A Survey of Linear Logic Programming
Computer Science Department
University of Pennsylvania
Philadelphia, PA 19106-6389 USA
dale @ saul.cis.upenn.edu
This survey will appear in the third issue of the
Newsletter of the Network of Excellence on Computational Logic.
It is also available in Postscript
and DVI formats.
I have written a more up-to-date
overview paper on this topic.
It is now common place to recognize the important role of logic in the
foundations of computer science in general and programming languages more
specifically. For this reason, when a major new advance is made in our
understanding of logic, we can expect to see that advance ripple into other
areas of computer science. Such rippling has been observed during the past
eight years since the first introduction of linear logic [Girard 1987]. This exciting advance
in logic provides new ways of embracing aspects of computation directly in
a rich logical framework. Since this development extends and enriches our
understanding of classical and intuitionistic logic, it provides new
insights into the many computational systems built on those two logics.
2. Applications of Linear Logic to Programming Languages
Both functional and logic programming have made extensive use of
classical and intuitionistic logic to motivate language designs and to
analyze programs and specifications.
One inspiration for the design of functional programming languages is the
Curry-Howard Isomorphism. This isomorphism states that programs and
proofs can be equated and that the normalization of proofs (say, by
beta-conversion or cut-elimination) can be seen as computation. Linear
logic supplies new proof structures, called proof nets, and the dynamics
of their normalization can be used to express some aspects of
Bellin & Scott 1992,
The Curry-Howard Isomorphism also states that the types of programs can be
seen as formulas, and the richer formulas of linear logic allow for more
expressive types. Such stronger types have been used to help provide
static analysis of such things as run-time garbage, aliases, reference
counters, and single-threadedness
[Guzmán & Hudak 1990,
Maraist et. al. 95,
Chirimar et. al.].
Linear Logic has also shown promise in helping with the analysis of
conventional logic programs. See, for example, the work of Cerrito
on specifying the semantics of various
aspects of Prolog using linear logic
and of Reddy in specifying modes using linear logic
The most active work on using linear logic in logic programming, however,
has been in the area of designing and using new logic programming
3. New Logic Programming Languages
In the field of logic programming there does not seem to be a
principle, like that of the Curry Howard Isomorphism, that is at once
simple, natural, deep, and generally accepted as a design principle.
Although most early work on logic programming had been fixed on a
particular logic, namely that of first-order, classical Horn clauses,
many researchers have recently adopted proof search in sequent calculi
as a setting for designing and exploring the dynamics and properties
of logic programs. In this setting, sequents are used to denote the
state of a computation and the transformations that occur to sequents
as cut-free proofs are incrementally constructed are used to model the
dynamics of computation.
When few logical connectives are needed, it is often straightforward
to define a logic that has a natural
operational semantics (meaning that it is easy for a programmer to
understand how proofs are attempted). The following three designs
are examples of such linear logic programming languages.
Each of these languages incorporate small subsets of linear logic and
use multisets of formulas to capture object structure or collections of
processes and messages, and use multiset rewriting to capture inheritance and
- LO (Linear Objects)
[Andreoli & Pareschi 1990,
Andreoli & Pareschi 1991]
was designed by Andreoli and Pareschi as an
extension to the Horn clause paradigm in which atomic formulas are
generalized to be multisets of atomic formulas connected by multiplicative
disjunctions ("pars"). In LO, backchaining
becomes multiset rewriting. This language has been used to specify
object-oriented programming and the coordination of processes.
- ACL by Kobayashi and Yonezawa is an asynchronous calculus in
which the send and read primitives were essentially identified to two
complementary linear logic connectives
[Kobayashi & Yonezawa 1993,
Kobayashi & Yonezawa 1994].
- Lincoln and Saraswat, in unpublished reports, developed
a linear version of concurrent constraint programming and used linear logic
connectives to extend previous languages in this paradigm
[Lincoln & Saraswat 1993,
One design principle that has been used in recent years states that
goal-directed search should be complete for logic programs.
Within intuitionistic logic, this was first formalized using the the proof
theoretic notion of uniform proofs [Miller et. al 1991]. Horn clause
logic and hereditary Harrop formulas (the logic underlying lambda Prolog)
are both examples of settings where goal-directed search is complete for
intuitionistic provability. This definition of goal-directed search
depends on the fact that sequents in intuitionistic logic are
single-conclusion; that is, they contain a single conclusion (the
goal) to be proved. It was therefore straightforward to extend the
definition of uniform proofs to intuitionistic linear logic (where sequents
again have a single conclusion), and Hodas and Miller have used that
extension to design the linear logic programming language Lolli
Hodas & Miller 1994].
Lolli can be seen as a modular extension to lambda Prolog that allows items
in the program to be use either once or an unlimited number of times.
Linear logic's connectives can be used to provide elegant and flexible
management of both kinds of program clauses.
In the sequent characterization of full linear logic, sequents can have
multiple formulas in the conclusion: that is, the goal to be proved may be a
multiset of formulas whose provability cannot be isolated from each
other and where each formula can assist each other in some (hopefully,
Given this structure of goals, the notion of goal directed search and
uniform proof needed to be extended. There appear to be two ways to make
this extension. In one approach a goal with multiple parts is required to
have some component goal that can be reduced. This approach, used
by Harland and Pym, is the weaker approach and goal-directed search
would be complete for many subsets of linear logic
[Pym & Harland 1994],
some of which have complex operational
semantics [Harland & Pym 1992].
See their Lygon language [Harland &
for example. Another approach requires that in a goal with multiple
component goals, all components must be simultaneously reducible.
Miller first used this definition in [Miller 1992]
to provide a linear logic encoding of the pi-calculus. He later
showed that by selecting a suitable and complete set of
connectives, all of linear logic can be seen as logic programming. This
particular presentation of linear logic, called Forum
[Miller 1994], can be seen (and
motivated) as an extension of LO, lambda Prolog, and Lolli (but not of Lygon).
Hodas is currently developing a prototype implementation of Forum based on
techniques used in the implementation of Lolli.
4. Applications for new languages
Not all of these designs have been undertaken as purely technical exercises
involving the structure of sequent calculus proofs. In fact, mnay of these
designs have been motivated by the need to make logic specifications more
expressive. Some of the resulting systems have supplied new and useful
specifications in various domains.
- Many of these programming languages were designed, at least in part,
to allow concurrent specifications
[Kobayashi & Yonezawa 1993,
Kobayashi & Yonezawa 1994,
Lincoln & Saraswat 1993,
[Bruscoli & Guglielmi 1995].
- Object-oriented programming
- Capturing state and inheritance was an early goal of the LO system
[Andreoli & Pareschi 1991]
and a motivation for the design of Lolli.
Another approach to state encapsulation can be found in
[Miller 1994] and in
[Delzanno & Martelli 1995].
- Operational semantics
- Forum has been successfully used to specify the operational semantics
of imperative and concurrent features such as those in Algol and ML
Miller 1994]. Chirimar has also
specified in Forum the operational semantics of a pipe-lined, RISC processor
- Natural language parsing
- Lolli has provided a declarative approach to
gap threading within English relative clauses
- Object-logic proof systems
- Lolli has been used to refine the usual, intuitionistic specifications
of object-level natural deduction systems
[Hodas & Miller 1994] and Forum
has been used to provide specifications of object-level sequent systems
5. Research in Sequent Calculus Proof Search
Since the majority of linear logic programming are described using
sequent calculus proof systems, a great deal of work in understanding and
implementing these languages has focused on properties of proofs, rather
than on model theoretic considerations.
In recent years, results in proof theory
have been developed specifically to support this foundation of the logic
Andreoli developed some deep results about proof
search in linear logic in his PhD thesis
[Andreoli 1990a] (see also
There is also related work by Galmiche, Boudinet, and Perrier
[Galmiche & Boudinet 1994,
Galmiche & Perrier 1994],
[Tammet 1994], and others.
A problem specific to proof search in linear logic
is how to effectively split resources
between conjunctive branches of a computation.
For Lolli, Hodas and Miller developed a lazy splitting approach,
called the input-output model of resource consumption
[Hodas & Miller 1994,
various researchers are actively refining and extending that model
(see, for example, [Cervesato
Possible future projects include exploring how to exploit Girard's LU proof
system [Girard 1993] and
definitional reflection [Schroeder-Heister 1993].
Also, since linear logic is a rich and expressive logic, finding
interesting subsets of it that can be given effective implementations is
currently an open problem.
7. Various WWW pages of related interest
I wish to thank I. Cervesato, F. Pfenning, and C. Schürmann for maintaining
on Linear Logic: it proved valuable in assembling this survey.
Josh Hodas and James Harland provided some useful comments on a draft of this document.
I am also pleased to acknowledge support from ONR N00014-93-1-1324,
NSF CCR-91-02753, NSF CCR-92-09224, and DARPA N00014-85-K-0018.