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Linear logic is one of the logical systems with special properties suitable for describing real processes used in computer science. It enables one to specify dynamics, non determinism, consecutive processes and important resources as memory and time on syntactic level. Moreover, its deduction system enables one to verify specified properties. Constructing an appropriate model based on categories can serve for modeling various program systems in the wide spectrum of computer science. Mainly, propositional linear logic is used for these purposes. The expression power of linear logic significantly grows by extending propositional logic with predicates and quantifiers. Our paper concerns itself with defining predicate linear logic together with its deduction system and our main aim is to construct a categorical model of predicate linear logic as a symmetric monoidal closed category.
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Journal of Applied Mathematics and Computational Mechanics 2015, 14(1), 27-42
www.amcm.pcz.pl p-ISSN 2299-9965
DOI: 10.17512/jamcm.2015.1.03 e-ISSN 2353-0588
A CATEGORICAL MODEL OF PREDICATE LINEAR LOGIC
Emília Demeterová, Daniel Mihályi, Valerie Novitzká
Department of Computers and Informatics, Faculty of Electrical Engineering and Informatics
Technical University of Košice
Košice, Slovak Republic
emilia.demeterova@tuke.sk, daniel.mihalyi@tuke.sk,
valerie.novitzka@tuke.sk
Abstract. Linear logic is one of the logical systems with special properties suitable for
describing real processes used in computer science. It enables one to specify dynamics, non
determinism, consecutive processes and important resources as memory and time on syn-
tactic level. Moreover, its deduction system enables one to verify specified properties.
Constructing an appropriate model based on categories can serve for modeling various pro-
gram systems in the wide spectrum of computer science. Mainly, propositional linear logic
is used for these purposes. The expression power of linear logic significantly grows by
extending propositional logic with predicates and quantifiers. Our paper concerns itself
with defining predicate linear logic together with its deduction system and our main aim is
to construct a categorical model of predicate linear logic as a symmetric monoidal closed
category.
Keywords: linear type theory, predicate linear logic, symmetric monoidal closed category
Introduction
Linear logic was introduced by Jean-Yves Girard in 1987 . Linear logic is the
only logic from the existing logical systems which is able to describe processes as
they behave in real world. It can describe dynamics of processes, external or inter-
nal non determinism, consecutive processes manage with resources on a syntactic
level. Linear logic can be considered as a bridge between computer science and
logic.
Propositional linear logic is often used for describing program systems [2, 3],
their behavior  and its extension with modal operators enables the modeling of
knowledge achievement .
In spite of these useful properties of propositional linear logic, its expressive
power is insufficient for describing properties of some objects and relations
between them. For these purposes are appropriate predicates together with quanti-
fiers as it provides predicate logic. Therefore in this paper we formulate predicate
linear logic, its syntax, deduction system and categorical model. Calculations can
be expressed by linear terms. Properties and relations of calculations can be
E. Demeterová, D. Mihályi, V. Novitzká
28
expressed by predicates. Quantifiers are able to specify a group of objects for
which some feature or relation is valid. In the future we plan to use predicate linear
logic for specifying component based systems , especially for describing inter-
actions and dependencies between components. In the next section we shortly
introduce propositional linear logic, its syntax and deduction calculus. In section 2
we define the linear type theory and its model as a symmetric monoidal closed cat-
egory. Section 3 contains syntax and deduction calculus of a multiplicative frag-
ment of predicate linear logic and in section 4 we construct a categorical model of
predicate linear logic.
1. Propositional linear logic
Linear logic has much greater expressive power than classical logic thanks to
more connectives with special properties. Besides, every formula of classical prop-
ositional logic can be expressed in linear logic, too.
One of the most important properties of linear logic is its ability to describe
dynamics of processes. By linear implication it is able to express changes,
sequentiality and causality of processes.
Another important property of linear logic is its ability to handle resources -
with logical space and logical time . It allows one to express the internal struc-
ture of the resources, their consumption together with a continuance of processes
by the incrementation of time. Resources used in linear logic, logical space and
logical time are the most important resources used in computer science, too. These
resources are used at calculating a running program.
In same cases it is advisable to use only some fragments  of linear logic, e.g.
multiplicative, additive. An interesting fragment is intuitionistic linear logic which
satisfies Curry-Howard correspondence [8, 9] with the computing system such as λ
- calculus.
Linear logic defines new logical connectives . Depending on the fragments
of linear logic there exist multiplicative conjunction , multiplicative disjunction
haustibility of resources can be described by modal operators ! of course and ? why
not.
1.1. Syntax of propositional linear logic
In this section we introduce the syntax of propositional linear logic. Let
Prop = {p
1
, p
2
, … , p
n
}
be a countable set of elementary sentences denoted by p
1
, p
2
, . Every formula
can be understood either as a resource or as an action. A linear formula φ has one
of the following forms defined by the BNF rule:
A categorical model of predicate linear logic
29
φ ::= p | 1 | 0 | T | | φφ | φφ | φ&φ | φϕ | φφ | φ
| !φ | ?φ
Formula φ ψ expresses a multiplicative conjunction and it has the neutral
element 1. This formula expresses that both actions are performed simultane-
ously or both resources are available at once.
Formula φ & ψ expresses an additive conjunction and it has the neutral
element T. This formula expresses that only one of the actions is performed but
we can anticipate or deduce which one from the environment. We can call this
external non determinism. In the case of resources only one of them is available.
Formula φ ψ expresses additive disjunction and it has the neutral element 0.
This formula expresses that only one of the actions is performed but we cannot
anticipate which one. We can call this internal non determinism.
Formula φ℘ψ expresses multiplicative disjunction and it has the neutral
element
. This formula expresses that if the first action is not performed then
the second one is performed or vice versa.
Expression φ
is called linear negation and it denotes a reaction of an
action φ or a consumption of a resource φ. Linear negation is involutive
φ
⊥⊥
φ.
Formula is a modal formula with the modal operator of course. It expresses
the potential inexhaustible resource φ.
Formula uses the modal operator why not and it expresses the actuality of
potential resource inexhaustibility. Modal operators of course and why not are
dual
(!φ)
≡ ?(φ
).
Formula φ ψ expresses linear implication. This formula expresses that the
first action is a cause of the second action or in the case of resources, it express-
es that the first resource is consumed after linear implication.
Linear implication expressed by the modal operator of course !
!φ φ
is an analogy to classical implication φ
φ.
1.2. Deduction calculus for linear logic
We describe the deduction system of linear logic by sequent calculus defined by
Gentzen. A sequent has the form
φ
1
,…, φ
n
ψ
1
,…, ψ
m
,
E. Demeterová, D. Mihályi, V. Novitzká
30
where formulae φ
1
,…, φ
n
are assumptions implying at least one of the formulae
ψ
1
,…, ψ
m
.
The deduction system of linear logic consists of the rules for the connectives, con-
stants and modal operators of linear logic.
The deduction rules have the form

, …,
where the sequence of sequents S
1
,...,S
n
contains conditions that have to be valid in
order to deduce the conclusion S.
The deduction system of propositional linear logic has the following deduction
rules:
() (1)
,,
,
,
() (2)
,,
,
(⊗
) (3)
,
,
,
,,
(⊗
) (4)

,
(⊤
) (5)

(⊤
) (6)
,,
,
,,
(℘
) (7)
,,
,
(℘
) (8)
(⊥
) (9)

,
(⊥
) (10)
,,
,
,,
(⊸
) (11)
,,
.
(⊸
) (12)
,
,!
(
) (13)
!,?
!⊢!,?
(
) (14)
!,⊢?
!,?⊢?
(
) (15)
,
⊢?,
(
) (16)

,!
(
) (17)

⊢?,
(
) (18)
,!,!
,!
(
) (19)
⊢?,?,
⊢?,
(
) (20)
,
,&
(&

) (21)
,
,&
(&

) (22)
,,
&,
(&
) (23)

,
(
) (24)
A categorical model of predicate linear logic
31
,,
,
(⊕
) (25)
,
,
(⊕

) (26)
,
,
(⊕

) (27)

,
(
) (28)


≡ ( 

)
(29)
Many approaches exist to express the semantics of predicate linear logic.
Primarily the semantics of linear logic was expressed by coherent spaces [1-10] or
by quantales . Phase semantics  does not say anything about connectives
and expresses only the truth of the statements. Heyting's semantics  is not
interested in the truth of the expressions, but their sense. This semantics is im-
portant only if there exists a provable formula.
In constructing a model of predicate linear logic, we follow the idea that every
elementary linear formula can be represented as a type. First, we define linear type
theory and then we construct a categorical model of predicate linear logic.
2. Linear type theory
To introduce predicates into linear logic, we have to define types and linear
terms. Every programming language includes some predefined basic types. Let
B = {X, Y, Z} be a set of basic types and I the unit type. The syntax of linear types
 are defined by the following BNF grammar:
σ ::= X | I | σσ | σσ
where
X is a basic type;
I is the unit type;
σσ is a linear product type;
σσ is a linear function type.
2.1. Linear terms
Linear terms are expressed by operations and variables. First, we define linear
preterms.
Let Var(σ) be a set of variables of type σ. Let Preterm σ be a set of preterms types
of σ constructed as
() Preterm (I) is an empty linear preterm;
x Preterm (σ) is a linear preterm of types σ, if xvar(σ) is a variable of
type σ;
E. Demeterová, D. Mihályi, V. Novitzká
32
(s,t)Preterm(στ) is a linear preterm of product type στ, if
sPreterm(σ) and t Preterm(τ) are linear preterms;
α(s)Preterm(τ) is a linear preterm of type τ if s Preterm(σ) is a linear
preterm and α: σ τ is a function.
Linear terms have the following syntax:
t ::= x | f(t,...,t)
i.e. a term is either a variable or application of function. Every term has associated
a unique type and we denote typed terms in sequent form as
Γ
t: σ
where Γ is a finite sequence of typed variables x
1
1
,...,x
n
:σ
n
.
Linear term s of type σ is a preterm sPreterm(σ) in which every variable
occurs only once. A linear combinator is a closed linear term, i.e. all variables
occuring in the term are bounded. We introduce several linear combinators
expressing special properties needed for modeling the linear type theory:
linear combinator Id
σ
expresses identity on the type σ
Id
σ
:σ → σ;
linear combinator assl expresses left associativity on types σ, τ, χ
assl
σ, τ, χ
:σ(τ χ) → (σ τ) χ;
linear combinator assr expresses right associativity on types σ, τ, χ
assr
σ, τ, χ
: τ)χ → σ(τ χ);
linear combinator swap expresses commutativity on type
swap
σ,τ
:σ τ → τ σ;
linear combinator open expresses left neutral element
open
σ
:σ → 1σ;
linear combinator close expresses right neutral element
close
σ
:1 σ → σ;
linear combinator eval expresses evaluation over types
eval
στ
:τ)σ→ τ.
A categorical model of predicate linear logic
33
We can construct generalized combinators with the rules of composition, prod-
uct and abstraction  from linear combinators and function symbols where α, β, γ
denote either function or combinator:
::
:()
Linear combinator composition expresses composition of functions α and β.
::
:( !")
Linear combinator product expresses tensor product between α and β, and the
result is a tensor product of corresponding types.
:
Λ(): ⟶ ()(#\$ #)
Linear combinator abstraction expresses a map from the argument σ on the
function type τ θ.
In order to define equivalence between terms, we introduce relation of equivalence
of terms s
σ
t. Equivalence means that terms will have the same value after the
evaluation, so for any functions f, g term( σ τ) and basic term x:σ hold
if eval
σ,τ
(f,x)
τ
eval
σ,τ
(g,x), then f ≡
σ
g.
Properties of defined linear combinators are defined by the following axioms:
%!
()
(&)() &(())
#'
,,
(, (,"))
#
,,
((,, )")
(#
,
(,)
)
()
')
((), )
)*#'
,
(+()(), )
(,), ")
(, (,"))
(,")
((), )
(,)
where s, t, u are terms with same types and variables, α, β, γ are combinators.
Free variables can be substituted by terms of same type in terms. Evaluation of
terms are gained by their substitution. If s is a linear term of type σ, sterm(σ),
and x is a variable of type σ, xvar(σ), then t[s/x] expresses linear term t, where
every free occurrence of variable x is substituted with a linear term s. Substitution
is defined as following:
E. Demeterová, D. Mihályi, V. Novitzká
34
,-[ .] = ,-;
=,-;
01 .
2=344. =0;
044. 0;
,,"-1 .
2
,-1 .
2=
=,[ .], "[ .]
⁄ -;
([ .]);
where t and u are linear terms of optional type, α is functional symbol or combina-
tor between linear types.
We call the expression t[s/x] linear term, if terms t and s do not have the same var-
iables.
2.2. Categorical model of linear type theory
As a basis for a categorical model of linear type theory we use symmetric
monoidal closed category [15, 16]. The reason for this choice was formulated by
the following facts:
a type can be represented as an object in Cartesian closed category . Symmet-
ric monoidal closed category is a generalization of it.
For arbitrary monoidal closed category it exists a linear type theory [14, 17],
whose model is this category.
Terms can be represented by the categorical morphisms .
Symmetric monoidal closed category [17, 19] is defined by a sixtuple
(ℭ,⊗, %,#,',,5(−, −)),
where
is a category;
: x is a tensor product;
I is the neutral element of the tensor product, it is an object in . This object
serves as terminal object of the category ;
a
X,Y,Z
:(XY) Z X (YZ) is a natural isomorphism which expresses left
associativity of tensor product, X, Y, Z are objects in ;
l: IX → X is a natural isomorphism expressing left neutral element of the ten-
sor product;
c
X,Y
: XY YX is a natural isomorphism expressing commutativity of the
tensor product;
for every object X in the functor ─X has a right adjoint Hom-functor Hom
(X,─) with natural transformations
ε
X,Y
: Hom(X,Y) X →Y
δ
X,Y
: X → Hom(Y, XY).
Now we construct symmetric monoidal closed category as a model of linear
type theory. Category objects of the are type contexts Γ,,..., which can be real-
ized as finite products of types
A categorical model of predicate linear logic
35
σ
1
... σ
n
.
Morphisms in are linear terms x
1
:σ
1
,..., x
n
: σ
n
t:τ expressed as morphisms:
σ
1
... σ
n
τ;
In order to express the semantics of linear type theory we have to define inter-
pretation functions for objects and morphisms. The interpretation function for the
objects is
i:7
obj
and it assigns an object in
obj
to every type from 7 as follows:
,-=89
,%-=8%9 ,-=8989
,-=5(89,89)
The interpretation function for morphism
j:7erm
morp
assigns a morphism in
morp
for every typed linear term from a set 7erm.
:: t: σ ⟶ 89:8σ989
Combinators of linear type theory are interpreted in as follows:
:(%!
) = !
:(#
,,
) = #
,,
:(#'
,,
)
:((#
,
)
:()
)
:(')
)
:()*#'
,
)
=
=
=
=
=
#
,,

,
'
'

;
,
We denote the composition of combinators α and β, where α: σ
τ and
β: τ θ, as βα. We interpret that as a composition of morphism in category
j(βα) = j(β) ∘ :(α)
The semantics of the linear type theory is a pair of functions
(i,j).
3. Multiplicative fragment of predicate linear logic
After defining linear type theory we can introduce predicate linear logic.
We use only multiplicative fragment of linear logic that can be modelled in sym-
metric monoidal closed category. The multiplicative fragment of predicate linear
E. Demeterová, D. Mihályi, V. Novitzká
36
logic  contains the multiplicative connectives with their neutral elements from
the propositional linear logic. A linear formula φ has a form defined by the follow-
ing BNF rule:
φ ::= p | 1 |
| φφ | φφ | φφ | φ
| !φ | ?φ | P(t
1
,…,t
n
) |
x
φ|
x
φ,
where:
formula P(t
1
,…,t
n
) is a predicate expressing relations between terms (t
1
,…,t
n
);
formula
x
φ expresses universal quantifier applied on the formula and
x
φ
expresses existential quantifier applied on the formula. Quantifiers bind occur-
rence of variables.
In accordance to arity we can distinguish predicate symbols as:
unary P(t) which expresses some property of a term t;
binary P(t
1
,t
2
) which expresses some binary relation between terms t
1
and t
2
;
n-ary P(t
1
,…,t
n
) which expresses n-ary relation between terms (t
1
,…,t
n
).
Quantifiers have the same meaning as in the predicate logic and De Morgan's
rules are valid:
(∀<=)
≡ ∃<=
(∃<=)
≡ ∀<=
(30)
De Morgan's rules are also valid for predicates:
(>(?))
>
(?)
(>
(?))
>(?) (31)
Because we added new forms for the formulae, we have to extend deduction sys-
tem with new deduction rules. For the quantifiers we have the following deduction
rules:
,[ 
]⊢
,(∀)
(∀<
) (32)
,
⊢(∀),
(∀<
) (33)
,
,(∃)
(∃<
) (34)
 
⁄ ,
,
(∃<
) (35)
where Γ is a finite sequence of linear formulae φ
1
,…, φ
n
and in the rules
∀x
L
and
∃x
R
x have no free occurrence in Γ and . These rules express how to introduce the
universal and the existential quantifier into formula φ.
4. Categorical model of multiplicative fragment of predicate linear logic
In the previous section we constructed the semantics of linear type theory as
symmetric monoidal closed category. Next we construct a categorical model of the
multiplicative fragment of predicate linear logic using the symmetric monoidal
closed category, defined in previous sections.
A categorical model of predicate linear logic
37
Any elementary sentence is interpreted as an object, basic type in category .
Neutral element
1%
is interpreted as terminal object I of the category . The neutral element is dual
to 1
1
,
therefore from the properties of category is interpreted as initial object of .
For the interpretation of negation we use the following equivalence:
8@
98@9
According to  we interpret every sequent
φ
1
,…,φ
n
ψ
as morphism
8@
9⊗ ⋯ ⊗ 8@
98A9
in , where 8@9expresses object, a representation of the formulae φ in .
Connectives are interpreted as morphisms in category as follows:
8@98A98@A9
8@98A98@A9
8@98A98A
9
Because the symmetric monoidal closed category is Cartesian closed category, the
existence of objects 8@A9, 8@A9and exponential object 8A
9 arises from its
properties .
Now we define interpretation of unary predicate P(t). A predicate P(t) is
a property of the value of a term t of type σ. Because 89 is an object in , the
interpretation of the predicate P(t)
8B()989
is a subset of the object 89 in the .
We define functors and adjoint functors in order to express the semantics of the
modal operator of course !, the existential and universal quantifiers.
A functor is morphism between categories. A functor F: ℭ C is a pair of
functions (F
0
,F
1
) 
D
!
: ℭ
"#\$
C
"#\$
D
: ℭ
%"&'
C
%"&'
,
E. Demeterová, D. Mihályi, V. Novitzká
38
for which holds:
if f : A → B is morphism in then F
1
(f) : F
0
(A) → F
0
(B) in C;
for any object A in holds F
1
(id
A
}) = id
F0(A)
;
if the composition gf is in then the composition F
1
(g) F
1
(f) is defined in C
and holds F
1
(gf) = F
1
(g) F
1
(f).
A functor D: ℭ ⟶ is called endofunctor over category . We define adjoint
endofunctors over category .
Let D: ℭ and E: ℭ be endofunctors  and Id be identity functor. We
say that:
F is a left adjoint to the endofunctor G, F G and
G is a right adjoint to the endofunctor F, G F
if there is a natural transformation
η:Id →G F
such that for any objects A, B in and any morphism f: A → G(B) it exists unique
morphism
g: F(A) → B
in , for which holds
f = G(F) ∘ ηA.
Adjunction means that there exact correspondence exists between morphisms
A → G(B) and B → F(A), i.e. the Homsets
GHI,J,K-,L-GHI(K,M,L-) (36)
are isomorphic. Adjunction can be illustrated also by the following commuting
diagram.
The property (36) is useful in defining semantics of modal operator of course !
as follows. Let F and G be a pair of adjoint endofunctors
F G
in . We define this modal operator as a composition
8!9:E D
A categorical model of predicate linear logic
39
such that for any object
  ! 
it returns an object isomorphic with , i.e. we can model unexhaustible resource
by composition of adjoint functors as it is illustrated in Figure 1.
To interpret quantifiers we use adjoint functors, too.
Fig. 1. Model of predicate linear logic
Let  be an interpretation of the unary predicate symbol, where
is an object in . We consider a variable y:τ and we construct a predicate P(t,y)
interpreted as
, ,
where y has no free occurrence in t. We construct an auxiliary endofunctor H as
follows:
H:   ⟶  ,
where ) is a power set over .
Now we define a left adjoint functor of H that is interpretation of existential
quantifier
H
as
:  ⟶ ,
E. Demeterová, D. Mihályi, V. Novitzká
40
that for quantified formula y.P(t,y) returns a value of type τ (if it exists) satisfying
predicate P(t,y):
80.B(,0)9=N8989| exists a value in 898B(,0)9}.
Because of duality between existential and universal quantifiers in (31), we
interpret universal quantifier as a right adjoint to the auxiliary functor H
H ⊣ ⟦∀⟧.
Interpretation of quantifiers by adjoint functors is illustrated in the Figure 1.
In the following text we explain how deduction and proofs can be interpreted by
morphisms in our model.
The identity rule
@@(id)
is interpreted as identical morphism:
!
(
:8φ98φ9.
The proofs
)
,
*
+
,
+
are interpreted as morphisms
4:8O98@9 F:8O98A&Δ′9
in category . The proof of the multiplicative conjunction
O@, Δ
,
O
A, Δ′
O,O
@A, Δ, Δ′ (⊗
-
)
is interpreted as morphism
8O98O9
./
P
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
R
8@A&S&S9
in category .
Our model of predicate linear logic is constructed as a symmetrical monoidal
closed category of types together with appropriate adjoint functors for modal oper-
ator and quantifiers. A proof of a formula is modeled as a finite path of category
morphism.
A categorical model of predicate linear logic
41
Conclusions
Linear logic has many important properties useful for describing and verifying
various program systems. Its dynamic nature, expressing causality, non
determinism and handling resources make it the most appropriate logical system
for computer science. In this paper we defined predicate linear logic, its deduction
system and we constructed categorical model based on symmetric monoidal closed
category. This category with types as objects enables direct connection with
computing, where types and typed data structures play an important role.
Categories provide many useful structures that we can also use in constructing
models of logical systems. In this paper we used special properties of adjunct
endofunctors for modeling the modal operators expressing non exhaustibility of
resources and for modeling quantifiers. The model of predicate linear logic defined
in this paper will serve for our further research, either for specifying contracts and
dependencies between components in modeling component based systems, or in
modeling observable behavior of such systems on the base of coalgebras.
Acknowledgment
This work has been supported by APVV-0008 0 Grant: Modelling, simulation
and implementation of GPGPU-enabled architectures of high-throughput network
security tools.
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