Math 381 / CS 360
Advanced Logic with Theory of Computation
Spring 2001

Test Info

Test 1

• syntax and semantics for FOL, including quantifiers
  • truth-functional completeness and unnecessary parts of FOL
• proof rules for our Fitch proof system (but not for quantifiers yet)
• differences and relationships between "truth" and "proof"
  • Soundness and Completeness (statements both with and without quantifiers, proof for FOL without quantifiers and without equality)
  • Compactness and how it follows from Soundness, Completeness, and some properties of our proof system
  • role of truth tables in propositional logic, models in FOL with quantifiers
  • terms like tautology, logical consequence, consistent, formally complete, etc. and how to establish them
• computability
  • computable, computably enumerable
  • Turing machines
  • Church's Thesis
• cardinality
  • definitions, basic properties
  • Cantor-Schroeder-Bernstein Thm
  • countability and uncountabilty
• Remember: Mathematics is a typed language! Please identify types of mathematical objects (sets, functions, sentences, wffs, models, etc.) correctly and use them appropriately. Some terms and notations are dependent upon knowledge of the types of objects involved.

Test 2

• syntax and semantics for FOL, including quantifiers
  • truth-functional completeness and unnecessary parts of FOL
• proof rules for our Fitch proof system (including quantifiers now)
• differences and relationships between "truth" and "proof"
  • Soundness and Completeness (statements both with and without quantifiers, full proof of Soundness)
  • Compactness and how it follows from Soundness, Completeness, and some properties of our proof system (and applications of Compactness)
  • role of truth tables in propositional logic, models in FOL with quantifiers
  • formal definitions of model, variable assignment, satisfaction; building models
  • idea of extending the language (by adding constants, for example) and then extending a model correspondingly. (This came up in the non-standard model of PA, proof of soundness, Proposition 1, etc.)
  • terms like tautology, first order consequence, consistent, etc., and how to establish them
• computability
  • computable, computably enumerable
  • Turing machines
  • Church's Thesis
• Set Theory
  • Naive set theory
    • Axioms (extension and comprehension)
    • Why naive set theory is unsatisfactory (Russel Paradox)
  • ZF
    • Intuition behind axioms (need to restrict comprehension somehow; try to avoid sets that are ``too big'' or ``too complicated'', but still have enough sets around to be useful)
    • Useful mathematical objects that can be constructed in ZF (unions, pairs, relations, functions, models, etc.).
    • How Regulariy precludes certain ``weird'' sets.
  • Special sets: empty set, powersets, omega
• Peano Arithmetic and Induction
  • Proofs by induction, especially structural induction
  • Why various forms of induction are all equivalent
  • How to prove basic facts about the natural numbers from Peano's axioms (like in PS 13).
• Remember: Mathematics is a typed language! Please identify types of mathematical objects (sets, functions, sentences, wffs, models, etc.) correctly and use them appropriately. Some terms and notations are dependent upon knowledge of the types of objects involved.

A slightly different form of this information is also available in pdf, and ps formats.

Final Exam

Newer Topics Not on Earlier Tests

• Compactness Theorem: statement, proof and applications (like non-standard models of PA)
• Computational Aspects of First Order Logic. You should be familiar with the following ideas and algorithms
  • Horn Sentences/Clauses and the Horn Clause Algorithm
  • Resolution and its variants (DPP; modifications to work with quantifiers -- Skolemization and Unification)
  • Soundness and Completeness for basic Resolution
  • Other important forms: CNF, 3CNF, prenex form
  • You will not be tested on Prolog or Otter
• Computability and Computational Complexity
  • Turing Machines and Church's Thesis
  • definitions of classes and closure properties (especially computable, computably enumerable, P and NP)
  • Halting problem: Computably enumerable but not Computable
  • many-one reductions and NP-completeness
  • SAT, 3SAT, Vertex Cover and NP-completeness, especially how to use one NP-complete set to show that other sets are NP-complete

Important Topics from Throughout the Course

• See the lists above and the calendar for topics covered earlier in the course.
• The basic notions and results of FOL are essential. Examples: Soundness, Completeness, Compactness, use of models and/or truth tables to define truth, expressing things in FOL, tautology/tautological consequence, first order consequence/validity, satisfiable, etc., etc. etc.
• Remember: Mathematics is a typed language! Please identify types of mathematical objects (sets, functions, sentences, wffs, models, etc.) correctly and use them appropriately. Some terms and notations are dependent upon knowledge of the types of objects involved.

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This page maintained by:
Randall Pruim
Department of Mathematics and Statistics
Calvin College
rpruim@calvin.edu