CS 15-317: Constructive Logic (Fall 2015)

## Description

This course is about logics that express algorithms or computation. These constructive logics permeate Computer Science, both as a foundation for the discipline and as concrete languages with eminently practical uses. Indeed, logical languages, tools and methods play an essential role in the design, specification and verification of computer hardware and software, a role that is growing as systems become more complex. The course provides a general exposure to the ideas underlying constructive logic and examines some specific logics and their applications. It is organized into four parts:
• We begin by giving a basic vocabulary and tool set to talk about logics (and to a large extent programming languages). This includes the judgmental infrastructure of rules and derivations, and the elementary properties that a logical system should have. We use them to study propositional and predicate logic in their natural deduction and sequent calculus presentations.
• We then briefly observe that, in some logics, derivations look just like functional programs, a correspondence known as the Curry-Howard isomorphism. Under this proof-as-program interpretation, computation is carried out as a form of proof reduction.
• Next, we show that, in some logics, formulas can themselves be understood as programs, while proof search is the vehicle of computation. This formula-as-programs interpretation underlies logic programming. We examine languages based on both top-down search, such as Prolog and λ-Prolog, and bottom-up search, e.g., Datalog.
• We dedicate the last few lectures of the course to sampling some advanced logics, which support special purpose forms of computation.
This course is a gateway to the fascinating and multifaceted world of computational logic. If you find it enjoyable, you may also like 15-312 Foundations of Programming Languages (a systematic, logic-based investigation to programming languages), 15-414 Bug Catching: Automated Program Verification and Testing (the use of a logical technique known as model-checking to verify complex systems), and 80-311 Computability and Incompleteness (a study of the logical foundations of Computer Science).

## Prerequisites

This course targets primarily (but not exclusively) Computer Science undergraduate students in their junior or senior year. To enroll, students may have completed CS 15-210 (Principles of Programming) or received explicit permission from the instructor.

There is no textbook for this class. However, each lecture is covered by a handout or an article (see schedule). Readings are required unless explicitly flagged as optional.

This course assumes proficiency with informal logic and logical reasoning. If you feel you need a review or are otherwise unsure, you may want to read through Discovering Logic, by Iliano Cervesato.

## Software

We will use several languages and software tools in the various parts of the course.

### Logic programming languages and applications

Tutch
Tutch (short for Tutorial Proof Checker) is a tool that checks the validity of formal proofs that users provide. We will use version 0.52 beta. To run Tutch,
• or log on `unix.qatar.cmu.edu`;
• then, type `tutch filename.tut` where `filename.tut` contains your proof or proofs.

See the documentation for further information on running Tutch. The examples from the overview are available here.

Prolog
We will gain experience with top-down logic programming using GNU Prolog, version 1.3.0 or later. To run Prolog,

See the GNU Prolog manual for further information.

Lollibot
To explore bottom-up logic programming we will be using Lollibot, a linear logic programming language which is a custom fragment of Ollibot.

Lollibot is installed on `unix.qatar.cmu.edu`. You can run it on a file `filename.lob` with the command

```  \$ lollibot filename.lob
```

See the Lollibot primer by William Lovas for a quick introduction. Several examples from lecture are available here.

### Standard ML

Some homework assignments will have you write a intepreter for a logic programming language. While you are free to use whatever language you want for this, we will be providing starter code in Standard ML. Refer to the 15-150 software page to remind yourself about how to install and run SML.

### Latex

We strongly encourage you to typeset the written part of your homeworks in LaTeX (we will take this into account as part of your participation grade). It takes some getting used to it, especially if MS Word is all you've been exposed to, but it enormously simplifies producing pleasant mathematics. Here are some useful references:
You may find this Latex template for homeworks useful.

This is a 9 unit course.

• Labs (mandatory): 10%
• 1 point for reaching the lab threshold
• 0 points if you don't reach the threshold or if you don't show up
• Daily exercises and participation: 5%
• At the end of the every lecture, you will be given daily exercises, a couple of simple exercises or questions collected during the next class or lab. Each will be graded, with negative points for missing or sloppy work.
• Participation has the following components:
• Class preparedness: you must have done the readings before coming to class
• Get extra points by using Latex
• 10 + 1 homework assignments: 45%
• 1 to 2 weeks each
• Due at 11:59pm Doha time — strict!
• Written and programming parts
• Graded ASAP based on rubric for programming components
• To encourage good work and integrity, the instructor may invite students to his office to explain their solutions. Should this happen, the students' explanations will become part of their grades for that homework
• No joint homework unless explicitly instructed
• Last homework assignment is bonus
• Midterm 1: 10%, in class, closed books
• Midterm 2: 10%, in class, closed books
• Final exam: 20%, 3 hours, closed books

### Evaluation Criteria

Your assignments and exams are evaluated on the basis of:
• Correctness: your arguments should make sense, your proofs should be valid, and your program should work in the reference environment
• Specification: say what you want to do before doing it. In the case of programs, use structured comments describing types, meaning of the returned value, invariants, and side-effects
• Elegance: written material should be of the same quality as what a professional would write. No typos, no bad grammar, clarity is paramount. See these notes about ML programming style.
• Bonus points:
• 2% of the earned grade for each 12-hour period a homework is submitted ahead of the deadline
• up to 10% for particularly elegant solutions
• Negative points: at least -100% if caught cheating
• Don't cheat!

### Late Policy

There are no late days. Assignments submitted past the deadline will get a grade of 0.

You are expected to comply with the University Policy on Academic Integrity and Plagiarism (see also The Word and Understanding Academic Integrity). Note that the policy now requires that students acknowledge any help received from the Academic Resource Center (ARC).

Collaboration is regulated by the whiteboard policy: you can bounce ideas about a homework with other students, but when it comes to typing it down for submission, you are on your own — no notes, files, snapshots, etc. Morever, you must wait at least 4 hours before writing down the solution.

## Feedback

It is my goal to make this course successful, stimulating and enjoyable. If at any time you feel that the course is not meeting your expectations or you want to provide feedback on how the course is progressing for you, please contact me. If you would like to provide anonymous comments, please use the feedback form on the course home page or slide a note under my door. Comments of general interest will be answered on the course discussion board.

## Assessment

### Course Objectives

This course seeks to develop students who:
1. demonstrate a high level of proficiency with logic fundamentals
2. are familiar with algorithmic uses of computational logic
3. are experienced with a declarative programming paradigm
4. approach problems logically and create innovative solutions relying on proven techniques
5. have foundational knowledge and an attitude of career-long learning to remain current as technologies evolve
6. communicate orally and in writing in effective and appropriate ways within the discipline

### Learning Outcomes

Upon successful completion of this course, students will:
1. know and appreciate advanced concepts in logic. Namely, students will
1. have a deep understanding of propositional and predicate logic, be familiar with the distinction between intuitionistic and classical logic, and have had exposure to some advanced logics
2. correctly use a natural deduction or sequent calculus presentation in appropriate situations
3. be familiar with fundamental properties such as soundness, completeness and consistency, as well as with cut-elimination
4. be able to use structural induction to prove elementary properties of logical systems
2. have a clear understanding of the logical mechanisms underlying several forms of computation. Namely, they will
1. be acquainted with the proofs-as-programs correspondence
2. be familiar with the formulas-as-programs paradigm and computation as proof search
3. understand the concept of abstract logic programming language and uniform proof
4. distinguish backward-chaining and forward-chaining modes of computation
5. extract algorithms from constructive proofs, for example a compiler from a proof of soundness
3. be able to write programs in a variety of styles within the logic programming paradigm. Specifically, they will
1. be familiar with the fundamental algorithms of top-down logic programming including resolution and unification
2. be able to write medium-sized programs in Prolog and related languages
3. understand the principles and applications of bottom-up logic programming as found in Datalog
4. have had exposure to elementary forms of rewriting logic as found in forward-chaining linear logic programming
4. are able to think abstractly and to combine methods to solve a problem. Namely, they will be able to
1. apply abstractions to solve problems
2. develop a systematic analysis method for solving problems
3. logically break down a problem into logical components
5. have the ability to use learned concepts to remain current in their field. Specifically, they will be able to
1. reformat information from one way to another
2. recognize how to apply the fundamentals learned to new or emerging technologies
6. be able to communicate clearly and effectively ideas, concepts and intentions within the field of computational logic, namely
1. be able to describe technical constructs (concepts) clearly, so as to be readily understood by their peers
2. be able to give an individual presentation on a technical subject to audience of peers within the discipline of computational logic
3. form a cogent, logical argument asserting and reiterating all technical concepts that lie within the bounds of the taught curriculum or their research within that curriculum

 2015 Iliano Cervesato