Assignment 6
15-816: Linear Logic
Frank Pfenning
Due
Wednesday, April 18, 2012
This assignment consists of several, somewhat open-ended problems.
You should pick one of them, or any of the problems from
Assignment 5 that you have not done yet.
If you have
Lecture Notes on
Resource Management
15-816: Linear Logic
Frank Pfenning
Lecture 18
March 28, 2012
Backward chaining, or other linear logic proof search procedures that work
backwards from the conclusion, have to deal with the problem of resource
manageme
Lecture Notes on
Combinatory Modal Logic
15-816: Modal Logic
Frank Pfenning
Lecture 9
February 16, 2010
1
Introduction
The connection between proofs and program so far has been through a
proof term assignment for natural deduction. Proof reduction then fo
Lecture Notes on
Soundness and Correspondence
15-816: Modal Logic
Andr Platzer
e
Lecture 7
February 2, 2010
1
Introduction to This Lecture
In this lecture, we will cover the question how axiomatics and semantics
of modal logic t together. For corresponden
Lecture Notes on
Sequent Calculus
15-816: Modal Logic
Frank Pfenning
Lecture 8
February 9, 2010
1
Introduction
In this lecture we present the sequent calculus and its theory. The sequent
calculus was originally developed by Gentzen [Gen35] as a means to e
Lecture Notes on
Classical Modal Logic
15-816: Modal Logic
Andr Platzer
e
Lecture 5
January 26, 2010
1
Introduction to This Lecture
The goal of this lecture is to develop a starting point for classical modal
logic.
Classical logic studies formulas that ar
Lecture Notes on
Judgments and Propositions
15-816: Modal Logic
Frank Pfenning
Lecture 1
January 12, 2010
1
Introduction to This Course
Logic is the study of reasoning. Since most mathematicians believe they are
working on the discovery of objective and a
Lecture Notes on
Proofs as Programs
15-816: Modal Logic
Frank Pfenning
Lecture 2
January 14, 2010
1
Introduction
In this lecture we investigate a computational interpretation of intuitionistic proofs and relate it to functional programming. On the proposi
Lecture Notes on
Categorical Judgments
15-816: Modal Logic
Frank Pfenning
Lecture 3
January 19, 2010
1
Introduction
The main basic judgments we have considered so far are:
A true
A
A
M :A
A is true
A has a verication
A may be used
M is a proof term for A,
Lecture Notes on
Soundness of Modal Tableaux
15-816: Modal Logic
Andr Platzer
e
Lecture 11
Februrary 23, 2010
1
Introduction to This Lecture
In the last lecture, we have seen the modal tableau calculus. Now we see
why it is sound. Again, we refer to Fitti
Lecture Notes on
Backward Chaining
15-816: Linear Logic
Frank Pfenning
Lecture 17
March 26, 2012
In the last lecture we saw that it is difcult to identify subcomputations in
pure forward chaining. By comparison, in functional programming calling a functio
Lecture Notes on
Ordered Forward Chaining
15-816: Linear Logic
Frank Pfenning
Lecture 16
March 21, 2012
In the last lecture we saw ordered logic, which is in some sense even more
primitive than linear logic. We also saw a focusing system for it, which
is
Chapter 1
Introduction
In mathematics, one sometimes lives under the illusion that there is just one
logic that formalizes the correct principles of mathematical reasoning, the socalled predicate calculus or classical rst-order logic. By contrast, in phil
Chapter 2
Linear Natural Deduction
Linear logic, in its original formulation by Girard [Gir87] and many subsequent
investigations was presented as a renement of classical logic. This calculus of
classical linear logic can be cleanly related to classical l
Chapter 3
Sequent Calculus
In the previous chapter we developed linear logic in the form of natural deduction, which is appropriate for many applications of linear logic. It is also
highly economical, in that we only needed one basic judgment (A true) and
Chapter 3
Sequent Calculus
In the previous chapter we developed linear logic in the form of natural deduction, which is appropriate for many applications of linear logic. It is also
highly economical, in that we only needed one basic judgment (A true) and
6.5 Exercises
131
llist2 = . 1 !(A ). Here we can observe directly if the list is empty or
A
not, but not the head or tail which remains unevaluated.
llist3 = . 1 (A !). Here we can observe directly if the list is empty or
A
not, and the head of the list
5.4 Some Example Programs
99
For the completness direction we need to generalize the induction hypothesis
somewhat dierently.
Theorem 5.2 (Completeness of I/O Resource Management)
1. If ; = A then ; [1], O\[0], O = A for any O .
2. If ; ; A = P then ; [1]
Assignment 5
15-816: Linear Logic
Frank Pfenning
Due
Monday, April 9, 2012
This assignment consists of several, somewhat open-ended problems. You
should pick one of them to do. If you would like to do a half-semester
project instead, please submit your pr
Final Exam
15-816 Linear Logic
Frank Pfenning
May 8, 2012
Name:
Andrew ID:
Instructions
This exam is closed-book, closed-notes.
You have 3 hours to complete the exam.
There are 6 problems.
Ordered
Classical
Resource
Forward
Logic
Lin. Logic
Semantics
C
Final Exam
15-816 Linear Logic
Frank Pfenning
May 8, 2012
Name:
Andrew ID: fp
Sample Solution
Instructions
This exam is closed-book, closed-notes.
You have 3 hours to complete the exam.
There are 6 problems.
Ordered
Classical
Resource
Forward
Logic
Lin
Assignment 7
15-816: Linear Logic
Frank Pfenning
Due
Wednesday, May 2, 2012
This assignment consists of several, somewhat open-ended problems.
You should pick one of them, or any of the problems from
Assignment 5 or Assignment 6 that you have not done yet
Lecture Notes on
Computational Interpretations of Modalities
15-816: Modal Logic
Frank Pfenning
Lecture 4
January 21, 2010
1
Introduction
In this lecture we present the rst two of many possible computational
interpretations of intuitionistic modal logic.