Scheme software is required to run the .scm files in this section.
This calendar shows the weekly schedule for the course, which usually includes three lecture and group problem solving sessions per week. This calendar provides links to course notes, problem sets, and other relevant resources. These materials are also provided separately in the readings, assignments, and tools sections of this course.
Week 1
Wednesday (First day of class)
- Course overview; no notes
Week 2
Readings
- Notes 1: Proving Arithmetic Equations, pp. 1–6 (PDF)
Assignments
- Due at the end of Week 2:
Monday
Wednesday
- Problems on arithmetic equations
Friday
- Problems on implementing catch/throw with call-cc
Week 3
Readings
- Notes 1: Proving Arithmetic Equations, final sections (PDF)
Wednesday
- Procedure to check a linear proof and convert between linear and tree proofs
Friday
- Exercise: Convert a tree proof to a substitution proof (PDF)
- Scheme code (SCM) for the conversion
- Arithmetic inequalities
- Pattern matcher match.scm (SCM)
Week 4
Readings
- Pattern matcher match.scm (SCM)
- Pattern-match based procedure proof-match.scm (SCM) for converting a sequence-of-equations proof into a tree proof
- Notes 2: Substitution into Arithmetic Expressions (PDF)
- Also, from last week, see Scheme code for converting a tree proof to a substitution proof (SCM)
Monday
- Structural induction proof of the Substitution Lemma and Soundness of Substitution Proofs (see Notes 2: Substitution into Arithmetic Expressions (PDF))
- Intro to pattern matching, with pattern-match based procedure proof-match.scm (SCM) for converting a sequence-of-equations proof into a tree proof
Wednesday
- Review of Notes 2: Substitution into Arithmetic Expressions (PDF)
- Problem for Friday: We extend Arithmetic Expressions with another case called an application
- “Doctor” program using unnested matching eliza.scm (SCM)
Friday
- Discussion of using match/rewrite rules to put arithmetic expressions (possibly extended with applications and a derivative operator) into canonical form. An example is in deriv-simplify-rules.scm (SCM); this version uses the alphabetized sum-of-monomials canonical form rather than the one-variable-polynomial-with-polynomial-coefficients canonical form of the Notes
- Intro to Scheme Substitution Model
Week 5
Readings
- Notes 3: A Scheme Substitution Model, Sections 1–6 (pp. 1–14) (PDF)
- Skim the scheme programs implementing the Scheme Substitution Model
Monday
- Bring your laptop loaded with the files for running the Substitution Model
- Observe the submodel running on expressions in test-submodel.scm (SCM)
Wednesday
- Discussion of control contexts in the Substitution Model
Friday
- Further example file for Substitution Model evaluation politician.scm (SCM)
Week 6
Readings
- Notes 4: Term Models (PDF)
Assignments
- Due on Monday of Week 7: Problems 1–3 in Notes 4: Term Models (PDF)
Monday
- Trivial decision procedure: Two terms are equal in all models iff they are identical
- Proving it: Introduction to term models
Wednesday
- Equational completeness theorem: If an equation follows logical from a set of equations, then the equation is provable using standard rules starting with the set of equations as axioms. Outline of proof using a term model
Friday
- Introduction to simple types
Week 7
Monday
- Environment models for simple types
Wednesday
- Combinator formulation of environment models
Friday
- Term models for simple types
Week 8
Assignments
- Due on Monday of Week 10: Problem Set 1 (PDF)
Monday
- Compiling Scheme to register machines with a memory array
Wednesday
- Reducing RegM’s with a memory array to 2-Counter machines
Friday
- Semigroup word problems: Rewrite rules for 2-CM simulation
Week 9
Monday
- Semigroup word problems: Confluence implies equations capture one-way rewrite rules for 2-CM simulation
- Diamond Lemma for confluence
Week 10
Assignments
- Due at the end of Week 10: Problem 18 from Notes 5: Scheme Computability, Part I (PDF)
Monday
- Computability on S-expressions
Wednesday
- Notes 5: Scheme Computability, Part I (PDF)
Friday
- Proof that productivity inherits up many-one reducibility (≦m). Further properties of ≦m
Week 11
Wednesday
- Notes 7: Counter Machines (PDF)
- Notes 8: Semigroup Word Problems (PDF)
- Notes 9: 1st-order Theory of Concatenation (PDF)
Friday
- Notes 3: Scheme Substitution Model (PDF)
- Notes 5: Scheme Computability, Part I (PDF)
- Notes 6: Scheme Computability, Part II (PDF)
Week 12
Assignments
- Due on Wednesday of Week 13: Final Problem Set, 6 of these 10 problem:
Monday
- Course projects (PDF)
Wednesday
- Halting Problem for 2-Counter Machines
≤m Th(∑*, ·)
≤m Th({1,2}*, ·)
≤m Th(N,+,×, ≦)
≡m Th(Z,+,-, ×, ≦),
where Th(M) denotes the 1st-order formulas valid in model M
Friday
- Complete proof that [2-CM Halting Problem ≤m Th(∑*, ·)]
- Observe that Halts reduces to the set of closed formulas of the form ∃z.G where all quantifiers in G range over subwords of z
Week 13
Assignments
- Due at the beginning of this week: Course project (PDF) proposals
Monday
- Topic: Diophantine sets over the integers and naturals; closure under intersection. Primes ≠ range(g) for any polynomial, g
Wednesday
- MINScheme ≡T Halts. Sketch of relative computability: The jump operation
Friday
- Diophantine Predicates closed under ∧, ∨, ∃, but not ¬ (negation)
- A Diophantine polynomial whose nonnegative range is the nonprimes
Week 14
Assignments
- Due at the end of this week: Term project (PDF)
Monday
- Excerpt: pp. 174–181 on Undecidability of Exponential Diophantine Polynomials, from Jones, Neil D. “Computability and Complexity: from a Programming Perspective,” MIT Press, c. 1997, 466pp.
Wednesday (Last class)
- Term project
