search; properties; probabilistic models, role and impact; other types of games.
Recitation: Adversarial search
Recitation PDF no solutions

8/21 Break, no class
8/23 Classical planning 1: Factored states; STRIPS/PDDL description language; states as conjuction of propositions; structure of actions; goals and sub-goals; planning problem: domain, instance, planner; complexity of planning; forward search for planning; quest for heuristics; ignore preconditions and domain-independent heuristics. PDF R&N Ch. 10
8/24 Classical planning 2: Domain-Independent heuristics, issues using ignore preconditions; backward search: regression on actions, action selection, state sets; planning graphs for domain-independent heuristics and solution finding: construction, mutex links, properties, use for domain-independent heuristics; GRAPHPLAN algorithm: extract solution and backward search, properties, solutions. PDF R&N Ch. 10

8/28 Constraint Satisfaction Problems (CSPs) 1: From search problems to CSPs, concepts of states, plans, constraints; definitions and taxonomy of CSPs and optimization problems; constraint graph; DFS search for CSPs; commutativity and backtracking. PDF R&N Ch. 4, 6
8/30 Constraint Satisfaction Problems (CSPs) 2, Local search: Arc consistency; constraint propagation (inference, message passing); pruning by inference-propagation; AC-3; Forward checking; Maintaining Arc Consistency (MAC); complexity and properties of pruning methods; Local search and local search methods for CSP; hill-climbing; WalkSAT
PDF R&N Ch. 4, 6
Paper on WalkSAT (Local search for max-SAT) in DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 26, 1996
8/31 Recitation: CSPs

9/4 Optimization models 1, Convex optimization: Optimization problems; general concepts and notation; taxonomy of optimization problems; applications; convex problems; convex sets; convex functions; properties of convex optimization; solution approaches. PDF
9/6 Optimization models 2, Integer optimization: Integer optimization problems; properties, modeling and solution approaches; branch-and-bound: divide-and-conquer idea, subproblems and relaxations, MIP gap, anytime properties, design strategies. PDF Branch-and-Bound, book chapter
9/7 Recitations: Optimization models and methods

9/11 Optimization 3, Solution methods: Use of derivatives and gradients for fnding stationary points; properties of gradient vectors; iterative numeric approach for unconstrained optimization; gradient descent / ascent; effect of step size; batch and stochastic/incremental gradient descent; projection operator for gradient descent in constrained optimization; simplex method for linear programming: concepts and steps of operations. PDF Survey on gradient descent algorithms
9/13 Review of concepts so far
9/14 Midterm Exam 1

9/18 Fall break
9/20 Fall break
9/21 Fall break

9/25 Bayesian Networks 1: Uncertaintiy and need for probabilistic models for prediction and control in the real-world; full joint distributions: from chain rule to product of conditional probabilities; Bayesian networks as grahical models for probabilistic inference; global and local semantics; Markov blanket; BN construction; information propagation and types of probabilistic relations among the nodes. PDF R&N Ch. 13, 14
9/27 Bayesian Networks 2: Exact inference by enumeration; complexity of the exact inference; expression trees for optimized computations; approximate inference methods based on sampling / Monte Carlo approaches; direct sampling from Bayesian network; sampling in presence of evidence; rejection sampling; likelihood weighting; limitations of incremental sample construction methods and state-based approaches; Gibbs sampling and Markov Chain Monte Carlo; sampling from a Markov blanket; notion of stationary distribution for state probabilities; Gibbs sampling at work (with exact inference). PDF R&N Ch. 13, 14
9/28 Recitations: Bayesian networks

10/2 Markov Chains 1:Random processes and probabilistic inference over a time index; discrete time/states vs. continous time/states; correlation and random chains; Markov property; Markov chains as Bayesian networks; transition probabilities and CPTs; n-steps state probability distribution; general 2-state and n-state chains, computation of limiting and stationary distributions; invariance w.r.t initial state distribution; stationary distribution and eigenvectors, fixed-point equation; different cases: limiting and invariant distribution; limiting but not invariant, no limiting but possibly stationary distribution. PDF R&N 15.2 - 15.6
10/4 Markov Chains 2, Hidden Markov Models (HMM):Review of different cases for limiting and stationary distribution; state classes: absorbing, periodic, aperiodic, transient, peristent, recurrent, ergodic; communication classes; irreducible chains; regular chains; ergodic chains and their properties.
Partial state observability; sensor model; reasoning over sequences and HMM formalism; probabilistic model as a Bayesian network; HMM for inference tasks: filtering and prediction; separate belief updates and joint belief updates.
PDF Markov chains 2
R&N 15.2 - 15.6
10/5 Recitations: Markov chains, HMMs

10/9 Markov Decision Processes 1: Categories of Markov processes; state-action time diagrams; policies; utilities, discounting; value function, q-value function; Bellman expectation equations. PDF R&N Ch. 17
10/11 Markov Decision Processes 2: Bellman optimality equations; Value Iteration; measures of convergence of Value Iteration; Policy Iteration PDF Bellman-Ford algorithm, deterministic dynamic programming, asynchronous, distributed
10/12 Recitations: MDPs
Markov Decision Processes 3: LP formulation, converge properties, different forms of implementing Value Iteration: asynchronous, in-place (Gauss-Siedel), prioritized, prioritized sweeping, partitioned
PDF Book extract on variants of Value Iteration

10/16 Reinforcement Learning 1: From MDPs to reinforcement learning; RL problem, challenges, core issues; passive RL, policy estimation: estimating the MDP and solving value iteration, estimating directly state values; Monte Carlo methods for direct episode-based estimation of state utilities / policy evaluation; MC vs. Bellman's equation: Temporal Differences (TD) for policy evaluation PDF R&N Ch. 21
Sutton and Barto RL book, 2020
10/18 Midterm review
10/19 Break, no class

10/23 Midterm Exam 2
10/25 Reinforcement Learning 2: Active RL; exploration vs. exploitation; GLIE; Monte Carlo action learning: exploring starts, generalized policy iteration, soft policies; on-policy and off-policy learning; introduction to SARSA and Q-Learning PDF
10/26 Reinforcement Learning 3: SARSA and Q-Learning, algorithms, properties, working examples; how to explore, exploration functions
Recitations: Tabular RL

10/30 Reinforcement Learning 4: RL in large-scale, continuous state scenarios; parametric function approximation; linear approximation; TD and Monte Carlo approximation of V and Q functions; incremental, online stochastic gradient minimization of approximation error; sample target as surrogate of labeled (supervised) target; formulas for incremental updates in approximate MC, TD (Sarsa, Q-Learning). PDF
11/1 Game Theory 1: General concepts and definitions of game theory (information, strategies, pure strategies, utilities, strategy profiles); strategy dominance; normal-form; payoff matrix; solution concepts and equilibria; Nash equilibrium; properties of NE; examples. PDF R&N Sec. 17.5-17.6
Book on Multi-agents systems, Ch. 3-4
11/2 Game Theory 2: Mixed strategies; Mixed strategies Nash equilibria: definition, existence, computation; game formalization and analysis using algebra and calculus; Nash equilibria as optima and saddle points of payoff function. PDF

11/6 Game Theory 3: Correlated equilibrium: motivation, models, solutions; Stackelberg, Leader-Follower games: LP-based model, solution, complexity; Leader-follower security games PDF
11/8 Recitation: Game Theory
11/9 Recitation, Q&A:

Homework Assignments

Topic Files Due Dates
HW 1: Problems and Agents, Uninformed Search 8/11
HW 2: Informed Search 8/18
HW 3: Adversarial Search 8/25
HW 4: Classical planning 9/1
HW 5: Constraint Satisfaction Problems 9/8
HW 6: Modeling and solving optimization problems 9/15
HW 7: Probabilistic modeling, Markov chains 9/29
HW 8: Bayesian networks 10/6
HW 9: Hidden Markov Models 10/13
HW 10: Markov Decision Processes for sequential-decision making 10/20
HW 11: Reinforcement learning with tabular methods 10/27
HW 12: Reinforcement learning with approximation methods 11/3
HW 13: Game Theory for multi-agent adversarial scenarios 11/10

Homework Policies

  • Homework is due by the posted deadline. Assignments submitted past the deadline will incur the use of late days.

  • You have 4 late days, but cannot use more than 2 late days per homework. No credit will be given for homework submitted more than 2 days after the due date. After your 4 late days have been used you will receive 40% off for each additional day late.

  • You can discuss the exercises with your classmates, but you should write up your own solutions. If you find a solution in any source other than the material provided on the course website or the textbook, you must mention the source. You can work on the programming questions in pairs, but theoretical questions are always submitted individually. Make sure that you include a README file with your andrew id and your collaborator's andrew id.

  • In general, for both assignments and exams, CMU's directives for academic integrity apply and must be duly followed. Students are required to refer to CMU's general policies on cheating and plagiarism: https://www.cmu.edu/policies/student-and-student-life/academic-integrity.html

Exam dates and policies

The class includes two midterm exams and one final exam. Midterms are set set for September 14 and October 23, the final is TBD.

During exams students are allowed to consult 1-page cheatsheet (written in any desired format). No other material is allowed, including textbooks, computers/smartphones, or copies of lecture handouts.

Students with special needs or disabilities are kindly asked to talk about it to the teacher at the beginning of the semester and make use of the assistance procedures provided by CMU https://scotty.qatar.cmu.edu/qword/student-affairs/office-of-health-and-wellness/assistance-for-individuals-with-disabilities

Office Hours

Samar Rahmouni - Office hours: Mon, Wed 6:00pm - 7:00pm

Name Email Hours Location
Gianni Di Caro gdicaro@cmu.edu Wednesdays 10:30-1:00pm + Send me an email to set up an appointment M 1007
Samar Rahmouni srahmoun@andrew.cmu.edu Monday and Wednesday: 6:00pm - 7:00pm ARC