Categorical and Algebraic Aspects of the Intuitionistic Modal Logic IEL − and its predicate extensions
aa r X i v : . [ m a t h . L O ] M a y Categorical and Algebraic Aspects of the IntuitionisticModal Logic IEL ´ and its predicate extensions Daniel Rogozin
Lomonosov Moscow State University Serokell O ¨U
Abstract
The system of intuitionistic modal logic
IEL ´ was proposed by S. Artemov and T.Protopopescu as the intuitionistic version of belief logic [3]. We construct the modal lambdacalculus which is Curry-Howard isomorphic to IEL ´ as the type-theoretical representationof applicative computation widely known in functional programming. We also provide acategorical interpretation of this modal lambda calculus considering coalgebras associatedwith a monoidal functor on a cartesian closed category. Finally, we study Heyting algebrasand locales with corresponding operators. Such operators are used in point-free topology aswell. We study compelete semantics `a la Kripke-Joyal for predicate extensions of IEL ´ andrelated logics using Dedekind-MacNeille completions and modal cover systems introducedby Goldblatt [31]. The paper extends the conference paper published in the LFCS’20 volume[59]. Keywords—
Intuitionistic modal logic, Modal type theory, Functional programming, Locales, Prenu-cleus, Cover systems
Intuitionistic modal logic study extensions of intuitionistic logic with modal operators. One may considersuch extensions from two directions. The first perspective is a consideration of intuitionistic modal logicas a branch of modal logic. Here, intuitionistic modalities might be interpreted as a constructive necessity,provability in Heyting arithmetics, intuitionistic knowledge, and so on. The second perspective is themodal type theory providing a more computational interpretation of intuitionistic modalities. Each valuein an arbitrary computation is annotated with the relevant data type and modalised type might be oneof them.The first perspective arises to Prior who introduced the system called
MIPC [57] to investigate modalcounterparts of intuitionistic predicate monadic logic. The relation between intuitionistic modalitiesand quantifiers was developed by Bull [15] and by Ono [53]. Monadic Heyting algebras were studiedcomprehensively by Bezhanishvili as well, see, for instance, [8].Fischer-Servi provided an intuitionistic analogue of the minimal normal modal logic containing and as mutually inexpressible connectives [60].Williamson [67] discussed the question of intuitionistic epistemic modalities considering the problemof an intuitionist knowledge in means of the capability of verification. Artemov and Protopopescudeveloped this direction further, see [3] and [58].We also emphasise briefly the direction related to Heyting algebras with operators. Heyting algebraswith Fischer-Servi modal operators have a topological duality piggybacked on Esakia’s results [22]. This uality that provides the characterisation of general descriptive frames for extensions of intuitionisticmodal logic containing the Fischer-Servi system, see the paper by Palmigiano [54]. Macnab examined theclass of Heyting algebras nuclei [49]. We discuss nuclei closely in Section 5. Here we merely claim thatlogic of Heyting algebras with nucleus and their predicate extensions was investigated by Bezhanishviliand Ghilardi [9]; Goldblatt [28] [31]; Fairtlough, Mendler, and Walton [24] [26].We refer the reader to this paper by Wolter and Zakharyaschev [69], the paper by Boˇzi´c and Doˇsen[14], and the monograph by Simpson [62]. These works contain the underlying results in model-theoreticaspects of intuitionistic modal logic. The second perspective we emphasised is related to intuitionistic modalities in a computational land-scape. The Curry-Howard correspondence provides bridges between intuitionistic proofs and programsunderstood in a type-theoretical sense [52] [63].Modal lambda calculi often correspond to certain intuitionistic modal logics, see the papers byArtemov [2]; Bierman and de Paiva [13]; Davies and Pfenning [18]; Fairtlough and Mendler[25], etc.De Paiva and Ritter [19] provided a categorical framework in semantics for those type systems. Onemay study modal operators within Homotopy Type Theory, see the recent paper by Rijke, Shulman,and Spitters [64].One may find a proof of concept for modal types in functional programming. Let us observe asort of computation called monadic. A monad is a concept in functional programming implementedin the functional language called Haskell. Moggi examined such monads type-theoretically [51]. Veryinformally, a monad is a method of structuring a computation as a linearly connected chain of actionswithin such types as the list or the input/output ( IO ). Such sequences are often called a pipeline in whichone passes a value from an external world and yield a result after the series of actions. There is a wayto consider computational monads logically within intuitionistic modal logic.Functional programming languages such as Haskell, Idris or Purescript have specific type classes for computation within an environment. By computational context (or, environment ), we mean some,roughly speaking, type-level map f , where f is a “function” from ˚ to ˚ : such a type-level map takes asimple type which has kind ˚ and yields another simple type of kind ˚ . For a more detailed descriptionof the type system with kinds implemented in Haskell see [63].Here, the underlying type class is Functor which has the following formal definition: c l a s s Functor f where fmap : : ( a ´ > b ) ´ > f a ´ > f b Functor provides a generalisation of higher-order functions as map . map merely yields an image of alist by a given function. Let us take a look at its implementation: map : : ( a ´ > b ) ´ > [ a ] ´ > [ b ] map f [ ] = [ ] map f ( x : xs ) = f x : ( map f xs )The first line claims that map is a two-argument function. The arguments of map are a unary functionof type a Ñ b and a list of elements belonging to a . The result of map is a list of b . This line of the pieceof code is the so-called type-signature. Type-signature describes the behaviour of the function in termsof types of input and output.The next two lines describe the recursive implementation of map . At first, we tell that an image ofthe empty list is empty. This part is the termination condition of a recursion. After that, we considerthe case with a non-empty list. A non-empty list is a list obtained by adding an element to the top ofthe list. Suppose one has a list xs and x is an element of type a . In the case of non-empty list x : xs ,one needs to call map recursively on the tail xs . We also apply a given function f to the head x . Finally,we add f x to the top of the list map f xs which is an image of the tail xs . In Haskell, type class is a general interface for some special group of data types. he list data type is one of the functor instances. Generally, Functor provides a uniform method tocarry unary functions through parametrised types. In other words, the notion of a functor in functionalprogramming is a counterpart of the category-theoretic one.One may extend a functor to the so-called monad which is a functional programming counterpartof Kleisli triples. In Haskell-like languages, one also has the type class called
Monad , a type class of anabstract data type of action in some computational environment. Here we define the
Monad type classas follows: c l a s s Functor m = > Monad m wherereturn : : a ´ > m a( >> =) : : m a ´ > ( a ´ > m b ) ´ > m b Monad is a type class that extends
Functor with two methods called return and (>>=) (a monadicbind).Monads present a uniform technique for miscellaneous computations such as computation with amutable state, many-valued computation, side effect input-output computation, etc. All those kindsof computation have an arrangement in the same fashion as pipelines. Historically, monads were im-plemented in Haskell to process side-effects that arise in the input/output world. The advantage of amonad is an ability to isolate side-effects within a monad remaining the relevant code purely functional.That is, one has a tool to describe a sequence of actions, where the result of each step depends on theprevious ones somehow. In other words, one has so-called monadic binding by which such a sequence ofactions with dependencies performs.Monadic metalanguage is the modal lambda calculus that describes a computation within an abstractmonad [51]. From a proof-theoretical point of view, this modal extension of the simply-typed lambdacalculus is Curry-Howard isomorphic to lax logic, the logic of Heyting algebras with a nucleus operatorwe discussed earlier. The typing rules for modalities of this metalanguage correspond to the return andthe monadic bind methods.Let us take a look at the example of a monad. There is a parametrised data type
Maybe in Haskell.The main application of
Maybe is making a partial function total: data Maybe a =
Nothing | Just aThe data type consists of two constructors. Suppose we deal with some computation that might terminatewith a failure.
Nothing is a flag that claims this failure arose. The second constructor
Just stores somevalue of a , a successful result of a considered computation.For example, one needs to extract the first element of a list. There might be an error if a given arrayis empty. This problem could be solved with the Maybe data type:safeH ead : : [ a ] ´ > Maybe asafeH ead [ ] =
Nothing safeH ead ( x : xs ) =
Just xThe
Maybe instance of
Monad is the following one: instance Monad Maybe wherereturn = Just ( Just x ) >> = f = f x Nothing >> = f = Nothing
Here, the return method merely embeds any value of a into the type Maybe a . The implementationof a monadic bind for Maybe is also quite simple. Suppose one has a function f of type a Ñ Maybe b and some value x of type Maybe a . Here we match on x . If x is Nothing , then the monadic bind yields
Nothing . Otherwise, we extract the value of type a and apply a given function.The monad interface for Maybe allows one to perform sequences of actions, where some values mightbe undefined. If all values are well defined on each step, then the result of an execution is a term ofthe form
Just n . Otherwise, if something went wrong and we have no required value somewhere, thenthe computation halts with
Nothing . The other examples of
Monad instances have more or less the sameexplanation since the monadic interface was proposed for effects processing. et us discuss the Applicative class. Paterson and McBride proposed this class to describe effectfulprogramming in an applicative style [50]. One may consider the
Applicative type class as an inter-mediate one between
Functor and
Monad . See this paper to have a more precise understanding of theconnection between applicative functors and monads [46].Here is the precise definition of
Applicative : c l a s s Functor f = > A p p l i c a t i v e f where pure : : a ´ > f a( < ∗ > ) : : f ( a ´ > b ) ´ > f a ´ > f bThe main aim of an applicative functor is a generalisation the action of a functor for functions havingan arbitrary arity, for instance:l i f t A 2: : A p p l i c a t i v e f = > ( a ´ > b ´ > c ) ´ > f a ´ > f b ´ > f cl i f t A 2 f x y = ( ( pure f ) < ∗ > x ) < ∗ > y liftA2 is a version of fmap for arbitrary two-argument function. It is clear that one may implement liftA3 , liftA4 , and liftAn for each n ă ω . In the case of lists, liftA2 passes a two-argument function,two lists, and yields the list obtained by applying to every possible pair the first element of which is anelement of the first list and the second element belongs to the second list.In this paper, we consider applicative computation type-theoretically. The modal axioms of IEL ´ and types of the Applicative methods in Haskell-like languages are quite similar. We investigatethe relationship between intuitionistic epistemic logic
IEL ´ and applicative computation providing themodal lambda calculus Curry-Howard isomorphic to IEL ´ .This calculus consists of the rules for simply-typed lambda-calculus extended via the special modalrules. We assume that the proposed type system axiomatises applicative computation. We provide aproof-theoretical view of this sort of computation in functional programming and prove such metatheo-retical properties as strong normalisation and confluence. The initial idea to consider applicative functorstype-theoretically belongs to Krishnaswami [41]. We are going to develop his ideas considering the IEL ´ from a computational perspective. Litak et. al. [47] made an observation that the logic IEL ´ might betreated as a logic of an applicative functor as well .In further sections, we study semantical questions of IEL ´ and related logics. We study categor-ical semantics for the provided modal lambda calculus and cover semantics for quantified versions ofintuitionistic modal logic with IEL ´ -like modalities. IEL ´ Intuitionistic modal logic
IEL ´ was proposed by S. Artemov and T. Protopopescu [3]. According to theauthors, IEL ´ represents beliefs agreed with BHK-semantics of intuitionistic logic. IEL ´ is a weakerversion of the system IEL that represents knowledge as provably consistent intuitionistic belief. Thislogic consists of the following axioms and derivation rules:
Definition 1.
Intuitionistic epistemic logic
IEL ´ :1. p ϕ Ñ p ψ Ñ θ qq Ñ pp ϕ Ñ ψ q Ñ p ϕ Ñ θ qq ϕ Ñ p ψ Ñ ϕ q ϕ Ñ p ψ Ñ p ϕ ^ ψ qq ϕ ^ ϕ Ñ ϕ i , i “ , John Connor (the City College of New York) also connected the intuitionistic epistemic logic
IEL ´ withpropositional truncation in Homotopy Type Theory. Those results were presented at the category theory seminar,the CUNY Graduate Centre. At the moment, there is only a video of that talk on YouTube. . p ϕ Ñ θ q Ñ pp ψ Ñ θ q Ñ p ϕ _ ψ Ñ θ qq ϕ i Ñ ϕ _ ϕ , i “ , K Ñ ϕ (cid:13) p ϕ Ñ ψ q Ñ p (cid:13) ϕ Ñ (cid:13) ψ q ϕ Ñ (cid:13) ψ
10. From ϕ Ñ ψ and ϕ infer ψ (Modus ponens). The ϕ Ñ (cid:13) ψ axiom is also called co-reflection . One may consider this axiom as the principleconnecting intuitionistic truth and intuitionistic knowledge. From a Kripkean point, IEL ´ is the logicof all frames x W, ď , E y , where x W, R y is a partial order and E is a binary “knowledge” relation, asubrelation of ď . The relation E obeys the following conditions:1. E p w q ĎÒ w for each w P W .2. E p u q Ď E p w q , if wRu .A model for IEL ´ is a quadruple M “ x W, ď , E, ϑ y , an extended intuitionistic Kripke model withthe additional forcing relation for modal formulas defined via the relation E . The (cid:13) connective has the“necessity” semantics: M , x , (cid:13) ϕ ô @ y P E p x q M , y , ϕ . IEL , the full epistemic intuitionistic logic, extends
IEL ´ as IEL “ IEL ´ ‘ (cid:13) ϕ Ñ ϕ . Thisadditional axiom is often called the intuitionistic relfection principle . An IEL -frame is an
IEL ´ framewith the condition E p u q ‰ H for each u P W . One has the following theorem proved by Artemov andProtopopescu [3] by the canonical model on prime theories: Theorem 1.
Let L P t
IEL ´ , IEL u , then Log p Frames p L qq “ L . V. Krupski and A. Yatmanov investigated proof-theoretical and algorithmic aspects of the logic
IEL .In this paper [42], they provided the sequent calculus for
IEL and proved that the derivability problemof this calculus is PSPACE-complete.
IEL ´ is decidable as well since this logic has the finite modelproperty, see the paper by Wolter and Zakharyaschev [68].For further purposes, we define the natural deduction calculus for IEL ´ that we call NIEL ´ . Forsimplicity, we restrict our language to Ñ , ^ , and (cid:13) . Definition 2.
The natural deduction calculus
NIEL ´ for IEL ´ is an extension of the intuitionisticnatural deduction calculus with the additional inference rules for modality: ax Γ , ϕ $ ϕ Γ , ϕ $ ψ Ñ I Γ $ ϕ Ñ ψ Γ $ ϕ Γ $ ψ ^ I Γ $ ϕ ^ ψ Γ $ ϕ (cid:13) I Γ $ (cid:13) ϕ Γ $ ϕ Ñ ψ Γ $ ϕ Ñ E Γ $ ψ Γ $ ϕ ^ ϕ ^ E , i “ , $ ϕ i Γ $ (cid:13) ÝÑ ϕ ÝÑ ϕ $ ψ (cid:13) I Γ $ (cid:13) ψ The first modal rule allows one to derive co-reflection and its consequences. The second modal ruleis a counterpart of (cid:13) I rule in natural deduction calculus for constructive K (see [40]). We will denoteΓ $ (cid:13) ϕ , . . . , Γ $ (cid:13) ϕ n and ϕ , . . . , ϕ n $ ψ as Γ $ (cid:13) ÝÑ ϕ and ÝÑ ϕ $ ψ respectively for brevity.It is straightforward to check that the second modal rule is equivalent to the K (cid:13) -rule: $ ϕ (cid:13) Γ $ (cid:13) ϕ Let us show that one may translate
NIEL ´ into IEL ´ as follows: Lemma 1. Γ $ NIEL ´ ϕ ñ IEL ´Ñ , ^ , (cid:13) $ Ź Γ Ñ ϕ .Proof. Induction on the derivation. Let us consider the modal cases.1. If Γ $ NIEL ´ ϕ , then IEL ´Ñ , ^ , (cid:13) $ Ź Γ Ñ (cid:13) ϕ . p q Ź Γ Ñ ϕ assumption p q ϕ Ñ (cid:13) ϕ co-reflection p q p Ź Γ Ñ ϕ q Ñ pp ϕ Ñ (cid:13) ϕ q Ñ p Ź Γ Ñ (cid:13) ϕ qq IPC theorem p q p ϕ Ñ (cid:13) ϕ q Ñ p Ź Γ Ñ (cid:13) ϕ q from (1), (3) and MP p q Ź Γ Ñ (cid:13) ϕ from (2), (4) and MP2. If Γ $ NIEL ´ (cid:13) ÝÑ ϕ and ÝÑ A $ ψ , then IEL ´Ñ , ^ , (cid:13) $ Ź Γ Ñ (cid:13) ψ . p q Ź Γ Ñ (cid:13) ϕ , . . . , Ź Γ Ñ (cid:13) ϕ n assumption p q Ź Γ Ñ n Ź i “ (cid:13) ϕ i IEL ´ theorem p q n Ź i “ (cid:13) ϕ i Ñ (cid:13) n Ź i “ ϕ i IEL ´ theorem p q Ź Γ Ñ (cid:13) n Ź i “ ϕ i from (2), (3) and transitivity p q n Ź i “ ϕ i Ñ ψ assumption p q p n Ź i “ ϕ i Ñ ψ q Ñ (cid:13) p n Ź i “ ϕ i Ñ ψ q co-reflection p q (cid:13) p n Ź i “ ϕ i Ñ ψ q from (5), (6) and MP p q (cid:13) n Ź i “ ϕ i Ñ (cid:13) ψ from (7) and normality p q Ź Γ Ñ (cid:13) ψ from (4), (8) and transitivity Lemma 2. If IEL ´Ñ , ^ , (cid:13) $ A , then NIEL ´ $ A .Proof. By straightforward derivation of modal axioms in
NIEL ´ . We will consider those derivationsvia terms below.One may enrich the observed natural deduction calculus with the well-known inference rules fordisjunction and bottom and prove the same lemmas as above. We build further the typed lambda-calculus based on the NIEL ´ by proof-assignment in the inference rules. IEL ´ logic Let us define terms and types.
Definition 3.
The set of terms:Let V “ t x, y, z, . . . u be the set of variables, the following grammar generates the set Λ (cid:13) of terms: Λ (cid:13) :: “ V | p λ V . Λ (cid:13) q | p Λ (cid:13) Λ (cid:13) q | px Λ (cid:13) , Λ (cid:13) yq | p π Λ (cid:13) q | p π Λ (cid:13) q | p pure Λ (cid:13) q | p let (cid:13) V ˚ “ Λ ˚ (cid:13) in Λ (cid:13) q here V ˚ and Λ ˚ (cid:13) denote the set of finite sequences of variables Y i ă ω V i and the set of finite sequencesof terms Y i ă ω Λ i (cid:13) respectively. In the term p let (cid:13) ÝÑ x “ ÝÑ M in N q , the sequence of variables ÝÑ x and thesequence of terms ÝÑ M should have the same length. Otherwise, such a term is not well-formed.As we discuss below, the terms of the form let (cid:13) ÝÑ x “ ÝÑ M in N correspond to the special localbinding. Definition 4.
The set of types:Let T “ t p , p , . . . u be the set of atomic types, the set T (cid:13) of types is generated by the grammar: T (cid:13) :: “ T | p T (cid:13) Ñ T (cid:13) q | p T (cid:13) ˆ T (cid:13) q | p (cid:13) T (cid:13) q A context has the standard definition [52][63] as a sequence of type declarations Γ “ t x : ϕ , . . . , x n : ϕ n ´ u . Here x i is a variable and ϕ i is a type for each i ă n ă ω . Definition 5.
The modal lambda calculus λ IEL ´ : ax Γ , x : ϕ $ x : ϕ Γ , x : ϕ $ M : ψ Ñ i Γ $ λx.M : ϕ Ñ ψ Γ $ M : ϕ Γ $ N : ψ ˆ i Γ $ x M, N y : ϕ ˆ ψ Γ $ M : ϕ (cid:13) I Γ $ pure M : (cid:13) ϕ Γ $ M : ϕ Ñ ψ Γ $ N : ϕ Ñ e Γ $ MN : ψ Γ $ M : ϕ ˆ ϕ ˆ e , i “ , $ π i M : ϕ i Γ $ ÝÑ M : (cid:13) ÝÑ ϕ ÝÑ x : ÝÑ A $ N : ψ let (cid:13) Γ $ let (cid:13) ÝÑ x “ ÝÑ M in N : (cid:13) ψ Γ $ ÝÑ M : (cid:13) ÝÑ ϕ is a short form for the sequence Γ $ M : (cid:13) ϕ , . . . , Γ $ M n : (cid:13) ϕ n and ÝÑ x : ÝÑ ϕ $ N : ψ is a short form for x : ϕ , . . . , x n : ϕ n $ N : B . We use this short form instead of let (cid:13) x , . . . , x n “ M , . . . , M n in N . The (cid:13) I -typing rule is the same as (cid:13) -introduction in monadic metalanguage [55]. (cid:13) I injects an object of type A into (cid:13) . According to this rule, it is clear that the type constructor pure reflects the method pure in the Applicative class.The rule let (cid:13) is similar to the (cid:13) -rule in typed lambda calculus for intuitionistic normal modal logic IK , see [38]. Informally, one may read let (cid:13) ÝÑ x “ ÝÑ M in N as a simultaneous local binding in N , whereeach free variable of a term N should be binded with term of modalised type from ÝÑ M . In other words,we modalise all free variables of a term N and ‘substitute‘ them to the terms belonging to the sequence ÝÑ M . As a matter of fact, our calculus extends the typed lambda calculus for IK with (cid:13) I -rule with theco-reflection rule allowing one to modalise any type of an arbitrary context.Here are some examples: x : ϕ $ x : ϕ (cid:13) I x : ϕ $ pure x : (cid:13) ϕ Ñ I $ p λx. pure x q : ϕ Ñ (cid:13) ϕf : (cid:13) p ϕ Ñ ψ q $ f : (cid:13) p ϕ Ñ ψ q x : (cid:13) ϕ $ x : (cid:13) ϕ g : ϕ Ñ ψ $ g : ϕ Ñ ψ y : ϕ $ ϕ : ψ Ñ e g : ϕ Ñ ψ, y : ϕ $ gy : ψ let (cid:13) f : (cid:13) p ϕ Ñ ψ q , x : (cid:13) ϕ $ let (cid:13) g, y “ f, x in gy : (cid:13) ψ Ñ I f : (cid:13) p ϕ Ñ ψ q $ λx. let (cid:13) g, y “ f, x in gy : (cid:13) ϕ Ñ (cid:13) ψ Ñ I $ λf.λx. let (cid:13) g, y “ f, x in gy : (cid:13) p ϕ Ñ ψ q Ñ (cid:13) ϕ Ñ (cid:13) ψ ere we provided the derivations for modal axioms of IEL ´ . In fact, we proved Lemma 2 usingproof-assignment.Now we define free variables and substitutions: Definition 6.
The set
F V p M q of free variables for a term M :1. F V p x q “ t x u .2. F V p λx.M q “ F V p M qzt x u .3. F V p MN q “ F V p M q Y F V p N q .4. F V px M, N yq “
F V p M q Y F V p N q .5. F V p π i M q “ F V p M q , i “ , .6. F V p pure M q “ F V p M q .7. F V p let (cid:13) ÝÑ x “ ÝÑ M in N q “ Y ni “ F V p M q , where n “ |ÝÑ M | . Definition 7.
Substitution:1. x r x : “ N s “ N , x r y : “ N s “ x .2. p MN qr x : “ N s “ M r x : “ N s N r x : “ N s .3. p λx.M qr y : “ N s “ λx.M r y : “ N s , y P F V p M q .4. p M, N qr x : “ P s “ p M r x : “ P s , N r x : “ P sq .5. p π i M qr x : “ P s “ π i p M r x : “ P sq , i “ , .6. p pure M qr x : “ P s “ pure p M r x : “ P sq .7. p let (cid:13) ÝÑ x “ ÝÑ M in N qr y : “ P s “ let (cid:13) ÝÑ x “ pÝÑ M r y : “ P sq in N . Substitutions and free variables for terms of the kind let (cid:13) ÝÑ x “ ÝÑ M in N are defined similarly to[38]. That is, we do not take into account free variables of N because those variables occur in the list ÝÑ x and are eliminated by the assignment ÝÑ x “ ÝÑ M .The reduction rules are the following ones: Definition 8. β -reduction rules for λ IEL ´ .1. p λx.M q N Ñ β M r x : “ N s .2. π x M, N y Ñ β M .3. π x M, N y Ñ β N .4. let (cid:13) ÝÑ x , y, ÝÑ z “ ÝÑ M, let (cid:13) ÝÑ w “ ÝÑ N in Q, ÝÑ P in R Ñ β let (cid:13) ÝÑ x , ÝÑ w , ÝÑ z “ ÝÑ M, ÝÑ N , ÝÑ P in R r y : “ Q s .5. let (cid:13) ÝÑ x “ pure ÝÑ M in N Ñ β pure N rÝÑ x : “ ÝÑ M s .6. let (cid:13) “ in M Ñ β pure M , where is an empty sequence of terms. If M reduces to N by one of these rules, then we write M Ñ r N . A multistep reduction ։ r is areflexive transitive closure of Ñ r . “ r is a symmetric closure of ։ r . Now we formulate the standardlemmas. Proposition 1.
The generation lemma for (cid:13) I .Let Γ $ pure M : (cid:13) ϕ , then Γ $ M : ϕ .Proof. Straightforwardly.
Lemma 3.
Basic lemmas.1. If Γ $ M : ϕ and Γ Ď ∆ , then ∆ $ M : ϕ . . If Γ $ M : ϕ , then ∆ $ M : ϕ , where ∆ “ t x : ψ | p x : ψ q P Γ & x P F V p M qu .3. If Γ , x : ϕ $ M : φ and Γ $ N : ϕ , then Γ $ M r x : “ N s : ψ .Proof. The items 1-2 are proved by induction on Γ $ M : ϕ . The third item is shown by induction on thederivation of Γ $ N : ψ . Theorem 2.
Subject reductionIf Γ $ M : ϕ and M ։ r N , then Γ $ N : ϕ .Proof. Induction on the derivation Γ $ M : ϕ and on the generation of Ñ β . The general statementfollows from transitivity of ։ β , Proposition 1, and Lemma 3. Theorem 3. ։ β is strongly normalising.Proof. Follows from Theorem 5 below, so far as reduction in the monadic metalanguage is stronglynormalising [7] and λ IEL ´ is sound with respect to the monadic metalanguage. Theorem 4. ։ r is confluent.Proof. By Newman’s lemma [63], if a given relation is strongly normalising and locally confluent, then thisrelation is confluent. It is sufficient to show that a multistep reduction is locally confluent.
Lemma 4. If M Ñ r N and M Ñ r Q , then there exists some term P , such that N ։ r P and Q ։ r P .Proof. Let us consider the following critical pairs and show that they are joinable:1. let (cid:13) x “ p let (cid:13) ÝÑ y “ pures ÝÑ N in P q in M β (cid:15) (cid:15) β , , ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ let (cid:13) ÝÑ y “ pure ÝÑ N in M r x : “ P s let (cid:13) x “ pure P rÝÑ y : “ ÝÑ N s in M let (cid:13) ÝÑ y “ pure ÝÑ N in M r x : “ P s Ñ β pure M r x : “ P srÝÑ y : “ ÝÑ N s let (cid:13) x “ pure P rÝÑ y : “ ÝÑ N s in M Ñ β pure M r x : “ P rÝÑ y : “ ÝÑ N ss ” Since x R ÝÑ y pure M r x : “ P srÝÑ y : “ ÝÑ N s let (cid:13) x “ p let (cid:13) “ in N q in M β (cid:15) (cid:15) β + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ let (cid:13) “ in M r x : “ N s let (cid:13) x “ pure N in M let (cid:13) “ in M r x : “ N s Ñ β let (cid:13) p M r x : “ N sq let (cid:13) x “ pure N in M Ñ β pure p M r x : “ N sq ne may consider four critical pairs analysed in the confluence proof for the lambda-calculus basedon the intuitionistic normal modal logic IK [38]. Those pairs are joinable in our calculus as well. The monadic metalanguage is the modal lambda-calculus based on the categorical semantics of compu-tation proposed by Moggi [51]. As we mentioned above, the monadic metalanguage might be consideredas the type-theoretical representation of computation with an abstract data type of action. In fact,the monadic metalanguage is a type-theoretical formulation for monadic computation implemented inHaskell. We show that λ IEL ´ is sound with respect to the monadic metalanguage. Definition 9.
The monadic metalanguageThe monadic metalanguage extends the simply-typed lambda calculus with the additional typing rules: Γ $ M : ϕ ∇ I Γ $ val M : ∇ ϕ Γ $ M : ∇ ϕ Γ , x : ϕ $ N : ∇ ψ let ∇ Γ $ let val x “ M in N : ∇ ψ The reduction rules are the following ones (in addition to the standard rule for abstraction andapplication):1. let val x “ val M in N Ñ β N r x : “ M s let val x “ p let val y “ N in P q in M Ñ β let val y “ N in p let val x “ P in M q let val x “ M in x Ñ η M Let us define the translation x . y from λ IEL ´ to the monadic metalanguage:1. x p i y “ p i , where p i is atomic2. x ϕ Ñ ψ y “ x ϕ y Ñ x ψ y x (cid:13) ϕ y “ ∇ x ϕ y x x y “ x , x is a variable2. x λx.M y “ λx. x M y x M N y “ x M yx N y x pure M y “ val x M y x let (cid:13) ÝÑ x “ ÝÑ M in N y “ let val ÝÑ x “ x ÝÑ M y in x N y where let val ÝÑ x “ x ÝÑ M y in N denotes let val x “ x M y in p . . . in p let val x n “ x M n y in N q . . . q It is clear that, if Γ “ t x : ϕ , . . . , x n : ϕ n u is a context, then x Γ y “ t x : x ϕ y , . . . , x n : x ϕ n y u .Let us denote $ λ IEL ´ as the derivability relation in λ IEL ´ in order to distinguish from the monadicmetalanguage derivability. Lemma 5. If Γ $ λ IEL ´ M : A , then x Γ y $ x M y : x A y in the monadic metalanguage.Proof. By induction on Γ $ λ IEL ´ M : A . One may prove the cases of I and let as follows: x Γ y $ x M y : x A yx Γ y $ val x M y : ∇ x A yx Γ y $ x ÝÑ M y : ∇ x ÝÑ A y ÝÑ x : x ÝÑ A y $ x N y : x B y ÝÑ x : x ÝÑ A y $ val x N y : ∇ x B yx Γ y $ let val ÝÑ x “ x ÝÑ M y in val x N y : ∇ x B y ow one may formulate the following lemma: Lemma 6. x M r x : “ N s y “ x M y r x : “ x N y s M ։ r N ñ x M y ։ β x N y Proof.
1. Induction on the structure of M .2. By the induction on Ñ r :(a) For simplicity, we consider the case with only one variable in let (cid:13) local binding, that canbe easily extended to an arbitrary number of variables in local binding: x let (cid:13) x “ p let (cid:13) ÝÑ y “ ÝÑ N in P q in M y “ let val x “ p let val ÝÑ y “ x ÝÑ N y in val x P y q in val x M y Ñ β let val ÝÑ y “ x ÝÑ N y in p let val x “ x P y in val x M y q Ñ β let val ÝÑ y “ x ÝÑ N y in val x M y r x : “ x P y s “ x let (cid:13) ÝÑ y “ ÝÑ N in M r x : “ P s y (b) x let (cid:13) ÝÑ x “ pure ÝÑ N in M y “ let val ÝÑ x “ val x ÝÑ N y in val x M y Ñ β val x M y rÝÑ x : “ x ÝÑ N y s “ x pure M rÝÑ x : “ ÝÑ N s y (c) x let (cid:13) x “ M in x y “ let val x “ x M y in val x Ñ η x M y Theorem 5.IEL ´ is sound with respect to the monadic metalanguage.Proof. Follows from the lemmas above.
In this subsection, we provide categorical semantics for the modal lambda calculus proposed aboveconsidering the co-reflection principle coalgebraically. Here we need a bit of category theory. We recall therequired definitions first. For the abstract definitions of category, functor, natural transformation see thebook by Goldblatt [32] or the book by MacLane and Moerdijk [48]. We piggyback the construction usedin the proof of the completeness for simply-typed lambda-calculus, see [1] and [44] to have comprehensivedetails.
Definition 10.
A category C is called cartesian closed if this category has products A ˆ B , exponentials B A and a terminal object satisfying the universal product and exponentiation properties. Following to Bellin et. al. [6] and Kakutani [38] [39], we interpret a modal operator as a monoidalendofunctor on a cartesian closed category. Monoidal endofunctors are introduced as morphisms of thosecategories that respect monoidal structure, products and a terminal object in our case. Here we refer tothe work by Eilenberg and Kelly for precise details [21]. We define a monoidal endofunctor on a cartesianclosed category as an underlying notion.
Definition 11.
Let C be a cartesian closed category and F : C Ñ C an endofunctor, F is called monoidalif there exists a natural transformation m consisting of components m A,B : F A ˆ F B Ñ F p A ˆ B q anda natural transformation u : Ñ F such that the well-known diagrams commute (MacLane pentagonand triangle identity). he abstract definition of a coalgebra is the following one : Definition 12.
Let C be a category and F : C Ñ C an endofunctor. If A P Ob p C q , then an F -coalgebrais a pair x A, α y , where α P Hom C p A, F A q . An F -coalgebra homomorpism from x A, α y to x A, β y is a map f P Hom C p A, B q such that the following square commutes: A α / / f (cid:15) (cid:15) F A F f (cid:15) (cid:15) B β / / F B Given a natural transformation α : Id C Ñ F , one may associate an F -coalgebra x A, α A y for each A P Ob p C q . Homomorphisms of such coalgebras are defined by naturality. Definition 13.
Let C be a cartesian closed category, F : C Ñ C a monoidal functor on C , and α : Id C Ñ F a natural transformation. An IEL ´ -category is a pair x C , F , α y such that the following coherenceconditions hold:1. u “ α , where α m A,B ˝ p α A ˆ α B q “ α A ˆ B , i.e. the following diagram commutes: A ˆ B α A ˆ α B / / α A ˆ B ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ F A ˆ F B m A,B (cid:15) (cid:15) F p A ˆ B q The following construction describes the standard construction of typed lambda-calculus semantics[1] [44]. First of all, let us define semantic brackets rr . ss , a semantic translation from λ IEL ´ to the IEL ´ -category x C , F , α y . Suppose one has an assignment ˆ ¨ that maps every primitive type to some object of C . Such semantic brackets rr . ss have the following inductive definition:1. rr p i ss : “ ˆ p i rr ϕ Ñ ψ ss : “ rr ϕ ss rr ψ ss rr ϕ ˆ ψ ss : “ rr ϕ ss ˆ rr ψ ss rr (cid:13) ϕ ss “ F rr ϕ ss We extend this interpretaion for contexts by induction too:1. rr ss “ , where is a terminal object of a given CCC2. rr Γ , x : ϕ ss “ rr Γ ss ˆ rr ϕ ss Typing rules have the interpretation as follows. We understand typing assignments Γ $ M : A asarrows of the form rr Γ $ M : ϕ ss “ rr M ss : rr Γ ss Ñ rr ϕ ss . π : rr Γ ss ˆ rr ϕ ss Ñ rr ϕ ssrr M ss : rr Γ ss ˆ rr ϕ ss Ñ rr ψ ss Λ prr M ssq : rr Γ ss Ñ rr ψ ss rr ϕ ss rr M ss : rr Γ ss Ñ rr ϕ ss rr N ss : rr Γ ss Ñ rr ψ ssxrr M ss , rr N ssy : rr Γ ss Ñ rr ϕ ss ˆ rr ψ ss rr M ss : rr Γ ss Ñ rr ϕ ssrr M ss ˝ η rr ϕ ss : rr Γ ss Ñ F rr ϕ ssrr M ss : rr Γ ss Ñ rr ψ ss rr ϕ ss rr N ss : rr Γ ss Ñ rr ϕ ssxrr M ss , rr N ssy ˝ ǫ rr ϕ ss , rr B ss : rr Γ ss Ñ rr ψ ssrr M ss : rr Γ ss Ñ rr ϕ ss ˆ rr ϕ ss i “ , rr M ss ˝ π i : rr Γ ss Ñ rr ϕ i ss Coalgebraic techniques are widely used in logic and computer science as well, see [12] [43] [66]. rr M ss , . . . , rr M n ssy : rr Γ ss Ñ n ś i “ F rr ϕ i ss rr N ss : n ś i “ rr ϕ i ss Ñ rr ψ ss F prr N ssq ˝ m rr ϕ ss ,..., rr ϕ n ss ˝ xrr M ss , . . . , rr M n ssy : rr Γ ss Ñ F rr ψ ss The let (cid:13) -rule has the intepretation similar to -rule in term calculus for intutionistic K [6]. Thesemantic brackets respect substitution and reduction: Lemma 7. rr M r x : “ M , . . . , x n : “ M n sss “ rr M ss ˝ xrr M ss , . . . , rr M n ssy .2. If Γ $ M : A and M Ñ r N , then rr Γ $ M : A ss “ rr Γ $ N : A ss .Proof.
1. By simple induction on M . Let us check only the modal cases. rr Γ $ p pure M qr ~x : “ ~M s : (cid:13) ϕ ss “ rr Γ $ pure p M r ~x : “ ~M sq : (cid:13) ϕ ss “ η rr A ss ˝ rrp M r ~x : “ ~M sqss “ α rr A ss ˝ prr M ss ˝ xrr M ss , . . . , rr M n ssyq “p α rr A ss ˝ rr M ssq ˝ xrr M ss , . . . , rr M n ssy “ rr Γ $ pure M : (cid:13) ϕ ss ˝ xrr M ss , . . . , rr M n ssyrr Γ $ p let (cid:13) ~x “ ~M in N qr ~y : “ ~P s : (cid:13) ψ ss “ rr Γ $ let (cid:13) ~x “ p ~M r ~y : “ ~P sq in N : (cid:13) ψ ss “ F prr N ssq ˝ m rr ϕ ss ,..., rr ϕ n ss ˝ rr Γ $ p ~M r ~y : “ ~P sq : (cid:13) ~ϕ ss “ F prr N ssq ˝ m rr ϕ ss ,..., rr ϕ n ss ˝ rr ~M ss ˝ xrr P ss , . . . , rr P n ssy “rr Γ $ let (cid:13) ~x “ ~M in N : (cid:13) ϕ ss ˝ xrr P ss , . . . , rr P n ssy
2. The cases with β -reductions for let (cid:13) are shown in [38]. Those cases are similar to our ones. Letus consider the cases with the pure terms that immediately follow from the coherence conditionsof an IEL ´ -category and the previous item of this lemma.(a) rr Γ $ let (cid:13) ~x “ pure ~M in N : (cid:13) ψ ss “ rr Γ $ pure N r ~x : “ ~M s : (cid:13) ψ ssrr Γ $ let (cid:13) ~x “ pure ~M in N : B ss “ F prr N ssq ˝ m rr ϕ ss ,..., rr ϕ n ss ˝ x α rr ϕ ss ˝ rr M ss , . . . , α rr ϕ n ss ˝ rr M n ssy “ F prr N ssq ˝ m rr ϕ ss ,..., rr ϕ n ss ˝ p α rr ϕ ss ˆ ¨ ¨ ¨ ˆ α rr ϕ n ss q ˝ xrr M ss , . . . , rr M n ssy “ F prr N ssq ˝ α rr ϕ ssˆ¨¨¨ˆrr ϕ n ss ˝ xrr M ss , . . . , rr M n ssy “ α rr ψ ss ˝ rr N ss ˝ xrr M ss , . . . , rr M n ssy “ α rr ψ ss ˝ rr Γ $ N r ~x : “ ~M s : ψ ss “ rr Γ $ pure p N r ~x : “ ~M sq : (cid:13) ψ ss (b) rr$ let (cid:13) “ in M : (cid:13) ϕ ss “ rr$ pure M : (cid:13) ϕ ssrr$$ let (cid:13) “ in M : (cid:13) ϕ ss “ F prr M ssq ˝ u “ F prr M ssq ˝ α “ α rr A ss ˝ rr M ss “ rr$ pure M : (cid:13) ϕ ss The following soundness theorem follows from the lemma above and the whole construction:
Theorem 6.
SoundnessLet Γ $ M : ϕ and M “ r N , then rr Γ $ M : ϕ ss “ rr Γ $ N : ϕ ss . The completeness theorem is proved via the syntactic model. We will consider term model for thesimply-typed lambda-calculus with ˆ and Ñ standardly described in [1] [44].Let us define a binary relation on lambda-terms „ ϕ,ψ Ď p V ˆ Λ q as: p x, M q „ ϕ,ψ p y, N q ô x : ϕ $ M : ψ & y : ϕ $ N : ψ & M “ r N r y : “ x s We will denote equivalence class as r x, M s ϕ,ψ “ tp y, N q | p x, M q „ ϕ,ψ p y, N qu (we will drop indicesbelow). Let us recall the definition of the category C p λ q , a model structure for the simply-typed lambdacalculus.The category C p λ q has the class of objects defined as Ob C p λ q “ t ˆ ϕ | ϕ P T u Y t u . For ˆ ϕ, ˆ ψ P Ob C p λ q ,the set of morphism has the form Hom C p λ q p ˆ ϕ, ˆ ψ q “ tr x, M s | x : ϕ $ M : ψ u . Let r x, M s P Hom C p λ q p ˆ ϕ, ˆ ψ q and r y, N s P Hom C p λ q p ˆ ψ, ˆ θ q , then r y, M s ˝ r x, M s “ r x, N r y : “ M ss . Identity morphisms are id ˆ ϕ “ r x, x s . he category C p λ q is cartesian closed since is a terminal object such that Hom C p λ q p , ˆ ϕ q “tr ‚ , M s | $ M : ϕ is provable u ; { ϕ ˆ ψ “ ˆ ϕ ˆ ˆ ψ ; and { ϕ Ñ ψ “ ˆ ψ ˆ ϕ . Canonical projections are de-fined as r x, π i x s P Hom C p λ q p ˆ ϕ ˆ ˆ ϕ , ˆ ϕ i q for i “ ,
2. The evaluation arrow is a morphism ev ˆ ϕ, ˆ ψ “r x, p π x qp π x qs P Hom C p λ q p ˆ ψ ˆ ϕ ˆ ˆ ϕ, ˆ ψ q .Let us define a map F : C p λ q Ñ C p λ q , such that forall r x, M s P Hom C p λ q p ˆ ϕ, ˆ ψ q , d pr x, M sq “ r y, let (cid:13) x “ y in M s P Hom C p λ q p d ˆ ϕ, d ˆ ψ q . The following functoriality condition might be easily checked with thereduction rules:1. d p g ˝ f q “ d g ˝ d f ;2. d p id ˆ A q “ id d ˆ A .We define the following maps. η : Id C p λ q Ñ F such that for each ˆ ϕ P Ob C p λ q one has η ˆ ϕ “r x, pure x s P Hom C p λ q p ˆ A, d ˆ A q . We express a monoidal transformation as m ˆ ϕ, ˆ ψ : F ˆ ϕ ˆ F ˆ ψ Ñ F p ˆ ϕ ˆ ˆ ψ q such that one has m ˆ ϕ, ˆ ψ “ r p, let (cid:13) x, y “ π p, π p in x x, y ys P Hom C p λ q p d ˆ ϕ ˆ d ˆ ψ, d p ˆ ϕ ˆ ˆ ψ qq . Also weexpress u as r ‚ , let (cid:13) “ in ‚ s . F is a monoidal endofunctor, see, e.g. [39]. Let us check the required coherence conditions: Lemma 8. F p f q ˝ α ϕ “ α β ˝ f p m ˆ ϕ, ˆ ψ q ˝ p α ϕ ˆ α β q “ α ϕ ˆ α β u “ η Proof. η ˆ ψ ˝ f “ r y, pure y s ˝ r x, M s “ r x, pure y r y : “ M ss “ r x, pure M s From the other hand, one has: d f ˝ η ˆ A “r z, let (cid:13) x “ z in M s ˝ r x, pure x s “ r x, let (cid:13) x “ z in M r z : “ pure x ss “r x, let (cid:13) x “ pure x in M s “ r x, pure M r x : “ x ss “ r x, pure M s m ˆ A, ˆ B ˝ p η ˆ A ˆ η ˆ B q “ η ˆ A ˆ ˆ B m ˆ A, ˆ B ˝ p η ˆ A ˆ η ˆ B q “r q, let (cid:13) x, y “ π q, π q in x x, y ys ˝ r p, x pure p π p q , pure p π p qys “r p, let (cid:13) x, y “ π q, π q in x x, y yr q : “ x pure p π p q , pure p π p qyss “r p, let (cid:13) x, y “ π px pure p π p q , pure p π p qyq , π px pure p π p q , pure p π p qyq in x x, y ys “r p, let (cid:13) x, y “ pure p π p q , pure p π p q in x x, y ys “r p, pure px x, y yr x : “ π p, y : “ π p sqs “ r p, pure x π p, π p ys “ r p, pure p s “ η ˆ A ˆ ˆ B
3. Immediately.The previous results imply completeness.
Lemma 9. x C p λ q , d , η y is an IEL ´ -category A Heyting algebra is a bounded distributive lattice H “ x H, ^ , _ , K , Jy with the binary operation ñ such that the following equivalence hold: a ^ b ď c iff a ď b ñ c ecall that a locale is a complete lattice L “ x L, ^ , Ž y satisfying the infinite distributive law: a ^ Ž B “ Ž t a ^ b | b P B u for each B Ď L .The notion of a locale coincides with the notion of a complete Heyting algebra since an implicationmight uniquely defined for each a, b P L as a ñ b “ Ž t c P L | a ^ c ď b u Here we note that the categories of complete Heyting algebras and locales are not the same since theirclasses of morphisms are different. We do not take into consideration these categories, so we assumethat locale and complete Heyting algebra are synonymical terms .A locale is a central object in point-free topology , where a locale is a lattice-theoretic counterpartof a topological space. The aim of this discipline is to study point-set topology concerning topologicalspaces only with the structure of their topologies as lattices of opens without mentioning points. For thefurther discussion see [36] [37] [48] [56]. In usual point-set topology, we are often interested in subspaces.In point-free topology, subspaces are characterised via operators on a locale called nuclei . A nucleus on aHeyting algebra is a multiplicative closure operator or a completion operator according to the Dragalin’sterminology [20]. Definition 14.
A nucleus on a Heyting algebra H is a monotone map j : L Ñ L such that1. a ď ja ja “ jja j p a ^ b q “ ja ^ jb One may consider a nucleus operator as a lattice-theoretic analogue of a Lawvere-Tierney topologythat generalises the notion of a Grothendieck topology on a presheaf topos. In its turn, Lawvere-Tierneytopology provides a modal operator often called a geometric modality [45]. Here, one may read jϕ as“it is locally the case that ϕ ”. The logic of Heyting algebras with a nucleus operator was studied byGoldblatt from Kripkean and topos-theoretic perspectives, see [28] and [32] as well.It is also well-known that the set of fixpoints of a nucleus on a Heyting algebra is a Heyting subalgebra.From a point-free topological view, nuclei characterise sublocales [56]. Those operators play a tremendousrole in a locale representation as well. In this monograph [20], Dragalin showed that any completeHeyting algebra is isomorphic to the locale of fixpoints of a nucleus operator on the algebra of up-sets.Moreover, any spatial locale (the lattice of open sets) is isomorphic to the complete Heyting algebra offixpoints of a nucleus operator generated by a suitable Dragalin frame. We recall that a Dragalin frameis a structure that generalises both Kripke and Beth semantics of intuitionistic logic. Bezhanishvili andHolliday strengthened this result for arbitrary complete Heyting algebras, see [10] and [11] as well.Goldblatt provided the alternative way of a locale representation [31] [33] with cover systems. Per-haps, Dragalin frames and Goldblatt cover systems may be connected to each other somehow, but itseems that the relationship between them is not investigated yet .We examine that framework closely. First of all, let us recall some relevant notions. Let x P, ďy bea poset. A subset A Ď P is called upwardly closed , if x P A and x ď y implies y P A . For A Ď P , Ò A “ t x P P | D y P A y ď x u . If x P P , then the cone at x is an up-set Ò x “Ò t x u . A subset Y Ď P refines a subset X Ď P if Y ĎÒ X . By Up p P, ďq we will mean the poset (in fact, the locale) of allupwardly closed subsets of a partial order x P, ďy . It is also clear that the set of all upwardly closed setsforms a locale.Here we consider triples S “ x P, ď , Źy , where x P, ďy is a poset and Ź is a binary relation between P and P p P q . Given x P P and C Ď P , then we say that x is covered by C ( C is an x -cover), if x Ź C ( C Ÿ x ). There is a third synonym for locales and complete Heyting algebras called frame , but we already used thisterm in means of Kripke semantics. Such topology is often called pointless , but we find the point-free topology term more appropriate. This note is based on the recent conversation between Prof. Valentin Shehtman and the author. over systems were presented to study local truth that comes from a topological and topos-theoreticintuition. A statement is locally true concerning some object as topological space or an open subsetif this object has an open cover in each member of which the statement is true. For instance, such astatement might be local equality of continuous maps, see [28] and also [29]. An abstract cover systemhas the following definition: Definition 15.
A triple S “ x P, ď , Źy as above is called cover system, if the following axioms hold for x P P :1. (Existence) There exists an x -cover C such that C ĎÒ x
2. (Transitivity) Let x Ź C and for each y P C y Ź C y , then x Ź Ť y P C C y
3. (Refinement) If x ď y , then any x -cover might be refined to a y -cover. That is, C Ź x implies thatthere exists an y -cover C such that C Ď C Let S be a cover system, let us define an operator j : P p P q Ñ P p P q as jX “ t x P P | D C x Ź C Ď X u If x P jX is called a local member of X . A subset X Ď P is called localised if jX Ď X . A localisedup-set is called a proposition . Prop p S q is the set of all propositions of a cover system. Goldblatt showedthat such an operator is a closure operator on a locale of all up-sets [31] that follows from the axioms ofa cover system. According to that, a subset X is a proposition iff X “Ò X “ jX . Definition 16.
A cover system is called localic, if the following axiom holdEvery x -cover can be refined to an x -cover that is included in Ò x .That is, x Ź C implies that there exists x Ź C such that C ĎÒ C and C ĎÒ x . This localic axiom makes that j -operator a nucleus. That is, if S “ x P, ď , Źy is a localic coversystem, then Prop p S q is a sublocale of Up p P, ďq since the set of fixpoints of nucleus is a sublocale ofUp p P, ďq . Here we strengthen the fourth axiom of a localic cover system as:Every x -cover is included in Ò x .Such a local cover system is called a strictly localic cover system . The stronger fourth axiom is builtin such generalisation of open-covers systems as Grothendieck topology and cover schemes [5] [48].The representation theorem for an abritrary locale is the following one [31]: Theorem 7.
Let L be a locale, then there exists a strictly localic cover system S such that L – Prop p S q .Proof. We provide a proof sketch in order to remain the paper self-contained.Given a locale L “ x L, Ž , ^y . Let us define S L “ x L, Ď , Źy such that x Ď y iff y ď x and x Ź C iff x “ Ž C in L . Then S L is a localic cover system. The strictness follows from the fact that if x Ź C , thatis, x “ Ž C , then C Ď p x s “ t y | y Ď x u . Every cone p x s “Ò x is localised, thus, p x s is a proposition. Itis not to so difficult to see that an arbitrary proposition of S L is a downset of Ď .An isomorphism itself is established with the map x ÞÑ p x s .As a consequence, one has a uniform embedding for arbitrary Heyting algebras as follows: Theorem 8.
Every Heyting algebra is isomorphic to a subalgebra of propositions of a suitable strictlylocalic cover system.Proof.
Every Heyting algebra has a Dedekind-Macneille completion H ã Ñ H , where H is a locale, see[31]. But H is isomorphic to the locale of propositions of a strictly localic cover system S H . trictly localic cover systems provide alternative model structures for intuitionistic predicate logic.Let S “ x P, ď , Źy be a strictly localic cover system and let D be a non-empty set, a domain of individuals.Let V be a valuation function that maps each k -ary predicate letter P to V p P q : D k Ñ Prop p S q . To in-terpret variables, we use D -assignments that have the form of infinite sequences σ “ x σ , σ , . . . , σ n , . . . y ,where σ i P D for each i ă ω . A D -assignment maps each variable x i to the corresponding σ i . Given anassignment σ and d P D , then σ p d { n q is a D -assignment obtained from σ replacing σ n to d .By IPL-model we will mean a structure M “ x S , D, V y , where S is a strictly localic cover system, D is a domain of individuals, and V is a D -valuation. Given a D -assignment and x P S , the truth relation S , x, σ |ù ϕ is defined inductively:1. M , x, σ , P p x n , . . . , x n k q iff x P V p P qp σ n , . . . , σ n k q .2. M , x, σ , ϕ ^ ψ iff M , x, σ , ϕ and M , x, σ , ψ .3. M , x, σ , ϕ ^ ψ iff there exists an x -cover C such that for each y P C M , x, σ , ϕ or M , x, σ , ψ .4. M , x, σ , ϕ Ñ ψ iff for all y PÒ x , if M , y, σ , ϕ implies M , y, , ψ .5. M , x, σ , @ x n ϕ iff for each d P D , M , x, σ p d { n q , ϕ .6. M , x, σ , D x n ϕ iff there exist an x -cover C and d P D such that for each y P C one has M , y, σ p d { n q , ϕ .Given a formula ϕ , one may associate a truth set || ϕ || M σ defined in means of locale operations onProp p S q :1. || P p x n , . . . , x n k q|| M σ “ V p P qp σ n , . . . , σ n k q || ϕ ^ ψ || M σ “ || ϕ || M σ X || ψ || M σ || ϕ _ ψ || M σ “ j p|| ϕ || M σ Y || ψ || M σ q || ϕ Ñ ψ || M σ “ || ϕ || M σ ñ || ψ || M σ ||@ x n ϕ || M σ “ Ź d P D || ϕ || M σ p d { n q ||D x n ϕ || M σ “ j p Ž d P D || ϕ || M σ p d { n q q where j is an associated nucleus on the locale of S -propositions.Thus, one has the completeness theorem : Theorem 9.
Intuitionistic first-order logic is sound and complete with respect to
IP L -models.
We define modal cover systems. Suppose one has a localic cover system S “ x S, ď , Źy . We seek toextend S with a binary relation R on S that yields an operator on P p S q : x R y A “ t x P S | D y P A xRy u “ R ´ p A q Definition 17.
A quadruple M “ x S, ď , Ź , R y is called a modal cover system, if a triple x S, ď , Źy is astrictly localic cover system and the following conditions hold:1. (Confluence) If x ď y and xRz , then there exists w such that yRw and z ď w .2. (Modal localisation) If there exists C such that x Ź C Ď x R y A , then there exists y P R p x q with a y -cover included in X . The first condition is a general requirement for intuitionistic modal logic allowing x R y A to be anup-set whenever A is. The modal localisation principle claims that Prop p M q is closed under x R y .There is a representation theorem for locales with monotone operators, see [31] to have a proof indetail: Here we note that this constuction admits generalisations and provides complete semantics for predicatesubstructural logics, see, e. g., [30]. heorem 10. Let L be a locale and m : L Ñ L a monotone map on L , then the algebra x L , m y isisomorphic to the algebra x Prop p S L q , x R m yqy Proof.
As we already know by Theorem 7, L “ x L, Ž , ^y is isomorphic to the locale Prop p S L q . S L “x L, Ď , Źy is a strictly localic cover system, where x Ď y iff y ď x and x Ź C iff x “ Ž C in L . Werecall that this isomorphism was established with map x ÞÑ p x s “ t y P L | y Ď x u . Let us put xR m y iff x ď my . The relation is well-defined and the confluence and modal localisation conditions holds. Thekey observation is that p ma s “ x R m yp a s .Goldblatt introduced modal cover systems to provide semantics for quantified lax logic and intuition-istic counterparts of modal predicate logics K and S IEL ´ -likemodalities. We discuss prenuclei operators, overview their use cases and provide representation for Heyting algebraswith such operators via suitable modal localic cover systems. A weaker version of nuclei operators isquite helpful in point-free topology as well.
Definition 18.
Let H be a Heyting algebra, a prenucleus on H is an operator monotone j : H Ñ H such that foreach a, b P H :1. a ď ja ja ^ b ď j p a ^ b q .A prenucleus is called multiplicative if it distributives over finite infima. By prenuclear algebra , we will mean a pair x H , j y , where j is a prenucleus on H . A prenuclear algebrais localic when its Heyting reduct is a locale. A prenuclear algebra is multiplitcative if its prenucleusis. Simmons calls multiplicative prenuclei merely as prenuclei [61], but this term is more spread foroperators as defined above, see, e.g. [56]. We introduce the term “multiplicative prenucleus” in order todistinguish all those operators from each other since we are going to consider both of them.Prenuclei operators have an application in point-free topology in factorising locales considering sublo-cales as quotients. See the paper by Banaschewski [4] and the monograph by Picado and Pultr [56] forthe discussion in detail. We just note that one may generate nucleus from a prenucleus the generatedby a sequence of prenuclei parametrised over ordinals.One may involve multiplicative prenuclei to the study of nuclei lattices. Infima are defined pointwisethere. Joins are more awkward to be defined explicitly. Multiplicative prenuclei provide a suitabledescription of nuclei joins in such locales. Multiplicative prenuclei form a locale as well and they areclosed under composition and pointwise directed joins. Thus, one may define joins of nuclei in means ofso-called nuclear reflection, an approximation of nucleus via prenuclei. Here we refer the reader to thispaper [23], where this aspect has a more comprehensive explanation.The other aspect of multiplicative prenuclei were studied by Haykazyan and Simmons [35] [61]. Theyconsider the special multiplicative prenucleus. Given a bounded distributive lattice L , one may introducea preorder ĺ defined as follows for each a, b P L : a ĺ b ô @ c P L a _ c “ J ñ b _ c “ J If this preorder on a locale is agreed with the parent order, then this complete Heyting algebra is calledsubfit. This preorder also has an associated map ξ : a ÞÑ Ž t b P L | b ĺ a u as it is observed by Coquand[16]. ξ is a prenucleus on an arbitrary locale as it is shown by Simmons [61], where he studies certainproperties of nuclei on the locale of complete Heyting algebra ideals. Moreover, one may associate acertain nucleus with the prenucleus ξ to measure the subfitness of a locale.Let us define prenuclear cover systems to have a suitable representation for prenuclear algebras. efinition 19. Let S “ x S, ĺ , Ź , R y be a modal cover system, then S is called prenuclear, if the followingtwo conditions hold:1. R is reflexive.2. Let x, y P S such that xRy , then there exists z PÒ y such that x ď z and x P R p z q . One may visualise the second condition with the following diagram: x R (cid:15) (cid:15) R / / ❴❴❴❴❴❴ ď / / ❴❴❴❴❴❴ D zy ď ♣♣♣♣♣♣♣ This lemma claims that a prenuclear cover system is well-defined as follows, the similar statementwas proved by Goldblatt for nuclear cover systems [31]:
Lemma 10.
Let S “ x P, ď , Ź , R y be a prenuclear cover system, then x R y is a prenucleus on Prop p S q ,that is for each A, B P Prop p S q :1. A Ď x R y A A X x R y B Ď x R yp A X B q Proof.
The condition A Ď x R y A holds according to the standard modal logic argument.Let us check the second condition. Let A X x R y B , then x P A and x P R p y q for some y P B . xRy implies there exists z PÒ y such that xRz and x ď z . A is an up-set, then z P A , so z P A X B , but xRz ,thus, x P x R yp A X B q .The lemma above allows one to extend the representation of arbitrary modal cover system describedin the proof of Theorem 10 to prenuclear ones: Theorem 11.
Every localic prenuclear algebra is isomorphic to the algebra of propositions associatedwith some modal prenuclear localic cover system.Proof.
Let L “ x L, Ž , ^y be a locale and L “ x L , ι y a localic prenuclear algebra. Then S L “ x L, Ď , Ź , R ι y is a modal cover system, where xR ι y iff x ď ιy . Let us ensure that this cover system is prenuclear one.The relation is clearly reflexive, xR ι x follows from the inflationary condition. The second prenuclear coversystem axiom also holds. xR ι y , then x ď ιy . Let us put z “ x ^ y , then xR ι z since x ď x ^ ιy ď ι p x ^ y q . y Ď z holds obviously.To embed an arbitrary prenuclear algebra, Heyting reduct of which is a non-necessarily completeone, one need to preserve prenuclei under Dedekind-MacNeille completions. First of all, we recall whatthat completion is. Given a bounded lattice L , a completion of L is a complete lattice L that contains L as a sublattice. A completion L is called Dedekind-McNeille if every element of L is both a join andmeet of elements of L , that is for each a P L (see [17] to read more about lattice completions): a “ Ž t b P L | a ď b u “ Ź t b P L | b ď a u .The class of all Heyting algebras is closed under Dedekind-MacNeille completions: if H is a Heytingalgebra, then H is a locale. An implication in an arbitrary Heyting algebra H has an extension asfollows, where a, b P H : a ñ b “ Ź t c Ñ d | a ě c P H & d ď b P H u .Given a lattice L and f : L Ñ L a monotone function on this lattice, let us define maps f ˝ , f ‚ : L Ñ L for a P L : Completions of Heyting algebras are interesting topic itself, we refer the reader to these papers [27] [34] forfurther discussion. ˝ p a q “ Ž t f p x q | a ě x P L u f ‚ p a q “ Ź t f p x q | a ď x P L u f ˝ and f ‚ both extend f and f ˝ ď f ‚ in means of the pointwise order. Generally, neither f ˝ ismultiplicative nor f ‚ , if f is. One the other hand, if f is a multiplicative function on a Heyting algebra,so f ˝ is, see [65]. Lemma 11.
Let ι be a prenucleus on a Heyting algebra H , then ι ‚ is a prenucleus on H .Proof. The proof is similar for the analogous statement about nuclei [31]. ι is inflationary, so ι ‚ is, it isreadily checked. Let us check that a ^ ι ‚ b ď ι ‚ p a ^ b q for each a, b P H .One may prove the following representation theorem for Heyting algebra with prenuclei operatorscombining Theorem 11 and Lemma 11: Theorem 12.
Every prenuclear algebra is isomorphic to the algebra of propositions obtained by someprenuclear localic cover system.
We consider the multiplicative case. Lower extensions respect multiplicativity and upper ones pre-serve inflationarity. We provide the equivalent definition of a multiplicative prenuclear algebra as followsto simplify the issue:
Proposition 2.
Let H be a Heyting algebra and j a function that preserves finite infima, then for each a, b P H one has a ď ja iff a ^ jb ď j p a ^ b q .Proof. Both implications are quite simple. One has a “ a ^ J “ a ^ j J ď j p a ^ Jq “ ja . On the otherhand, a ^ jb ď ja ^ jb “ j p a ^ b q . Lemma 12.
Let H be a Heyting algebra and ι a multiplicative prenucleus on H , then ι ˝ is a multiplicativenucleus on H .Proof. According to Proposition 2, one may equivalently replace the inflationarity condition to j J “ J and a ^ ιb ď ι p a ^ b q . In fact, one needs to check that the inequation x ^ ι ˝ y ď ι ˝ p x ^ y q holds for each x, y P H . One has: x ^ ι ˝ y “ Ž t a P H | a ď x u ^ Ž t ιb P H | b P H , b ď y u “ Ž t a ^ ιb | a ď x, b ď y, a, b P H u ď Ž t ι p a ^ b q | a ď x, b ď y, a, b P H u ď Ž t ιc | c P H , c ď x ^ y u “ ι ˝ p x ^ y q ι ˝ is multiplicative since ι is multiplicative. Thus, ι ˝ is a multiplicative prenucleus on H .Let us define a suitable cover system. Definition 20.
Let M “ x S, ď , Ź , R y be a modal cover system, then M is called multiplicative prenuclearif the following conditions hold:1. R is serial, that is, for each x P S there exists y P S such that xRy .2. if xRy and xRz then there exists w PÒ x X Ò y such that xRw .3. Let x, y P S such that xRy , then there exists z PÒ y such that x ď z and x P R p z q . One may consider a multiplicative prenuclear frame as an R -reduct of a CK-modal cover system [31]with the added principle that corresponds to the second postulate of a prenuclear cover system. Such acover system describes the logic with modal axioms (cid:13) J , (cid:13) p ^ (cid:13) q ñ (cid:13) p p ^ q q , and p ^ (cid:13) q ñ (cid:13) p p ^ q q plus the (cid:13) -monotonicity rule. It is not so hard to see that this logic is deductively equivalent to IEL ´ over intuitionistic logic. Lemma 13.
Let M “ x S, ď , Ź , R y be a multiplicative prenuclear cover system, then x R y is a multiplica-tive prenucleus on Prop p S q . roof. x R y is clearly serial. The multiplicativity follows from the second postulate of a multiplicativeprenuclear cover system. The third equation is proved similarly to Lemma 10. Theorem 13.
1. Every localic multiplicative prenuclear algebra is representable as a modal locale of the propositionsobtained by a suitable modal cover system.2. Every multiplicative prenuclear algerba is isomorphic to the subalgerba to the algebra of propositionsobtained by some multiplicative prenuclear localic cover system.Proof.
1. The proof is similar to Theorem 11 concerning Lemma 13.2. Follows from the previous item and Lemma 12.Finally, we consider
IEL -cover systems and corresponding multiplicative prenuclear algebras, wherethe equation j K “ K is satisfied. We call such multiplicative prenuclear algebras dense . In particular, j K “ K implies j ˝ K “ K [65]. Thus, if an operator on a Heyting algebra is a dense multiplicativeprenucleus, so its lower Dedekind-MacNeille completion is.An
IEL -cover system is a multiplicative prenuclear system S “ x P, ď , Ź , R y such that for each x, y P S if xRy and y Ź H implies x Ź H . This condition yields x R yH “ H .Thus, one may immediately extend Theorem 13: Theorem 14.
1. Every localic dense multiplicative prenuclear algebra is isomorphic to the algerba of propositions ofsome
IEL -cover system.2. Every dense multiplicative prenuclear algebra is isomorphic to the subalgebra of propositions ofsome
IEL -cover system.
In this subsection, by
IEL ´´ we mean the set of formulas defined as the closuse of this set IPC ‘ ϕ Ñ (cid:13) ϕ ‘ ϕ ^ (cid:13) ψ Ñ (cid:13) p ϕ ^ ψ q under the monotonicity rule: from ϕ Ñ ψ infer (cid:13) ϕ Ñ (cid:13) ψ .Let us define first intuitionistic modal predicate logic QIEL ´´ as an extension of intuitionistic predi-cate logic with modal axioms that correspond to the conditions of a prenucleus operator. We consider asignature consisting of predicate symbols of an arbitrary arity lacking function symbols and individualconstants.1. IEL ´´ -axioms2. @ xϕ Ñ ϕ p t { x q ϕ p t { x q Ñ D xϕ
4. The inference rules are Modus Ponens, Bernays rules, and (cid:13) -monotonicity.Then
QIEL ´ “ QIEL ´´ ‘ (cid:13) p ϕ Ñ ψ q Ñ p (cid:13) ϕ Ñ (cid:13) ψ q and QIEL “ QIEL ´ ‘ (cid:13) K , where ϕ “ ϕ Ñ K .In this section we show that the logics above are complete with respect to their suitable coversystems. Let L be a logic above QIEL ´ , let us define their models. Let C be a prenuclear cover system M “ x S, ď , Ź , R y , V a valuation function, and D a set of individuals, then an L -cover model is a triple M “ x M , V, D y . Given a D -assignment and x P S , a modal operator has the following semantics: M , x, σ |ù (cid:13) ϕ iff there exists y P R p x q such that M , y, σ |ù ϕ . n contrast to Kripkean semantics of IEL -like logics, we interpret modality in terms of “possibility”.Indeed, one may reformulate the truth condition above in means of an x R y -operator on the locale ofpropositions: || (cid:13) ϕ || M σ “ x R y|| ϕ || M σ The completeness theorem converts the Lindenbaum-Tarksi algebra to a suitable locale with a certainoperator via Dedekind-MacNeille completion. After that, we represent this algebra as an algebra ofpropositions of a localic system by the representation theorem we proved. To be more precise, one has:
Theorem 15.
Let L P t
IEL ´´ , IEL ´ , IEL u , then Q L is sound and complete with repsect to their coversystems.Proof. Let us consider the
QIEL ´´ -case, the rest two cases are shown in the same fashion via relevantrepresentation and Dedekind-MacNeille completions. Let Fm be the set of all formulas and V the setof all variables, then one has an equivalence relation ϕ „ ψ QIEL ´´ $ ϕ Ñ ψ and QIEL ´´ $ ψ Ñ ϕ .Then, one has an ordering on Fm { „ defined as | ϕ | ď | ψ | iff $ QIEL ´´ ϕ Ñ ψ . The operations on Fm { „ are defined as: | ϕ ^ ψ | “ | ϕ | ^ | ψ || ϕ _ ψ | “ | ϕ | _ | ψ || ϕ Ñ ψ | “ | ϕ | ñ | ψ ||@ xϕ | “ Ź x P V | ϕ ||D xϕ | “ Ž x P V | ϕ || (cid:13) ϕ | “ (cid:13) | ϕ |J “ | ϕ | , where IEL ´´ $ ϕ This algebra is clearly prenuclear, but its Heyting reduct is not necessarily complete. By Lemma 12,one may embed the Lindenbaum-Tarksi algebra L QIEL ´´ to the prenucleus (cid:13) ‚ on F { „ . A localicprenuclear algebra x F { „ , (cid:13) ‚ y is isomorphic to some prenuclear cover system. Thus, by Theorem 11, onehas an isomorphism f : x F { „ , (cid:13) ‚ y – x Prop p S QIEL ´´ q , x R (cid:13) y , y , where S QIEL ´´ is an obtained prenuclearcover system. Let define a QIEL ´´ cover model M “ x S QIEL ´´ , D, V y putting D “ V. A valuation V is defined as V p P qp x n , . . . , x n k q “ f p| P p x n , . . . , x n k q|q . Here, a D -assignment σ is merely an identityfunction. Here, the key observation is || ϕ || M σ “ f | ϕ | that might be shown by easy induction on ϕ . Then,if ϕ is true in every QIEL ´´ -model, then || ϕ || M σ “ J “ S , thus, f | ϕ | “ J . Hence, QIEL ´´ $ ϕ . Thus, QIEL ´´ is sound and complete with respect models on prenuclear cover systems.The QIEL ´ ( QIEL ) case follows from the same construction using Theorem 13 (Theorem 14).
The author is grateful to Sergei Artemov, Lev Beklemishev, Neel Krishnaswami, Vladimir Krupski,Valerii Plisko, Anil Nerode, Ilya Shapirovsky, Valentin Shehtman, and Vladimir Vasyukov for consulting,helpful advice and convesations, feedback, and valuable suggestions. Some of the results were presentedat the conference Logical Foundations in Computer Science 2020. The author thanks Anil Nerode andSergei Artemov for organisation and opportunity to give an online talk.This paper is dedicated in honour of the late Alexander Rogozin (1949 – 2014). eferences [1] Samson Abramsky and Nikos Tzevelekos. Introduction to categories and categorical logic. In Newstructures for physics , pages 3–94. Springer, 2010.[2] Sergei Artemov. Embedding of the modal λ -calculus into the logic of proofs. Trudy Matematich-eskogo Instituta imeni VA Steklova , 242:44–58, 2003.[3] Sergei Artemov and Tudor Protopopescu. Intuitionistic epistemic logic.
The Review of SymbolicLogic , 9(2):266–298, 2016.[4] Bernhard Banaschewski. Another look at the localic tychonoff theorem.
Commentationes Mathe-maticae Universitatis Carolinae , 29(4):647–656, 1988.[5] John L Bell. Cover schemes, frame-valued sets and their potential uses in spacetime physics. Tech-nical report, 2003.[6] Gianluigi Bellin, Valeria De Paiva, and Eike Ritter. Extended curry-howard correspondence for abasic constructive modal logic. In
Proceedings of methods for modalities , volume 2, 2001.[7] P. N. Benton, G. M. Bierman, and V. de Paiva. Computational types from a logical perspective.
Journal of Functional Programming , 8(2):177–193, 1998.[8] Guram Bezhanishvili. Varieties of monadic heyting algebras. part i.
Studia Logica , 61(3):367–402,1998.[9] Guram Bezhanishvili and Silvio Ghilardi. An algebraic approach to subframe logics. intuitionisticcase.
Annals of Pure and Applied Logic , 147(1-2):84–100, 2007.[10] Guram Bezhanishvili and Wesley H. Holliday. Locales, nuclei, and dragalin frames.
Advances inmodal logic , 11, 2016.[11] Guram Bezhanishvili and Wesley H. Holliday. A semantic hierarchy for intuitionistic logic.
Indaga-tiones Mathematicae , 30(3):403 – 469, 2019.[12] Nick Bezhanishvili, Jim de Groot, and Yde Venema. Coalgebraic geometric logic. arXiv preprintarXiv:1903.08837 , 2019.[13] Gavin M. Bierman and Valeria CV de Paiva. On an intuitionistic modal logic.
Studia Logica ,65(3):383–416, 2000.[14] Milan Boˇzi´c and Kosta Doˇsen. Models for normal intuitionistic modal logics.
Studia Logica ,43(3):217–245, 1984.[15] RA Bull. Mipc as the formalisation of an intuitionist concept of modality.
The Journal of SymbolicLogic , 31(4):609–616, 1997.[16] Thierry Coquand. Compact spaces and distributive lattices.
Journal of Pure and Applied Algebra ,184(1):1–6, 2003.[17] Brian A Davey and Hilary A Priestley.
Introduction to lattices and order . Cambridge universitypress, 2002.[18] Rowan Davies and Frank Pfenning. A modal analysis of staged computation.
Journal of the ACM(JACM) , 48(3):555–604, 2001.[19] Valeria [de Paiva] and Eike Ritter. Fibrational modal type theory.
Electronic Notes in TheoreticalComputer Science , 323:143 – 161, 2016. Proceedings of the Tenth Workshop on Logical and SemanticFrameworks, with Applications (LSFA 2015).[20] Albert Grigorevich Dragalin and Elliott Mendelson.
Mathematical intuitionism . American Mathe-matical Society, 1988.[21] Samuel Eilenberg and G. Max Kelly. Closed categories. In S. Eilenberg, D. K. Harrison, S. MacLane,and H. R¨ohrl, editors,
Proceedings of the Conference on Categorical Algebra , pages 421–562, Berlin,Heidelberg, 1966. Springer Berlin Heidelberg.[22] Leo Esakia.
Heyting algebras: Duality theory , volume 50. Springer, 2019.
23] Mart´ın H Escard´o. Joins in the frame of nuclei.
Applied Categorical Structures , 11(2):117–124, 2003.[24] Matt Fairtlough and Michael Mendler. Propositional lax logic.
Information and Computation ,137(1):1–33, 1997.[25] Matt Fairtlough and Michael Mendler. On the logical content of computational type theory: Asolution to currys problem. In
International Workshop on Types for Proofs and Programs , pages63–78. Springer, 2000.[26] Matt Fairtlough and Matt Walton. Quantified lax logic, 1997.[27] Mai Gehrke. Canonical extensions, esakia spaces, and universal models. In
Leo Esakia on dualityin modal and intuitionistic logics , pages 9–41. Springer, 2014.[28] Robert Goldblatt. Grothendieck topology as geometric modality.
Mathematical Logic Quarterly ,27(31-35):495–529, 1981.[29] Robert Goldblatt.
Mathematics of modality . Number 43. Center for the Study of Language (CSLI),1993.[30] Robert Goldblatt. A kripke-joyal semantics for noncommutative logic in quantales.
Advances inmodal logic , 6:209–225, 2006.[31] Robert Goldblatt. Cover semantics for quantified lax logic.
Journal of Logic and Computation ,21(6):1035–1063, 2011.[32] Robert Goldblatt.
Topoi: the categorial analysis of logic . Elsevier, 2014.[33] Robert Goldblatt. Representing and completing lattices by propositions of cover systems. In
Philo-sophical Logic: Current Trends in Asia , pages 1–18. Springer, 2017.[34] John Harding and Guram Bezhanishvili. Macneille completions of heyting algebras.
Houston Journalof Mathematics , 30:937–952, 2004.[35] Levon Haykazyan. More on a curious nucleus.
Journal of Pure and Applied Algebra , 224(2):860–868,2020.[36] Peter T Johnstone.
Stone spaces , volume 3. Cambridge university press, 1982.[37] Andr´e Joyal and Myles Tierney.
An extension of the Galois theory of Grothendieck , volume 309.American Mathematical Soc., 1984.[38] Y. Kakutani. Call-by-name and call-by-value in normal modal logic. In Zhong Shao, editor,
Program-ming Languages and Systems , pages 399–414, Berlin, Heidelberg, 2007. Springer Berlin Heidelberg.[39] Yoshihiko Kakutani. Calculi for intuitionistic normal modal logic. arXiv preprint arXiv:1606.03180 ,2016.[40] G. A. Kavvos. The many worlds of modal λ -calculi: I. curry-howard for necessity, possibility andtime. CoRR , abs/1605.08106, 2016.[41] Neel Krishnaswami. A computational lambda calculus for applicative functors. http://semantic-domain.blogspot.com/2012/08/a-computational-lambda-calculus-for.html.[42] Vladimir N Krupski and Alexey Yatmanov. Sequent calculus for intuitionistic epistemic logic iel.In
International Symposium on Logical Foundations of Computer Science , pages 187–201. Springer,2016.[43] Clemens Kupke and Dirk Pattinson. Coalgebraic semantics of modal logics: an overview.
TheoreticalComputer Science , 412(38):5070–5094, 2011.[44] Joachim Lambek and Philip J Scott.
Introduction to higher-order categorical logic , volume 7. Cam-bridge University Press, 1988.[45] F William Lawvere. Quantifiers and sheaves. In
Actes du congres international des mathematiciens,Nice , volume 1, pages 329–334, 1970.[46] Sam Lindley, Philip Wadler, and Jeremy Yallop. Idioms are oblivious, arrows are meticulous,monads are promiscuous.
Electronic notes in theoretical computer science , 229(5):97–117, 2011.
47] Tadeusz Litak, Miriam Polzer, and Ulrich Rabenstein. Negative translations and normal modality.In . Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017.[48] Saunders MacLane and Ieke Moerdijk.
Sheaves in geometry and logic: A first introduction to topostheory . Springer Science & Business Media, 2012.[49] DS Macnab. Modal operators on heyting algebras.
Algebra Universalis , 12(1):5–29, 1981.[50] C. McBride and R. Paterson. Applicative programming with effects.
Journal of Functional Pro-gramming , 18(1):1–13, 2008.[51] Eugenio Moggi. Notions of computation and monads.
Information and computation , 93(1):55–92,1991.[52] R. Nederpelt and H. Geuvers.
Type Theory and Formal Proof: An Introduction . Cambridge Uni-versity Press, New York, NY, USA, 1st edition, 2014.[53] Hiroakira Ono. On some intuitionistic modal logics.
Publications of the Research Institute forMathematical Sciences , 13(3):687–722, 1977.[54] Alessandra Palmigiano. Dualities for some intuitionistic modal logics, 2004.[55] F. Pfenning and R. Davies. A judgmental reconstruction of modal logic.
Mathematical Structuresin Computer Science , 11(4):511–540, 2001.[56] Jorge Picado and Aleˇs Pultr.
Frames and Locales: topology without points . Springer Science &Business Media, 2011.[57] Arthur N Prior.
Time and modality . OUP Oxford, 2003.[58] Tudor Protopopescu. Intuitionistic epistemology and modal logics of verification. In
InternationalWorkshop on Logic, Rationality and Interaction , pages 295–307. Springer, 2015.[59] Daniel Rogozin. Modal type theory based on the intuitionistic modal logic
IEL ´ . In InternationalSymposium on Logical Foundations of Computer Science , pages 236–248. Springer, 2020.[60] Gis`ele Fischer Servi. On modal logic with an intuitionistic base.
Studia Logica , 36(3):141–149, 1977.[61] Harold Simmons. A curious nucleus.
Journal of Pure and Applied Algebra , 214(11):2063–2073, 2010.[62] Alex K Simpson. The proof theory and semantics of intuitionistic modal logic. 1994.[63] Morten Heine Sørensen and Pawel Urzyczyn.
Lectures on the Curry-Howard isomorphism . Elsevier,2006.[64] Bas Spitters, Michael Shulman, and Egbert Rijke. Modalities in homotopy type theory.
LogicalMethods in Computer Science , 16, 2020.[65] Mark Theunissen and Yde Venema. Macneille completions of lattice expansions.
Algebra Universalis ,57(2):143–193, 2007.[66] Yde Venema. Algebras and coalgebras.
Handbook of modal logic , 3:331–426, 2006.[67] Timothy Williamson. On intuitionistic modal epistemic logic.
Journal of Philosophical Logic , pages63–89, 1992.[68] Frank Wolter and Michael Zakharyaschev. Intuitionistic modal logics as fragments of classicalbimodal logics.
Logic at work , pages 168–186, 1997.[69] Frank Wolter and Michael Zakharyaschev. Intuitionistic modal logic. In
Logic and Foundations ofMathematics , pages 227–238. Springer, 1999., pages 227–238. Springer, 1999.