App eared in SIGACT
Linear Logic
Patrick Lincoln
lincoln csl sri com
SRI and Stanford University
Linear Logic
Linear logic was intro duced by Girard in Since then many results have supp orted
Girard s claims such as Linear logic is a resource conscious logic Increasingly computer
scientists have recognized linear logic as an expressive and p owerful logic with deep connec
tions to concepts from computer science The expressive p ower of linear logic is evidenced by
some very natural enco dings of computational mo dels such as Petri nets counter machines
Turing machines and others
This note presents an intuitive overview of linear logic some recent theoretical results
and some interesting applications of linear logic to computer science Other intro ductions
to linear logic may b e found in
Linear Logic vs Classical and Intuitionistic Logics
Linear logic di ers from classical and intuitionistic logic in several fundamental ways Clas
sical logic may b e viewed as if it deals with static prop ositions ab out the world where each
prop osition is either true or false Because of the static nature of prop ositions in classical
logic one may duplicate prop ositions P implies P and P Implicitly we learn that one
P is as go o d as two Also one may discard prop ositions P and Q implies P Here the
prop osition Q has b een thrown away Both of these sentences are valid in classical logic
for any P and Q
In linear logic these sentences are not valid A linear logician might ask Where did the
second P come from and Where did the Q go Of course these questions are nonse
quiturs in the classical setting since prop ositions are assumed to b e static unchanging facts
ab out the world On the other hand the rules of linear logic imply that linear prop ositions
stand for dynamic prop erties or nite resources
For example consider the prop ositions D M and C conceived as resources
D One Dollar
M A pack of Marlb oros
C A pack of Camels
Consider the following axiomatization of a vending machine
D implies M
D implies C
Then in classical or intuitionistic logic one is able to deduce
D implies M and C
Which may b e read as With one dollar I may buy b oth a pack of Marlb oros and a pack of
Camels Although this deduction is valid in classical logic it is nonsense in the intended
interpretation of prop ositions as resources one cannot buy two packs of cigarettes with one
dollar from the vending machine describ ed This paradox arises out of the confusion in
classical and intuitionistic logic b etween two kinds of conjunction one intuitively meaning
I have b oth which is written in linear logic as and another meaning I have a choice
written in linear logic
Linear logic avoids such paradoxes by distinguishing two kinds of conjunction two kinds
of disjunction and by intro ducing a mo dal storage op erator that explicitly marks those
prop ositions that can b e arbitrarily reused
? ? ?
Linear negation A is involutive that is A A but is yet constructive This is one
of the fascinating asp ects of linear logic
Linear logic multiplicative conjunction A B stands for the prop osition that one has
b oth A and B at the same time The linear logic additive conjunction A B stands for
one s own choice b etween A and B but not b oth Dually there are two disjunctions
The multiplicative disjunction written A B stands for the prop osition if not A then B
Perhaps this disjunction can b e more easily understo o d by considering linear implication
?
A B which is de ned by A B The formula A B can b e thought of as can B b e
derived using A exactly once The additive disjunction A B stands for the p ossibility of
either A or B but you don t know which That is someone else s choice
For each of these connectives there is a unit is the unit of so A A and
A A is the unit of is the unit of and is the unit of
Exp onentials
To complete the logic there is a mo dal storage op erator of course and its dual why
not The formula A may b e thought of as a printing press for A s which can generate any
numb er of A s For example the U S government can b e thought to have D ol l ar s and
do esn t need to balance its budget while citizens do not have D ol l ar s and thus have to
balance their budgets
Example
To illustrate the use of linear connectives and mo dal op erators here is an example inspired
by Girard and Lafont Supp ose for a xed price a restaurant will provide a hamburger
a Coke as many french fries as you like onion soup or salad your choice and pie or ice
cream some else s choice One may enco de this information in the linear logic formula
b eside the menu
Fixed Price Menu D D D D D
Hamburger
Coke H C F O S P I
All the french fries you can eat
Onion Soup or Salad
Pie or Ice Cream dep ending on availability
Sequent Calculus Notation for Linear Logic
The entire set of Gentzen style sequent rules for linear logic are given at the end of this note
As explained ab ove the rules de ne two conjunctions and two disjunctions as well as mo dal
and constant op erators One could add quanti ers to form rst or higher order linear logic
but for this pap er we will restrict attention to prop ositional linear logic
The sequent calculus notation due to Gentzen uses roman letters for prop ositions
and greek letters for sequences of formulas A sequent is comp osed of two sequences of
formulas separated by a or turnstile symb ol One may read the sequent as asserting
that the multiplicative conjunction of the formulas in together imply the multiplicative
disjunction of the formulas in
A sequent calculus proof rule consists of a set of hyp othesis sequents displayed ab ove a
horizontal line and a single conclusion sequent displayed b elow the line as b elow
Hyp othesis Hyp othesis
Conclusion
Connections to Other Logics
The most interesting features of linear logic arise from the absence of the rules of contraction
and weakening In classical or intuitionistic logic the following rules are allowed
A A
Contraction Left Weakening Left
A A
Direct and A ne logic share with linear logic the elimination of the contraction rule
i e prop ositions cannot b e arbitrarily copied However b oth of these logics allow weakening
Relevance and Pertinent logic disallow weakening but allow contraction i e prop ositions
cannot b e arbitrarily discarded but can b e copied Pertinent logic is decidable but
Relevance logic adds a distributivity axiom which is absent from linear logic which makes
Relevance logic undecidable Linear logic disallows b oth weakening and contraction in
general although they are allowed for mo dal and formulas
Linear logic arose partly out of a study of intuitionistic implication Girard found that
the intuitionistic implication A B could b e decomp osed into two separate connectives
A B Girard showed that one could thus translate intuitionistic and also classical logic
into linear logic directly simply app ending mo dals to certain subformulas and making the
right choice as to which sort of conjunction and disjunction should b e used Here we see a
rst glimpse of the substance b ehind the slogan Linear logic is a logic b ehind logics
Connections to Computer Science
There has b een much recent excitement ab out linear logic in the logic based theoretical
computer science community Most of this excitement stems from the newfound ability to
capture di cult resource problems logically For example linear logic provides a natural
and simple enco ding of Petri net reachability In linear logic the formula a c b may
b e used to enco de a Petri net transition taking tokens from place a and c and adding a token
to place b Similarly the formula b d d c d may b e seen as a transition taking
one token from b and two tokens from d and adding one token to c These transitions are
presented graphically b elow
A B
a c b
b d d c d
J
a c d d
J
J
J
J
J
C D
J
J
Thus one can enco de Petri net transitions as reusable linear implications Tokens are
represented as atomic prop ositions and a reachability problem may b e presented as a se
quent
a c b b d d c d a c d d c d
This sequent is provable in linear logic if and only if there is a sequence of Petri net
rule applications that transform the token set fa c d dg to fc dg This connection has b een
well studied and extended to cover other mo dels of concurrency
Linear logic has also b een applied to several other areas of computer science One key
application of the resource sensitive asp ect of the logic was the development of a functional
programming language implementation in which garbage collection was replaced by explicit
duplication op erations based on linear logic More recent work has attempted to nd
a linear logical basis for many optimizations in lazy functional programming language
implementations by concentrating on linear logic as a typ e system
Other applications include analyzing the control structure of logic programs general
ized logic programming and natural language pro cessing A natural characteri
zation of p olynomial time computations can b e given in a b ounded version of linear logic
obtained by limiting reuse to sp eci ed b ounds i e by b ounding the numb er of references
to each datum in memory
We now turn our attention to some questions of a more theoretical nature
Complexity Results for Linear Logic
Although prop ositional linear logic was known to b e very expressive for some time it was
thought to b e decidable However prop ositional linear logic was recently shown to b e un
decidable Several other complexity results are given in including the pspace
completeness of prop ositional linear logic without or
A key op en complexity problem is the decision problem for the fragment of
linear logic An equivalent fragment is which su ces for the enco ding of Petri
net reachability questions as shown ab ove and thus is at least expspace hard It is
currently unknown if this multiplicative exp onential fragment of linear logic is decidable or
not
Undecidability of Prop ositional Linear Logic
The full pro of of undecidability is presented in and is sketched b elow
The pro of of the undecidability of full linear logic pro ceeds by reduction of a form of alter
nating counter machine to prop ositional linear logic An and branching two counter machine
ACM is a nondeterministic machine with a nite set of states A con guration is a triple
hQ A B i where Q is a state and A and B are natural numb ers the values of two coun
i i
ters An ACM has a nite set of instructions of ve kinds Increment A Increment B
Decrement A Decrement B and Fork The Increment and Decrement instructions
op erate as they do in standard counter machines The Fork instruction causes a ma
chine to split into two indep endent machines from state hQ A B i a machine taking the
i
transition Q ForkQ Q results in two machines hQ A B i and hQ A B i Thus an instan
i j k j k
taneous description is set of machine con gurations which is accepting only if all machine
con gurations are in the nal state and all counters are zero ACM s have an undecidable
halting problem One may notice that ACM s are essentially alternating Petri nets It is
convenient to use ACM s as opp osed to standard counter machines to show undecidability
since zero test has no natural counterpart in linear logic but there is a natural counterpart
of Fork the additive conjunction The remaining ACM instructions may b e enco ded
using techniques very similar to the Petri net rechability enco ding describ ed earlier
Linear logic has a great control over resources through the elimination of weakening and
contraction and the explicit addition of a resuable mo dal op erator Although the logic
do es not have quanti ers the combination of these features yields a great deal of expressive
p ower
Complexity of Fragments of Linear Logic
PSPACE completeness of Linear Logic Without
The multiplicative additive fragment of linear logic MALL excludes the reusable mo dals
Thus every formula is used at most once in any branch of any cut free MALL
pro of Also in every non cut MALL rule each hyp othesis sequent has a smaller numb er
of symb ols than the conclusion sequent This provides an immediate linear b ound on the
depth of cut free MALL pro ofs Since MALL enjoys a cut elimination prop erty there is a
nondeterministic PSPACE algorithm to decide MALL sequents
To show that MALL is PSPACE Hard one can enco de classical quanti ed b o olean for
mulas QBF For simplicity one may assume that a QBF is presented in prenex form The
quanti er free formula may b e enco ded using truth tables but the quanti ers present some
? ?
di culty One may enco de quanti ers using the additives x as x x and x as x x
This enco ding has incorrect b ehavior in that it do es not resp ect quanti er order but using
multiplicative connectives one can enforce an ordering up on the enco ding of quanti ers to
achieve soundness and completeness The full pro of of pspace completeness is presented
in
NP completeness of Multiplicative Linear Logic
The multiplicative fragment of linear logic contains only the connectives and or equiv
alently and a set of prop ositions and the constants and The decision problem
for this fragment is in np since an entire cut free multiplicative pro of may b e guessed and
checked in p olynomial time The decision problem is np hard by reduction from Partition
a problem which requires a p erfect partitioning of groups of ob jects in much the same way
that linear logic requires a complete accounting of prop ositions Somewhat surpris
ingly there is an alternate enco ding of Partition in multiplicative linear logic that do es not
use any prop ositions that is using only the constants and and the connectives and
Thus this multiplicative constant only fragment of linear logic is also np complete
The full pro of of np completeness is presented in
Summary of Linear Logic Complexity
The following table summarizes some of the results thus far achieved in the study of the
complexity of the decision problems for fragments of linear logic The rst two fragments
listed are those discussed in some detail ab ove the full prop ositional logic and the prop osi
tional fragment without mo dals MALL The decidability of the next problem is in some
question An enco ding of Petri net reachability problems in this fragment has b een studied
in but although expspace hard Petri net reachability is known to b e decidable
It is not known how much more expressive this fragment of linear logic might b e The fourth
fragment containing only the multiplicative connectives is NP Complete
Connectives Linear Logic
In Fragment Complexity
Undecidable
PSPACE Complete
unknown
NP Complete
In summary linear logic is an expressive logic with an intrinsic accounting of resources
Although a non classical logic linear logic has the pleasant features of cut elimination and in
volutive negation In practical use much mileage can b e gained from the resource sensitivity
of linear logic to enco de di cult problems in even the prop ositional fragment of linear
logic Current work is progressing to exploit the unique features of linear logic for use
as a typ e system to study computational complexity and compiler optimization tech
niques as well as uses in logic programming natural language
pro cessing and concurrency These recent contributions are developing
linear logic from a theoretical curiosity into a to ol that already has practical use within
mainstream computer science
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De nition of Linear Negation
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p p p p
i i
i i
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A B B A A B B A
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A B A B A B A B
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A A A A
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Sequent Calculus Rules for Linear Logic
` A A `
1 1 2 2
Cut Identity
A ` A
`
1 2 1 2
` A B A B `
1 2 1 2
Exch Right Exch Left
B A ` ` B A
1 2 1 2
A B ` ` A ` B
1 1 2 2
Left Right
A B ` ` A B
1 2 1 2
` A B ` A ` B
1 1 2 2
Left Right
A B ` ` A B
1 2 1 2
` A B A ` B `
1 1 2 2
Right Left
A B ` ` A B
1 2 1 2
` A ` B
A ` B `
Right Left
A B ` A B `
` A B
A ` B `
` A ` B
Right Left
` A B ` A B
A B `
` A A `
W C
A ` A `
` A A `
S D
A ` ` A
` ` A A
W C
` A ` A
` A A `
D S
` A A `
A ` ` A
? ?
Right Left
? ?
A ` ` A
Left ` ` > > Right
`
? Right ? Left ? `
` ?
`
Left ` Right
`