Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Chapter 5
Linear Logic Programming
When we think of logic we generally rst consider it as a discipline concerned
with the study of propositions, truth, and inference. This may appear at rst
to be independent from any notion of computation. However, there
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Invited talk at TLDI12
Towards Concurrent Type Theory
Lus Caires
Frank Pfenning
Bernardo Toninho
Universidade Nova de Lisboa
[email protected]
Carnegie Mellon University
[email protected]
Carnegie Mellon University &
Universidade Nova de Lisboa
btonin
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Focuspreserving Embeddings of Substructural Logics in Intuitionistic Logic
Jason Reed
University of Pennsylvania
Philadelphia, Pennsylvania, USA
Frank Pfenning
Carnegie Mellon University
Pittsburgh, Pennsylvania, USA
Abstract
a signature of function symb
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
4.3 Unication
77
3. If ; ; = A then ; = A.
4. If ; ; A = C then ; (, A) = C
Proof: By straightforward simultaneous induction on the structure of the given
deductions.
2
Focusing eliminates a lot of nondeterminism regarding the choices among
possible infe
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
6.5 Termination
6.5
133
Termination
As the example at the end of the previous section shows, unrestricted recursive
types destroy the normalization property. This also means it is impossible to
give all recursive types a logical interpretation. When we ex
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
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
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
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
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
2.5 An Example: Finite Automata
2.5
27
An Example: Finite Automata
One of the simplest example of computation with state is provided by nite
automata. In this section we discuss possible ways to model nondeterministic
nite automata in linear logic.
We re
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Midterm Exam
15816 Linear Logic
Frank Pfenning
March 7, 2012
Name:
Andrew ID: fp
Sample Solution
Instructions
This exam is closedbook, closednotes.
You have 80 minutes to complete the exam.
There are 4 problems.
Lights
Substructural
Out
Logics
Focus
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
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]
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Midterm Exam
15816 Linear Logic
Frank Pfenning
March 7, 2012
Name:
Andrew ID:
Instructions
This exam is closedbook, closednotes.
You have 80 minutes to complete the exam.
There are 4 problems.
Lights
Substructural
Out
Logics
Focusing
Quotations
Prob
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
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
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
7.3 Logical Frameworks
151
from a domain with a decidable equality theory. This appears to be a reasonable
compromise that can make the expressive power of dependent types available
to the programmer without sacricing decidable and ecient typechecking.
7
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
116
6.2
Linear Calculus
Linear Type Checking
The typing rules for the linear calculus are syntaxdirected in that the principal
term constructor determines the typing rule which must be used. Nonetheless,
the typing rules are not immediately suitable fo
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Chapter 6
Linear Calculus
In philosophy we distinguish between the notion of analytic and synthetic judgment [ML94], a terminology which goes back to Kant. Briey, an analytic
judgment can be seen to be evident by virtue of the terms contained in it. A
sy
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Chapter 7
Linear Type Theory
The distinction between logic and type theory is not always clear in the literature. From the judgmental point of view, the principal judgments of logic are
A is a proposition (A prop) and A is true (A true). This may be diere
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
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 rstorder logic. By contrast, in phil
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
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
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Ecient Resource Management
for Linear Logic Proof Search
Iliano Cervesato1 , Joshua S. Hodas2 , and Frank Pfenning1
1
Department of Computer Science, Carnegie Mellon University
Pittsburgh, PA 152133891, USA
Email: [email protected]
2
Computer Sci
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Final Exam
15816 Linear Logic
Frank Pfenning
May 8, 2012
Name:
Andrew ID:
Instructions
This exam is closedbook, closednotes.
You have 3 hours to complete the exam.
There are 6 problems.
Ordered
Classical
Resource
Forward
Logic
Lin. Logic
Semantics
C
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Final Exam
15816 Linear Logic
Frank Pfenning
May 8, 2012
Name:
Andrew ID: fp
Sample Solution
Instructions
This exam is closedbook, closednotes.
You have 3 hours to complete the exam.
There are 6 problems.
Ordered
Classical
Resource
Forward
Logic
Lin
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Chapter 4
Proof Search
Linear logic as introduced by Girard and presented in the previous chapter is a
rich system for the formalization of reasoning involving state. It conservatively
extends intuitionistic logic and can therefore also serve as the logic
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Subnets of Proofnets in
MLL?
G. Bellin
J. van de Wiele
December 21, 1994
Abstract
The paper studies the properties of the subnets of proofnets. Very
simple proofs are obtained of known results on proofnets for MLL? ,
Multiplicative Linear Logic without
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Assignment 7
15816: Linear Logic
Frank Pfenning
Due
Wednesday, May 2, 2012
This assignment consists of several, somewhat openended problems.
You should pick one of them, or any of the problems from
Assignment 5 or Assignment 6 that you have not done yet
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Assignment 5
15816: Linear Logic
Frank Pfenning
Due
Monday, April 9, 2012
This assignment consists of several, somewhat openended problems. You
should pick one of them to do. If you would like to do a halfsemester
project instead, please submit your pr
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Lecture Notes on
Classical Linear Logic
15816: Linear Logic
Frank Pfenning
Lecture 25
April 23, 2012
Originally, linear logic was conceived by Girard [Gir87] as a classical system, with onesided sequents, an involutive negation, and an appropriate
law o
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Assignment 6
15816: Linear Logic
Frank Pfenning
Due
Wednesday, April 18, 2012
This assignment consists of several, somewhat openended problems.
You should pick one of them, or any of the problems from
Assignment 5 that you have not done yet.
If you have
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Lecture Notes on
Concurrent Monadic Computations
15816: Linear Logic
Frank Pfenning
Lecture 22
April 11, 2012
In this lecture we explore the parallelism inherent in forwardchaining logic
programming. As we have seen in the last lecture, forward chaining
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Lecture Notes on
Embedding Linear Logic in Intuitionstic Logic
15816: Linear Logic
Frank Pfenning
Lecture 24
April 18, 2012
In this lecture we continue our investigation of the resource semantics
for linear logic from the previous lecture. We rst conside
Birla Institute of Technology & Science, Pilani  Hyderabad
linear logic
MATH 2551

Spring 2014
Lecture Notes on
Resource Semantics
15816: Linear Logic
Frank Pfenning
Lecture 23
April 16, 2012
In this lecture we explore a new presentation of linear logic, one where resources are explicitly tracked in the judgments. It is a new form of semantics, gi