Math and Logic for Cognitive Science

Fall 2012

General information

Lecture: Monday and Wednesday 9:30AM - 10:45AM Business BU400.
Lab: Friday 9:05AM - 9:55AM Geological Sciences GY226.
Course website:

Instructor: Eduardo Izquierdo.
Office: Eigenmann EG841.
Email: (include "Q250" in the subject line).
Office hours: Wednesday 10:45am - 11.45am (immediately after class). Or by appointment.

Assistant instructor: Aaron Fath.
Office: Psychology A326
Email: (include "Q250" in the subject line).
Office hours: Friday 9:55am - 10.55am (immediately after lab session). Or by appointment.

Associate instructor: Ruth Eberle.
Office: Eigenmann
Phone: 856-5722
Email: (include "Q250" in the subject line).

Course description

Cognitive science aims to provide rigorous explanations of the processes that underlie intelligent behavior. To this end, cognitive scientists employ a wide variety of tools and concepts from logic and various branches of mathematics. In this course, we explore a range of topics in math and logic that are particularly important for cognitive science: propositional and predicate logic, automata theory, and probability theory.

For each topic, our goal is twofold: first, to understand the basic mathematical and conceptual underpinnings; second, to understand the relevance for cognitive science. To address the latter, we consider each mathematical topic as it fits within a certain area of cognitive science. Thus, we consider logic and automata theory as they relate to computational models of the mind, and probability theory in the context of Bayesian modeling. Ideally, then, students will come away from this course with two things: (1) a skill set of basic mathematical tools and (2) appreciation and enthusiasm for how these tools can be used to model and understand the mind. The material for the course is self-contained and no prerequisites beyond a sound high school mathematics background are needed.


There is no assigned textbook for this course. I will instead draw on a number of resources in planning the lectures. Most of these resources will be open access, i.e. freely available online., so that I can share them with you all as well. As I come across useful resources I will direct your attention to them here. Also, if anyone else finds additional resources that are not listed here, please let me know so that I can share them with everyone else!

FC: Carol Critchlow & David Eck (2011). Foundations of Computation. Available free online (download PDF). Used for Units 1 and 2.

NN: Rolf Pfeifer, Dana Damian, and Rudolf Fuchslin (2010). Neural Networks. Available free online (download PDF). Used for Unit 3.

IP: Charles M. Grinstead & J. Laurie Snell (2006), Introduction to Probability. Available free online (download PDF).Used for Unit 4.


Red/Pink/Fucsia means "new link added recently." Blue means "link added a few days ago, or longer."

Week Day Topic Reading Assignment
Unit 1: Logic and Proof
Week 1 M Introduction, course logistics, and overview. H1 assigned

Propositional logic.

FC §1.1, §1.2
F Practice (exercises from FC §1.1, §1.2, and some solutions).
Week 2 M Predicate logic. FC §1.4 H2 assigned
H1 due (see answers)
W Deduction.FC §1.5
F Practice (exercises from FC §1.4, §1.5). Example problems and solutions.
Week 3 M No class: Labor day
W Proof. H3 assigned
H2 due
F Unit 1 review exercises (most solutions).
Week 4 M Knowledge-based agents.

Wumpus world §7.1,§7.2,§7.7

W Review. H3 due
F Review.
Unit 2: Theory of Computation
Week 5 M Exam 1: Logic and Proof (answers*)(stats)
W Sets and Functions

FC §2.1, §2.2, §2.4

F Practice H4 assigned
Week 6 M Language and Regular expressions

FC §3.1, §3.2

W Finite State AutomataFC §3.4
F Practice H4 due, H5 assigned
Week 7 M Turing machines 1FC §5.1
W Turing machines 2 Examples.
F Review

H5 due

Practice exam

Week 8 M Review TBA

Exam 2: Theory of computation

F No classes: Fall break
Unit 3: Linear Algebra and Neural Networks
Week 9 M Introduction to neural networks NN: Ch.1, Ch.2
W Perceptron and matrices

Matrix algebra

H6 assigned
F Practice
Week 10 M Neural networks

IN: Ch. 3

W Neural networksIN: Ch.4 H6 due,
F Learning (Delta rule) H7 assigned
Week 11 M Backpropagation
W Review TBA
F Discussion of final project H7 due (Ans #1, Ans #2, Ans #3), H8 assigned
Unit 4: Probability Theory and Bayesian Inference
Week 12 M Exam 3
W Introduction to probability IP: §1.2
F Practice

Final project topic due

H9 assigned

Week 13 M Random variables IP: §6.1
W Combinatorics IP: §3.1, §3.2
F Practice H9 due
Thanksgiving week
Week 14 M Conditional probability IP: §4.1, §4.3 H10 assigned
W Bayesian inference TBA
F Practice. H10 due
Week 15 M Review
W Exam 4.
F Presentation by students.
Final week
Week 16 M: 8:00am-10:00am Presentation by students. Final project due.

Class Policies

All students are expected to attend every class and lab section. Tardiness and unexcused absences will decrease the participation grade.

Labs will begin with an opportunity for students to ask questions related to material covered in the preceding week. The rest of the lab will then be devoted to presentation and discussion of materials designed to supplement that week’s lectures, including computer demonstrations, additional problems that we can work through together, etc.

Homework assignments will be given weekly or biweekly, with approximately eight or nine assignments in total. Each assignment will typically be due a week from the date that it is assigned and assignments will not be accepted late for any reason. Instead, the lowest homework grade will be dropped, so it would be wise to plan on using this drop in the case of illness, emergencies, etc. Assignments shall be handed in at the beginning of class on the due date. 

There will be four exams, covering, in order, the major topics of the course:
1. Logic and proofs.
2. Theory of computation.
3. Linear algebra and neural networks.
4. Probability theory and Bayesian models.
The exams will be distributed every three/four weeks (see schedule for details).

Final written report

Choose a topic that you are interested in related to math and logic for cognitive science. Write a short paper on this topic, approximately 1000 words (three pages in length, 1.5 spaced, 12 point font), making sure to discuss how the topic incorporates ideas from math and logic to study cognition. Your paper should also include references (at least four from books, journals, or conference proceedings), demonstrating that you have adequately researched the topic.

Student presentation

You are to present the topic that you are writing about for your final project. Essentially, you should show that you understand how a certain set of mathematical tools are being used to model and understand a certain aspect of cognition. You will have 7 minutes to present. You are encouraged to use slides for your presentation. If so, please send me the file for the presentation before the class starts, so that I can have it ready to go for you. Everyone will grade everyone else's presentation. I will give you a sheet with the name of the presenters. You will have to rate each presentation on: (a) Clarity of the presentation (communication skills); (b) Relevance to the course (does it apply math and logic to cognition?); (c) Interestingness of the ideas presented; and (d) Level of difficulty of the topic. For each one of those you can assign a value from 1 to 5 (1, not very clear/relevant/interesting/difficult, and 5 very clear/relevant/interesting/difficult). Your grading will be anonymous to the rest of your classmates, but they will not be anonymous to me. I may take liberty to re-interpret / normalize your evaluations.

10% Participation
20% Homework assignments (2.5% each, ×8, lowest dropped)
56% Exams (14% each, ×4)
14% Final project (8% written report, 6% oral presentation)

Supplementary material

Links to previous versions of the course: 2007, 2008, 2009, 2010, 2011.

Although not officially assigned for the course, the following two textbooks cover many of the topics that we will discuss. In particular, they both discuss propositional and predicate logic, the theory of computation, and probability theory. The text by Russell and Norvig also covers the basics of Neural Networks and Bayesian theory. Both of these books are available on reserve in the Swain Hall library.

Rosen, K. H. Discrete Mathematics and Its Applications, fifth edition. McGraw-Hill, 2003. QA39.3 .R67 2003.

Russell, S. and Norvig, P. Artificial Intelligence: A Modern Approach, second edition. Prentice Hall, 2003. Q335 .R86 2003

Additional resources by topic

Logic and computation:

James Hein, Discrete Structures, Logic, and Computability. The Student Study Guide is freely available online.

Linear algebra:

Jim Hefferon. Linear Algebra (2012). Available free online (download PDF).

Neural networks:

Anil K. Jain and Jianchang Mao (1996). Artificial Neural Networks: A Tutorial. Used for Unit 3.

David Kriesel (2005). A Brief Introduction to Neural Networks. Available free online (download PDF).