A Cook's tour of duality in logic: from quantifiers, through Vietoris, to measures
aa r X i v : . [ c s . L O ] J u l A Cook’s tour of duality in logic: from quantifiers,through Vietoris, to measures
Mai Gehrke , Tom´aˇs Jakl , and Luca Reggio CNRS and Universit´e Cˆote d’Azur, Nice, France Department of Computer Science and Technology, University of Cambridge, UK Department of Computer Science, University of Oxford, UK
Abstract
We identify and highlight certain landmark results in Samson Abramsky’s workwhich we believe are fundamental to current developments and future trends. Inparticular, we focus on the use of • topological duality methods to solve problems in logic and computer science; • category theory and, more particularly, free (and co-free) constructions; • these tools to unify the ‘power’ and ‘structure’ strands in computer science. Boole wanted to view propositional logic as arithmetic. This idea, of seeing logic asa kind of algebra, reached a broader and more foundational level with the work ofTarski and the Polish school of algebraic logicians. The basic concept is embodiedin what is now known as the
Lindenbaum-Tarski algebra of a logic. In the classicalcases, this algebra is obtained by quotienting the set of all formulas F by logicalequivalence, that is, L = F / ≈ where ϕ ≈ ψ if, and only if, ϕ and ψ are logically equivalent.When the equivalence relation ≈ is a congruence for the connectives of the logic, L may be seen as an algebra in the signature given by the connectives. This is thecase for many propositional logics as well as for first-order logic. There is, however,a fundamental difference in how well this works at these two levels of logic. This project has been funded by the European Research Council (ERC) under the European Union’sHorizon 2020 research and innovation program (grant agreement No.670624). Tom´aˇs Jakl has received par-tial support from the EPSRC grant EP/T007257/1. Luca Reggio has received funding from the EuropeanUnion’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agree-ment No.837724. or example, for Classical Propositional Logic (CPL), Intuitionistic PropositionalCalculus (IPC) and modal logics, the Lindenbaum-Tarski algebra is the free algebra over the set of primitive propositions of the appropriate variety. In the above men-tioned cases, these are Boolean algebras, Heyting algebras, and modal algebras ofthe appropriate signature, respectively. Further, for algebras in these varieties, con-gruences are given by the equivalence classes of the top elements which, logicallyspeaking, are the theories of the corresponding logics. Consequently, we have thatthe Lindenbaum-Tarski algebras of theories, in which one quotients out by logicalequivalence modulo the theory, account for the full varieties of Boolean algebras,Heyting algebras and modal algebras.The picture is not always quite this simple, even at the propositional level. E.g.the Lindenbaum-Tarski algebra of positive propositional logic (i.e. the fragment ofCPL without negation, which we will denote PPL) is indeed the free bounded dis-tributive lattice over the set of primitive propositions. However, since there arelattices with multiple congruences giving the same filter, we do not have the samenatural correspondence between the full variety of distributive lattices and the the-ories of PPL. This sort of problem can be dealt with and this is the subject of thefar-reaching theory of Abstract Algebraic Logic, see (Font and Verd´u, 1991) for theexample of PPL.Let us now consider (classical) first-order logic. Here also, logical equivalenceis a congruence for the logical connectives. We have the Boolean connectives, andunary connectives ∃ x and ∀ x , a pair for each individual variable x of the logicallanguage. The latter give rise to pairs of unary operations that are inter-definableby conjugation with negation. Thus, in the Boolean setting, it is enough to considerthe ∃ x operations. These are (unary) modal operators .In its most basic form, modal propositional logic corresponds to the variety ofmodal algebras (MAs), which are Boolean algebras augmented by a unary operationthat preserves finite joins. The algebraic approach is a powerful tool in the studyof modal logics, see e.g. (Rautenberg et al., 2006) for a survey. In particular, theLindenbaum-Tarski algebra for this logic is the free modal algebra over the proposi-tional variables, the normal modal logic extensions correspond to the subvarieties ofthe variety of MAs, and theories in these logics correspond to the individual algebrasin the corresponding varieties.The Lindenbaum-Tarski algebra of first-order formulas modulo logical equivalenceis a multimodal algebra, with modalities ♦ x , one for each variable x in the first-orderlanguage. These modalities satisfy some equational properties such as ϕ ≤ ♦ x ϕ ♦ x ( ϕ ∧ ♦ x ψ ) = ♦ x ϕ ∧ ♦ x ψ ♦ x ♦ y ϕ = ♦ y ♦ x ϕ. A fundamental problem, as compared with the propositional examples given above,is that these Lindenbaum-Tarski algebras are not free in any reasonable setting.Tarski and his students introduced the variety of cylindric algebras of which these Typically one also considers some named constants, which we are not mentioning here. Throughout, if no confusion arises, we write ϕ for the corresponding element of the Lindenbaum-Tarskialgebra, i.e. the logical equivalence class [ ϕ ] ≈ of the formula ϕ . re examples, see (Monk, 1986) for an overview. However, not all cylindric algebrasoccur as Lindenbaum-Tarski algebras for first-order theories. For one, when we havean infinite set of variables, and thus of modalities, for every element ϕ in the algebrathere is a finite set V ϕ of variables such that ♦ x ϕ = ϕ for all x V ϕ .Even though cylindric algebras have been extensively studied, little is knownspecifically about the ones arising as Lindenbaum-Tarski algebras of first-order the-ories. A notable exception is the paper (Myers, 1976) characterising the algebrasfor first-order logic over empty theories. Another important insight, due to Rasiowaand Sikorski, is the fact that the completeness theorem for first-order logic may beobtained using the Lindenbaum-Tarski construction (Rasiowa and Sikorski, 1950).Their proof uses the famous Rasiowa-Sikorski Lemma. This lemma, which may beseen as a consequence of the Baire Category Theorem in topology, states that, givena specified countable collection of subsets with suprema in a Boolean algebra, one canseparate the elements of the Boolean algebra with ultrafilters that are inaccessibleby these suprema.The lack of freeness of the Lindenbaum-Tarski algebras of first-order logic is over-come by moving from lattices with operators to categories and categorical logic. Inthe equational setting, algebraic theories can equivalently be described as Lawveretheories, i.e. categories with finite products and a distinguished object X such thatevery object is a finite power of X . Similarly, theories in a given fragment of first-order logic correspond to a certain class of categories.For instance, theories in the positive existential fragment of first-order logic, alsocalled coherent theories, correspond to coherent categories. Every coherent theory T yields a coherent category, the syntactic category of T , which may be seen as a gener-alisation of the Lindenbaum-Tarski construction, and which is free in an appropriatesense. Central to this construction is the fundamental insight, of Lawvere, that quan-tifiers are adjoints to substitution maps. Thus, existential quantifiers are encodedin coherent categories as lower adjoints to certain homomorphisms between latticesof subobjects. Further, there is some sense in which the correspondence betweentheories and quotients is regained (at the level of so-called classifying toposes of thetheories). See (Makkai and Reyes, 1977). Other fragments of first-order logic can bedealt with in a similar fashion, e.g. intuitionistic first-order theories correspond toHeyting categories, and classical first-order theories to Boolean coherent categories.See (Johnstone, 2002) for a thorough exposition.To make the relation between syntactic categories and Lindenbaum-Tarski al-gebras more explicit, we recall the notion of Boolean hyperdoctrines, tightly relatedto Boolean coherent categories. Consider the category Con of contexts and sub-stitutions. A context is a finite list of variables x , and a substitution from x to acontext y = y , . . . , y n is a tuple h t , . . . , t n i of terms with free variables in x . Givena first-order theory T , let P ( x ) be the Lindenbaum-Tarski algebra of first-order for-mulas with free variables in x , up to logical equivalence modulo T . A substitution h t , . . . , t n i : x → y induces a Boolean algebra homomorphism P ( y ) → P ( x ) sending For a variety of algebras V , the associated Lawvere theory is the dual of the category of finitely gener-ated free V -algebras with homomorphisms; the distinguished object is the free algebra on one generator. formula ϕ ( y ) to ϕ ( h t , . . . , t n i /y ). This yields a functor P : Con op → BA . The product projection π y : x, y → x in Con induces the Boolean algebra embedding P ( π y ) : P ( x ) ֒ → P ( x, y ), which admits both lower and upper adjoints: ∃ y ⊣ P ( π y ) , ∃ y ( ϕ ( x, y )) = ∃ y.ϕ ( x, y ) ,P ( π y ) ⊣ ∀ y , ∀ y ( ϕ ( x, y )) = ∀ y.ϕ ( x, y ) . This accounts for the
Boolean hyperdoctrine structure of P . The syntactic category ofthe theory T can be obtained from P by means of a 2-adjunction between Boolean hy-perdoctrines and Boolean categories, cf. (Pitts, 1983) or (Coumans, 2012, Chapter 5).While the categorical perspective solves a number of problems, it is not easilyamenable to the inductive point of view that we want to highlight here. We will getback to this in Section 2.2. Topological methods in logic have their origin in the work of M. H. Stone. The paper(Stone, 1936) established what is nowadays presented as a dual equivalence betweenthe category BA of Boolean algebras with homomorphisms and a full subcategory BStone of the category of topological spaces with continuous maps. The objectsof
BStone are the so-called
Boolean (Stone) spaces , i.e. compact Hausdorff spaceswhose collection of clopen (simultaneously closed and open) subsets forms a basisfor the topology. Usually referred to as
Stone duality for Boolean algebras , this isthe prototypical example of a dual equivalence induced by a dualizing object, i.e. anobject sitting at the same time in two categories. In fact, the quasi-inverse functorsproviding the equivalence between BA op and BStone are given by enriching the setof homomorphisms into the appropriate structure on the two-element set = { , } ,which can be seen either as the two-element Boolean algebra or as the two-elementBoolean space when equipped with the discrete topology.Given a Boolean algebra B , the space X B obtained by equipping the set of ho-momorphisms hom BA ( B, )with the subspace topology induced by the product topology on B is a Booleanspace, the (Stone) dual space of B . Under the correspondence sending a Booleanalgebra homomorphism h : B → to the subset h − (1) ⊆ B , the points of X B can beidentified with the ultrafilters on B . In logical terms, these are the complete consist-ent theories over B . Conversely, given a Boolean space X , the set of continuous mapshom BStone ( X, ) More precisely, the morphisms in
Con are defined as equivalence classes of substitutions, by identifyingtwo tuples h s , . . . , s n i and h t , . . . , t n i if they give rise to the same homomorphism. orms a Boolean subalgebra B X of the product algebra X , where is now viewedas a Boolean algebra. When equipped with the induced Boolean operations, B X is called the dual algebra of X . Upon identifying a continuous function f : X → with the clopen subset f − (1) ⊆ X , the Boolean algebra B X can be described as thefield of clopen subsets of X with the set-theoretic Boolean operations. Stone dualitystates that these object assignments extend to functors, and there are isomorphisms B ∼ = B X B and X ∼ = X B X (natural in B and X , respectively). Throughout, theelement of B X B corresponding to a ∈ B will be denoted by b a .Shortly after his seminal work in 1936, Stone generalised the duality to boundeddistributive lattices (Stone, 1938); there, the relevant category of spaces consistsof spectral spaces with perfect maps. A different formulation of the duality fordistributive lattices, induced by the dualizing object regarded either as a lattice oras a discrete ordered space where 0 <
1, was later introduced in (Priestley, 1970).When combined with the algebraic semantics, as outlined in the previous section,Stone duality yields a powerful framework for developing and applying topologicalmethods in logic. The potential advantages of applying duality are of two types.For one, duality theory often connects syntax and semantics. To wit, in the caseof CPL, the Lindenbaum-Tarski algebra is the free Boolean algebra on the set V ofpropositional variables, and its dual space is the Cantor space V of all valuationsover V . The second type of advantage is that it often is easier , technically, to solvea problem on the dual side.The use of duality is not restricted to the Boolean setting. Indeed, generalisationsand extensions of Stone duality have been exploited to study fragments and exten-sions of CPL. Many other special cases have since been developed based on Stone’sand Priestley’s dualities for bounded distributive lattices (corresponding to PPL).Here we just mention the duality for Heyting algebras, the algebraic semantics ofIPC, mainly developed by Leo Esakia (Esakia, 1974, 2019). Stone duality was alsoextended by J´onsson and Tarski to Boolean algebras with operators by introducingthe powerful framework of canonical extensions (J´onsson and Tarski, 1951, 1952).This was a crucial step for many applications, e.g. in modal logic.In theoretical computer science, the link between syntax and semantics providedby Stone-type dualities is particularly central as the two sides correspond to specific-ation languages and to spaces of computational states, respectively. The ability totranslate faithfully between these two worlds has often proved itself to be a powerfultheoretical tool as well as a handle for solving problems. A prime example is Ab-ramsky’s seminal work (Abramsky, 1987, 1991) linking program logic and domaintheory via Stone duality for bounded distributive lattices, which was awarded theIEEE LICS “Test of Time” Award in 2007. Other examples include large parts ofmodal and intuitionistic logics, where J´onsson-Tarski duality yields Kripke semantics(Blackburn et al., 2001). For a particular example, see Ghilardi’s work in modal andintuitionistic logic on unification (Ghilardi, 2004) and normal forms (Ghilardi, 1995).By contrast, Stone duality has not played a significant role, at least overtly, inmore algorithmic areas of theoretical computer science until recently. In the theory ofregular languages, finite and profinite monoids are an important tool, in particular for roving decidability, ever since their introduction in the 1960s and 1980s, respectively,see (Pin, 2009) for a survey. While it was observed as early as 1937 by Birkhoffthat profinite topological algebras are based on Boolean spaces (Birkhoff, 1937), theconnection with Stone duality was not used in automata theory until much morerecently. It was exploited first in an isolated case by (Pippenger, 1997), and thenmore structurally by (Gehrke et al., 2008). Further, realising that these methods areinstances of Stone duality provides an opportunity to generalise them to the setting ofcomputational complexity and the search for lower bounds (Gehrke and Krebs, 2017).This line of work connects tools from semantics, such as Stone duality, with problemsand methods on the algorithmic side of computer science, such as decidability andEilenberg-Reiterman theory. Similarly, recent work of Samson Abramsky and co-workers connects categorical tools from semantics, such as comonads, with conceptsfrom finite model theory, such as tree-width and tree-depth (Abramsky et al., 2017;Abramsky and Shah, 2018).Finite model theory, computational complexity theory and the theory of regularlanguages all belong to the branch of computer science where the use of resources incomputing is the main focus, whereas category theory and Stone duality have longbeen central tools in semantics of programming languages. While the trend of makingconnections and seeking unifying results that bridge the gap between semantics andalgorithmic issues has long been on the way (e.g. in the form of semantic work onresource sensitive logics), making this overt and placing it front and center stageis a recent phenomenon in which Samson Abramsky has played a central role. Inparticular, one may mention the 2017 semester-long program at The Simons Institutefor the Theory of Computing on Logical Structures in Computation of which he was aco-organiser, and the ensuing work and ongoing project with Anuj Dawar focussing onbridging what they aptly call the Structure versus Power gap in theoretical computerscience. The 2014 ERC project Duality in Formal Languages and Logic – a unifyingapproach to complexity and semantics (DuaLL), in which our recent work has takenplace, shares these goals.In Section 2.1, we highlight some of the ideas and concepts from Samson Ab-ramsky’s work in semantics that are playing an important role in our recent work onthe DuaLL project, which we will describe in Section 3. In Section 2.2, we brieflyreview two settings from logic pertinent to our work, and give a duality-centric de-scription of the treatment of the function space construction in Abramsky’s DomainTheory in Logical Form. This allows us to make a connection to the profinite methodsin automata theory.
An important contribution of Samson Abramsky’s is to use the duality betweensyntax and semantics, combined with a step-wise description of connectives in logicapplications. This phenomenon is the driving force behind his sweeping and elegantgeneral solution to domain equations in the paper Domain Theory in Logical Form(DTLF), (Abramsky, 1991). We will get back to this with a few more details inSection 2.2. In (Abramsky, 2005), which is the published version of various talks given uring the genesis of DTLF, Abramsky gives a simpler example of this general idea.The setting is non-well-founded sets, and the object he considers is the free modalalgebra (over the empty set). Other early uses of similar methods are due to Ghilardi(Ghilardi, 1992, 1995). Subsequently, the treatment of the free modal algebra given inAbramsky’s talks, in particular his talk at the 1988 British Colloquium on TheoreticalComputer Science in Edinburgh, has been identified as an important contribution tomodal logic in its own right, see e.g. (Rutten and Turi, 1993; Kupke et al., 2004;Venema and Vosmaer, 2014), and it is also very pertinent to the duality theoretictreatment of quantifiers which we will discuss in Section 3.The step-wise description of an algebra from a set of generators is what is oftencalled Noetherian induction in algebra and induction on the complexity of a formula in logic: The algebra is generated layer by layer, starting with the generators —which are said to be of rank 0 — by adding consecutive layers of the operations toobtain higher rank elements. Also, instead of doing this with all the operations, wemay do it relative to a fragment. In the case of modal algebras, for example, wemay consider as rank 0 all Boolean combinations of generators, rank less than orequal to 1 any element which may be expressed as a Boolean combination of rank 0and diamonds of rank 0 elements, and so on. This is a fine tool for the purpose ofinduction, but it is not a good tool for constructing algebras in general. However, ifthe operation is freely added modulo some equations which are of pure rank 1, thenit is in fact a powerful method of construction . This is exactly the situation for freemodal algebras, which are Boolean algebras with an additional operation satisfyingthe equations ♦ ≈ ♦ ( x ∨ y ) ≈ ♦ x ∨ ♦ y. These equations are both of pure rank 1. That is, in each equation, all occurrencesof each variable are in the scope of exactly one layer of modal operators.From a categorical point of view, one may see algebras in a variety as Eilenberg-Moore algebras for a finitary monad, but having a pure rank 1 axiomatisation meansthat these are also presentable as the algebras for an endofunctor , see (Kurz and Rosick´y,2012) where this is studied in greater generality. In the case of MAs, define the en-dofunctor M on Boolean algebras which takes a Boolean algebra B to the Booleanalgebra freely generated by elements ♦ a , for every a ∈ B , subject to the equationsfor modal algebras viewed as relations on these generators: ♦ ≈ ♦ ( a ∨ b ) ≈ ♦ a ∨ ♦ b ( ∀ a, b ∈ B ) . Then B , equipped with a unary operation f : B → B , is a modal algebra if and only ifthe map ♦ a f ( a ) extends to a Boolean algebra homomorphism h : M ( B ) → B . Italso follows that the free modal algebra over a Boolean algebra B may be constructedinductively , as the colimit of the sequence B B B . . . i i i where B = B , B n +1 is the coproduct B ⊕ M ( B n ), the map i is the embedding of B in the coproduct, and i n +1 = id B ⊕ M ( i n ). Note that, if B is finite, then so are allthe algebras in the sequence. Moreover, if we start with the free Boolean algebra on set V , then the colimit of the sequence is the free modal algebra over V , and B n isthe Boolean subalgebra consisting of all formulas of rank at most n .Further, we may of course dualize M to get a functor on BStone and a co-inductive description of the dual of free modal algebras. This dual endofunctor is theVietoris functor. Recall that, given a Boolean space X , the Vietoris hyperspace of X is the collection V ( X ) of closed subsets of X equipped with the topology generatedby the sets of the form ♦ U = { C ∈ V ( X ) | C ∩ U = ∅} and ( ♦ U ) c for U a clopen subset of X . With respect to this topology, V ( X ) is again a Booleanspace. See (Vietoris, 1922; Michael, 1951). Furthermore, for every continuous map f : X → Y , the forward-image map f (-) : V ( X ) → V ( Y ) is continuous. Hence, weobtain a functor V : BStone → BStone . Abramsky showed that the dual Stone space of the free modal algebra on no gener-ators coincides with the final coalgebra for the functor V . In general, the dual of thesequence of embeddings given above is X X × V ( X ) = X X × V ( X ) = X . . . π X id X ×V ( π X ) This result provides also a coalgebraic perspective on the duality between modalalgebras and descriptive general Kripke frames. As such, it has had a strong influenceon the very active coalgebraic approach to modal logic. The Vietoris hyperspaceconstruction also appeared earlier in modal logic in the work (published in Russian)of Leo Esakia, cf. (Esakia, 1974). See also (Esakia, 2019) for the recent Englishtranslation of Esakia’s 1985 book.
In this section we discuss duality methods in logic in three settings: classical first-order logic, B¨uchi’s logic on words, and Domain Theory in Logical Form.
First-order logic and spaces of types.
For classical first-order logic, the dualspace of the Lindenbaum-Tarski algebra of formulas is fairly easy to describe. Fixa countably infinite set of first-order variables v , v , . . . and a first-order signature σ , i.e. σ may contain relation symbols as well as function symbols and constants.Denote by FO ω the set of all first-order formulas in the signature σ over the set ofvariables. Given a theory T , that is, any set of first-order sentences in the signature σ , consider the collectionMod ω ( T ) = { ( A, α : ω → A ) | A is a σ -structure and A | = T } of models of T equipped with an assignment of the variables. The satisfaction relation | = ⊆ Mod ω × FO ω induces the equivalence relations of elementary equivalence andlogical equivalence on these sets, respectively:( A, α ) ≡ ( A ′ , α ′ ) iff ∀ ϕ ∈ FO ω A, α | = ϕ ⇐⇒ A ′ , α ′ | = ϕ nd ϕ ≈ ψ iff ∀ ( A, α ) ∈ Mod ω ( T ) A, α | = ϕ ⇐⇒ A, α | = ψ. The quotient FO ω ( T ) = FO ω / ≈ , i.e. the Lindenbaum–Tarski algebra of T , carriesa natural Boolean algebra structure. On the other hand, Typ ω ( T ) = Mod ω / ≡ isnaturally equipped with a topology, generated by the sets J ϕ K = { [( A, α )] | A, α | = ϕ } for ϕ ∈ FO ω , and is known as the space of types of T . G¨odel’s completeness theoremmay now be stated as follows:the space Typ ω ( T ) is the Stone dual of FO ω ( T ).For every n ∈ N , we can consider the Boolean subalgebra FO n ( T ) of FO ω ( T ) con-sisting of the equivalence classes of formulas with free variables in v , . . . , v n . Thedual space of FO n ( T ) is then the space of n -types of T . In particular, for n = 0, wesee that the dual space of the Lindenbaum-Tarski algebra of sentences FO ( T ) is thespace of elementary equivalence classes of models of T .Methods based on spaces of types play a central role in model theory. Their usecan be traced back to Tarski’s work, but the functorial nature of the construction wasbrought out and exploited nearly thirty years later by Morley in (Morley, 1974). Infact, it has been suggested that the notion of type space may be more fundamentalthan the notion of model (Macintyre, 2003). This point of view is related to thecategorical approach, as the type space functor of a theory T can be essentiallyidentified with the (pointwise) dual of the hyperdoctrine associated with T .This approach relies on the presentation of the algebra FO ω ( T ) as the colimit ofthe following diagram of Boolean algebra embeddings:FO ( T ) FO ( T ) FO ( T ) . . . Interestingly, this presentation does not fit with the inductive treatment of modallogic in Section 2.1, as the sentences, which is what we want to understand, belongto all the algebras in the chain. If we want to construct the Lindenbaum-Tarskialgebra FO ω ( T ) inductively, by adding a layer of quantifier ∃ at each step, we shouldstart from the Boolean subalgebra FO ( T ) of FO ω ( T ) consisting of the quantifier-free formulas. The algebra FO ( T ) sits inside the algebra FO ( T ) of formulas withquantifier rank at most 1, and so forth. The colimit of the diagramFO ( T ) FO ( T ) FO ( T ) . . . is again the algebra FO ω ( T ). In Section 3, we will illustrate how the inductivemethods used in B¨uchi’s logic apply in the general first-order setting (and beyond)using the ideas set forth in Section 2.1. ¨uchi’s logic on words and profinite monoids. The connection betweenlogic and automata goes back to the work of B¨uchi, Elgot, Rabin and others in the1960s. In particular, B¨uchi’s logic on words provides a powerful tool for the studyof formal languages. The basic idea consists in regarding words on a finite alphabet A , i.e. elements of the free monoid A ∗ , as finite models for so-called logic on words .That is, a word w ∈ A ∗ is seen as a relational structure on the initial segment of thenatural numbers { , . . . , | w |} , where | w | is the length of w , equipped with a unary relation P a for each a ∈ A whichsingles out the positions in w where the letter a appears. B¨uchi’s theorem statesthat the Lindenbaum-Tarski algebra of monadic second-order sentences for logic onwords with the successor relation (interpreted over finite words) is isomorphic to theBoolean subalgebra of P ( A ∗ ) consisting of the regular languages (B¨uchi, 1966).Since we are beyond first-order logic, and we have restricted to the finite models,the dual of the Lindenbaum-Tarski algebra is not A ∗ , i.e. the collection of (elementaryequivalence classes of) finite models. For the FO fragment of logic on words we canidentify the dual with a space of models provided we allow for pseudofinite words.See e.g. (van Gool and Steinberg, 2017). However, this is not the case for monadicsecond-order logic and duality guides the right choice for the space of generalisedmodels as the dual of the Lindenbaum-Tarski algebra. The latter coincides with (theunderlying space of) the profinite completion c A ∗ of the monoid A ∗ , or equivalently,the free profinite monoid on the set A .The observation that the space underlying the free profinite monoid is the dual ofthe Boolean algebra of languages recognised by finite monoids essentially goes back to(Birkhoff, 1937), and was rediscovered by Almeida in the setting of automata theory(Almeida, 1989). Further, the fact that the monoid multiplication of c A ∗ also arisesfrom duality for Boolean algebras with operators as the dual of certain quotientingoperations on regular languages was shown in (Gehrke et al., 2008).This type space tells us what generalised models for these logics should be, namelythe points of the free profinite monoids. The realisation that these are an import-ant tool in automata theory came in the 1980s (Reiterman, 1982; Almeida, 1994).However, it was introduced, not via logic and duality, but rather via the connectionbetween automata and finite semigroups, where the multiplication available on theprofinite monoid also plays a fundamental role.An essential insight in the proof of B¨uchi’s theorem is the fact that every monadicsecond-order formula is equivalent on words to an existential monadic second-orderformula, and thus the iterative approach is not relevant as the hierarchy collapses. See(Ghilardi and van Gool, 2016) for a duality and type-theoretic approach via modelcompanions. However, for the first-order fragment the iterative approach is verypowerful. The first, and still prototypical application, is Sch¨utzenberger’s theoremwhich applies an iterative method, similar to the one of Section 2.1, to character-ise the first-order fragment via duality. To be more precise, (Sch¨utzenberger, 1965)shows that the star-free languages are precisely those recognised by (finite) aperiodicmonoids. To prove this, Sch¨utzenberger identified a semidirect product construc-tion which captures dually the application of concatenation product on languages. he fact that star-free languages are precisely those given by first-order sentencesof B¨uchi’s logic was subsequently shown in (McNaughton and Papert, 1971), thoughsome passages in the introduction of (Sch¨utzenberger, 1965) suggest that Sch¨utzen-berger was aware of this connection when he proved his result. Domain Theory in Logical Form.
In denotational semantics one seeks math-ematical models of programs, which should be assigned in a compositional way. Thecompositionality means that program constructors should correspond to type con-structors, and solutions to domain equations should correspond to program specific-ations. Scott’s original solution to the domain equation X ∼ = [ X, X ] , seeking a domain X which is isomorphic to the domain of its endomorphisms, was ob-tained by constructing a profinite poset, that is, a spectral space. Much further workconfirmed that categorical methods, topology and in particular duality are central tothe theory, cf. (Scott and Strachey, 1971; Plotkin, 1976; Smyth and Plotkin, 1982;Smyth, 1983; Larsen and Winskel, 1991). Rather than seeing Stone duality and itsvariants as useful technical tools for denotational semantics, Abramsky put Stoneduality front and center stage: A program logic is given in which denotational typescorrespond to theories and the ensuing Lindenbaum-Tarski algebras of the theoriesare bounded distributive lattices, whose dual spaces yield the domains as types. Theconstructors involved in the domain equations thus have duals under Stone duality,and solutions are obtained as duals of the solutions of the corresponding equationon the lattice side. In (Abramsky, 1987) Stone duality is restricted to the so-calledScott domains. That is, algebraic domains that are consistently complete. Theseare fairly simple and are closed under many constructors, including function space.In (Abramsky, 1991) the larger category of bifinite domains, which, in addition, isclosed under powerdomain constructions, is used. We will say a bit more about bifin-ite domains later, but for now, we illustrate with a simple example at the level ofspectral spaces.The Smyth powerdomain, S ( X ), is the space whose points are the compact andsaturated subsets of X equipped with the upper Vietoris topology (Smyth, 1983).That is, the topology is generated by the subbasis given by the sets (cid:3) U = { K ∈ S ( X ) | K ⊆ U } , for U ⊆ X open . At first sight, this may seem like quite an exotic object to pull out of a hat to studynon-determinism. However, in Abramsky’s duality with program logic, this constructis the Stone dual of adding a layer of (demonic) non-determinism. Indeed, if X is aspectral space, then so is S ( X ), and if L is the dual of X , then S ( X ) is the dual of F (cid:3) ( L ) = F DL ( (cid:3) L ) / ≈ . A subset K ⊆ X is saturated provided it is an intersection of opens, or equivalently, it is an up-set inthe specialisation order of the space X . ere, F DL ( (cid:3) L ) denotes the free distributive lattice on the set of formal generators (cid:3) L = { (cid:3) a | a ∈ L } , and ≈ is the congruence given by the following scheme ofrelations between the generators: (cid:3) ( ^ G ) ≈ ^ (cid:3) G for G ⊆ L finite . Note that the Smyth powerdomain generalises the Vietoris hyperspace constructionfor Boolean spaces and, indeed, when L = B is a Boolean algebra, the Booleanizationof the lattice F (cid:3) ( B ) coincides with the Boolean algebra M ( B ) from Section 2.1.Now the domain equation X = S ( X ) is solved by the final coalgebra for S . How-ever, a priori, there is no guarantee that it exists. On the other hand, the dualequation L = F (cid:3) ( L ) is solved by the initial algebra, i.e. the free (cid:3) -algebra over theempty set. As explained in Section 2.1, the latter algebra is guaranteed to exist sincealgebraic varieties are closed under filtered colimits.Even though the duality theoretic paradigm supplied by the program logic makesit clearer why S ( X ) is the right object, one may still wonder how difficult it is todiscover that F (cid:3) ( L ) and S ( X ) are dual to each other. But this also is made quitealgorithmic by duality: The dual of a free distributive lattice, such as F DL ( (cid:3) L ), issimply the Sierpinski cube (cid:3) L . Indeed, a subset S ⊆ (cid:3) L corresponds to the uniquehomomorphism h S : F DL ( (cid:3) L ) → extending the characteristic map χ S : (cid:3) L → .Viewed as a theory (or prime filter) it is F S = { ϕ | ∃ S ′ ⊆ S finite with V S ′ ≤ ϕ } .Also, a quotient of F DL ( (cid:3) L ) such as F (cid:3) ( L ) is dual to a subspace of (cid:3) L , namely theone consisting of all those S ⊆ (cid:3) L such that (cid:3) ( ^ G ) ∈ F S ⇐⇒ ^ (cid:3) G ∈ F S , for G ⊆ L finite . By the definition of F S , this is equivalent to (cid:3) ( ^ G ) ∈ S ⇐⇒ (cid:3) G ⊆ S, for G ⊆ L finite . Note that (cid:3) L is homeomorphic to P ( L ) with the topology generated by the sets e a = { S ∈ P ( L ) | a ∈ S } for a ∈ L . Viewed as subsets of L , the elements thatbelong to the dual of F (cid:3) ( L ) are precisely the filters of L . That is, S ( X ) is homeo-morphic to the space Filt( L ) equipped with the topology generated by the sets e a for a ∈ L . This algorithmic method, using duality for quotients of free algebras andthen inductively adding layers of a connective, has been applied widely in the set-ting of propositional logics, see e.g. (Ghilardi, 1992; Gehrke and Bezhanishvili, 2011;Ghilardi, 2010; Coumans and van Gool, 2012).In (Abramsky, 1991) a large number of constructors such as S are treated, includ-ing the function space which, given two spaces X and Y , yields the space [ X, Y ] of allcontinuous functions X → Y in the compact-open topology. This case is more subtle,but it is closely related to the one above, and to the duality between lattices with All distributive lattices are assumed to be bounded, and lattice homomorphisms preserve these bounds. In this section, the dualizing object is regarded as either a distributive lattice, or a spectral spaceby equipping the two-element set with the Sierpinski topology. esiduation and Stone topological algebras, which is at the heart of the duality the-ory of profinite methods in automata theory. For these reasons, we go in a bit moredetail. The following are extracts of a book in preparation (Gehrke and van Gool,2020).Consider the duality as above but for the operator type of implication. That is,given distributive lattices (DLs) L and M , define F → ( L × M ) = F DL ( → ( L × M )) / ≈ , where → ( L × M ) = { a → b | a ∈ L, b ∈ M } are the formal generators and ≈ is thecongruence given by the following two schemes of relations between the generators:(i) a → V G = V { a → b | b ∈ G } for a ∈ L and G ⊆ M finite;(ii) W F → b = V { a → b | a ∈ F } for F ⊆ L finite and b ∈ M .Going through the same exercise as outlined above to identify the elements of L × M which are compatible with the schemes (i) and (ii), one obtains the following result. Theorem 2.1.
Let L and M be DLs, and let X and Y be their respective dual spaces.The dual of F → ( L × M ) is the space [ X, S ( Y )] of continuous functions from X to theSmyth powerspace of Y , in the compact-open topology. This provides a dual description of [ X, S ( Y )], but we are interested in [ X, Y ]which is a subspace of [ X, S ( Y )]. However, it is not in general a closed subspace inthe patch topology, reflecting the fact that [ X, Y ] is not in general a spectral space.One would need to move to frames, sober spaces and geometric theories to describe[
X, Y ] as the dual of a quotient. However, we have the following approximation.
Proposition 2.2.
Let L and M be DLs, and X, Y their respective dual spaces. Thedual of the quotient of F → ( L × M ) by a congruence θ is a subspace of [ X, Y ] if andonly if for all x ∈ X , a ∈ F x , and finite subset G ⊆ M , there is a ′ ∈ F x such that [ a → ( _ G )] θ ≤ [ _ { a ′ → b | b ∈ G } ] θ . Here, F x denotes the prime filter of L corresponding to the point x ∈ X . The above property may be thought of as saying that the operations x → (-), for x ∈ X , preserve finite joins. For this reason, it has been called ‘preserving joins atprimes’. Cf. Section 3.2 of (Gehrke, 2016), where it is used to characterise the latticeswith residuation that are dual to topological algebras based on Boolean spaces.There is a special case in which we can get our hands on the property of preservingjoins at primes with a finitary scheme of relations between generators. This is the casewhere the lattice L has enough join prime elements, i.e. every a ∈ L is a finite join ofjoin prime elements of L . This is for example true in free distributive lattices (wherethe meets of finite sets of generators are join prime), and it is intimately related tothe interaction of domain theory and Stone duality as we have the following theorem. Theorem 2.3. (Abramsky, 1991, Theorem 2.4.5) A lattice has enough join primesif, and only if, its dual space endowed with the Scott topology is a domain. et L be a lattice with enough join primes, and X its dual space. If P = J ( L )is the subposet of join prime elements of L , the free distributive lattice on the poset P is isomorphic to L . Further, X ∼ = Idl( P op ), the free directed join completion of P op in the Scott topology, while P op ∼ = Comp( X ), the set of compact elements of X .In particular, X is an algebraic domain. Accordingly, we see that everything, i.e. L , X , and the compact elements of X , is determined by P . The posets P that occurin this way were described already in (Plotkin, 1976), where the profinite domainswere characterised as those algebraic domains for which the set of compact elementsform a ‘MUB-complete poset’ in the nomenclature of (Abramsky and Jung, 1995).We now have a corollary of Proposition 2.2. Corollary 2.4.
Let L and M be DLs with dual spaces X and Y , respectively. Suppose L has enough join primes and let P = J ( L ) . Then the quotient of F → ( L × M ) bythe congruence θ given by the following scheme is dual to the function space [ X, Y ] : p → _ G ≈ _ { p → b | b ∈ G } for p ∈ P and G ⊆ M finite . In the above, we have just talked about spectral spaces and domains, but in or-der to have a class of spectral domains not only closed under function spaces andproducts, but also under the various versions of powerdomain, one must restrictoneself to the so-called bifinite domains. These were introduced (in the setting of do-mains with a least element) in (Plotkin, 1976) as generated by special MUB-completeposets P now known as Plotkin orders (Abramsky and Jung, 1995, Definition 4.2.1).These also have a beautiful very self-dual description relative to Stone duality.The following definition applies to categories concrete over the category Pos ofposets and monotone maps, such as the category of DLs or that of spectral spaces andspectral maps (w.r.t. the specialization order) with the obvious forgetful functors.
Definition 2.5.
Let C be a category equipped with a faithful functor U : C →
Pos . Apair of morphisms C f −→ D g −→ C in C is an embedding-retraction-pair (e-r-p) provided( U ( f ) , U ( g )) is an adjoint pair, and U ( f ) is injective. Further, such an e-r-p is saidto be finite if U ( C ) is finite.We have the following easy duality result. Proposition 2.6.
In Stone duality, the dual of a (finite) embedding-retraction-pairon either side of the duality is a (finite) embedding-retraction-pair on the other side.
We may then define bifiniteness in the setting of spectral spaces, rather than inthe setting of domains as it is customarily done.
Definition 2.7.
Let X be a spectral space, and L its dual lattice. We say that X and L are bifinite provided the following two equivalent conditions are satisfied:1. X is the cofiltered limit of the retractions of its finite e-r-p’s;2. L is the filtered colimit of the embeddings of its finite e-r-p’s. It follows from these two conditions that U ( f ) is an embedding with left inverse U ( g ). he following proposition, which clearly implies that a bifinite lattice must haveenough join primes, allows us to conclude that bifinite spectral spaces are bifinitedomains. Thus, the above definition is no more general than the standard one. Proposition 2.8.
Let L be a distributive lattice and K ⊆ L a finite sublattice. Thenthe following conditions are equivalent:1. There is a lattice homomorphism h : L → K making ( i, h ) an embedding-retraction-pair, where i : K → L is the inclusion;2. (i) For all b ∈ L , ↓ b ∩ K is a principal downset;(ii) J ( K ) ⊆ J ( L ) . In the categorical logic approach, cf. Sections 1 and 2.2, the stratification of thealgebra of formulas (up to logical equivalence modulo T ) provided by the hyper-doctrine P : Con op → BA is in a sense impredicative. Indeed, it starts from thealgebra of sentences P ( ∅ ), which is what we ultimately want to understand, to buildall formulas on a countably infinite set of variables. This contrasts with the step-wiseconstruction of algebras of formulas outlined in Section 2.1.We want to understand quantification as a step-by-step construction. To this end,in this section we analyse from a duality theoretic viewpoint the inductive processof applying a layer of quantifiers in three settings. First, we focus on existentialquantification in first-order logic over arbitrary structures. Then, on semiring andprobabilistic quantifiers in first-order logic over finite structures.As explained in Section 1, Lindenbaum-Tarski algebras of predicate logics typic-ally fail to be free algebras. The challenge then consists, in a sense, in building freeobjects which approximate the Lindenbaum-Tarski algebra we are interested in. Weillustrate this idea in the following examples. For existential quantification in first-order logic, the framework can be loosely de-scribed as follows. Assume we are given a Boolean algebra of formulas B , and webuild a new Boolean algebra B ∃ x by adding a layer of the quantifier ∃ x to the formulasin B . We then have a quotient map M ( B ) B ∃ x sending ♦ ϕ to ∃ x.ϕ , where M ( B ) is the Boolean algebra obtained by freely adding onelayer of modality as described in Section 2.1. Dually, we get a continuous embedding V ( X ) X ∃ x where X and X ∃ x are the dual spaces of B and B ∃ x , respectively. We have approxim-ated the space B ∃ x by means of the Vietoris space V ( X ), whose dual is a free object namely, the free modal algebra on B ). The problem then consists in characterising X ∃ x as a subspace of V ( X ). This is addressed by observing that X ∃ x is the imageof a continuous map into V ( X ) constructed in a canonical way. In the remaining ofthis section we provide the necessary details.Recall from Section 2.2 that a first-order formula ϕ ∈ FO ω ( T ) can be identifiedwith the set J ϕ K ⊆ Mod ω / ≡ consisting of the (equivalence classes of) models withassignments satisfying ϕ . If the free variables of ϕ are contained in v , . . . , v n , wecan restrict the variable assignments accordingly. WriteMod n = { [( A, α : { v , . . . , v n } → A )] | A is a σ -structure and A | = T } , where [( A, α )] = [( A ′ , α ′ )] if and only if A, α | = ϕ ⇔ A ′ , α ′ | = ϕ for every ϕ ∈ FO n ( T ).Henceforth, we abuse notation and denote an arbitrary element of Mod n by ( A, α )instead of [(
A, α )]. Then, FO n ( T ) embeds into P (Mod n ) via the mapFO n ( T ) ֒ → P (Mod n ) , [ ϕ ] J ϕ K n = { ( A, α ) ∈ Mod n | A, α | = ϕ } . The projection map π i : Mod n ։ Mod n \ i which forgets the value of the assignments on the variable v i induces a Booleanalgebra embedding π − i : P (Mod n \ i ) ֒ → P (Mod n )by applying the contravariant power-set functor. As in the hyperdoctrine approach,the homomorphism π − i has a lower adjoint and it is given by taking direct imagesunder π i . P (Mod n \ i ) ⊤ P (Mod n ) π − i π i (-) This lower adjoint map can be thought of as the quantifier ∃ v i . Indeed, it is readilyseen that π i ( J ϕ K n ) = J ∃ v i .ϕ K n \ i . More generally, abstracting away from the Booleansubalgebra FO n ( T ) ֒ → P (Mod n ), we can consider any Boolean algebra embedding j : B ֒ → P (Mod n )and regard it as a ‘semantically given logic’. The Boolean algebra obtained by addinga layer of the quantifier ∃ v i to B can be identified with the Boolean subalgebra B i ∃ of P (Mod n \ i ) generated by the set of direct images { π i ( j ( ϕ )) | ϕ ∈ B } . We now focus on the dual of the transformation B B i ∃ . Let f : β (Mod n ) ։ X be the continuous map dual to j : B ֒ → P (Mod n ). Here, β (Mod n ) denotes the ˇCech-Stone compactification of Mod n regarded as a discrete space, and is the dual Stonespace of P (Mod n ). We obtain a continuous map R : β (Mod n \ i ) V ( β (Mod n )) V ( X ) . β ( π i ) − V ( f ) he first component of R is the preimage map x β ( π i ) − ( x ), where the function β ( π i ) : β (Mod n ) → β (Mod n \ i ) is the Stone dual of π − i : P (Mod n \ i ) → P (Mod n ).The map β ( π i ) − is continuous because π − i has a lower adjoint. Indeed, the join-semilattice homomorphism π i (-) : P (Mod n ) → P (Mod n \ i ) induces a Boolean algebrahomomorphism M ( P (Mod n )) → P (Mod n \ i ), whose dual map is precisely β ( π i ) − .We then have the following result. Proposition 3.1.
The image of the continuous map R : β (Mod n \ i ) → V ( X ) is thedual space of B i ∃ .Proof. It is not difficult to verify that R − ( ♦ b ϕ ) = \ π i ( j ( ϕ )) for every ϕ ∈ B , see e.g.Corollary 3.2 of (Borlido and Gehrke, 2019). Consequently, the Boolean algebra dualto the image of R can be identified with the subalgebra of P (Mod n \ i ) generated bythe elements of the form π i ( j ( ϕ )) for ϕ ∈ B , which is precisely B i ∃ .To sum up, the transformation B B i ∃ which adds one layer of quantifier ∃ v i dually corresponds to taking the image of the continuous map R : β (Mod n \ i ) → V ( X ),canonically constructed from the continuous function f : β (Mod n ) ։ X . For a step-by-step treatment of quantifiers, we now want to add to B i ∃ the formulas which werealready in B . Hence, we take the Boolean subalgebra of P (Mod n ) generated bythe union B ∪ B i ∃ , which coincides with the image of the obvious Boolean algebrahomomorphism B + B i ∃ → P (Mod n ). This corresponds, dually, to taking the imageof the continuous product map β (Mod n ) V ( X ) × X. ( R ◦ β ( π i )) × f An essential obstacle to a two-sided duality theory for quantifiers is the lack of acharacterisation of the continuous maps β (Mod n ) → V ( X ) × X arising this way. Wewill return to this point in Section 4. The existential quantifier ∃ captures the existence, or non-existence, of an elementsatisfying a property. As such, it is a two-valued query. Semiring quantifiers, asstudied for instance in logic on words, generalise ∃ by allowing us to count the numberof witnesses in a given semiring. Recall that a semiring is a tuple ( S, + , · , ,
1) where( S, + ,
0) is a commutative monoid, ( S, · ,
1) is a monoid, the operation · distributesover +, and 0 · s = 0 = s · s ∈ S . If S is a fixed finite semiring, every element k ∈ S determines a quantifier ∃ k . Given a first-order formula ϕ with one free variable v and a finite structure A , the semantics of the sentence ∃ k v.ϕ ( v ) is given as follows: A | = ∃ k v.ϕ ( v ) iff 1 + · · · + 1 (repeated m -times) is equal to k in S where m is the number of elements a ∈ A such that A | = ϕ ( a ). A particular class of semiring quantifiers is given by the modular quantifiers, which count in a finitecyclic ring Z /q Z . These were introduced in logic on words in (Straubing et al., 1995). otice that A must be finite, for otherwise the set { a ∈ A | A | = ϕ ( a ) } may be infiniteand the sum 1 + · · · + 1 undefined. This problem could be overcome by requiringthat S be complete in an appropriate sense. The existential quantifier ∃ is recoveredby letting S = be the two-element Boolean ring and k = 1.Let Fin n be the subset of Mod n consisting of the finite models with assignments.Given a Boolean algebra embedding j : B ֒ → P (Fin n ) we can construct, akin to thecase of ∃ , a Boolean algebra B i ∃ S obtained by adding a layer of semiring quantifiers ∃ k v i for k ∈ S . For every ϕ ∈ B and ( A, α ) ∈ Fin n \ i , write m ϕ, ( A,α ) for the numberof elements a in A such that ( A, α ∪ { v i a } ) belongs to j ( ϕ ). Then, B i ∃ S can bedefined as the Boolean subalgebra of P (Fin n \ i ) generated by the sets { ( A, α ) ∈ Fin n \ i | · · · + 1 ( m ϕ, ( A,α ) -times) is equal to k } , for ϕ ∈ B and k ∈ S. In order to describe the dual of the transformation B B i ∃ S , we need to un-derstand which construction plays the role of the Vietoris hyperspace in the case ofsemiring quantifiers. For this purpose, notice that the Vietoris space V ( X ) can beidentified with a space of two-valued finitely additive measures on X , whenever X isa Boolean space. Regard X as a measurable space where the measurable subsetsare precisely the clopens, i.e. the elements of the Boolean algebra B dual to X . Afinitely additive -valued measure on X is then a function µ : B → satisfying µ (0) = 0 and µ ( a ∨ b ) ∨ µ ( a ∧ b ) = µ ( a ) ∨ µ ( b ) ∀ a, b ∈ B. Denote by M ( X, ) the collection of all finitely additive -valued measures on X ,and equip it with the subspace topology induced by the product topology on B . Proposition 3.2.
For every Boolean space X , the Vietoris hyperspace V ( X ) ishomeomorphic to M ( X, ) via the map V ( X ) → M ( X, ) , C µ C , where µ C ( a ) = ( if b a ∩ C = ∅ , otherwise.Proof. It is straightforward to verify that the map in the statement is a continuousbijection, with inverse M ( X, ) → V ( X ), µ T { b a ⊆ X | µ ( ¬ a ) = 0 } . Every con-tinuous bijection between compact Hausdorff spaces is a homeomorphism, hence thestatement follows.For semiring quantifiers, the hyperspace V ( X ) will thus be replaced by M ( X, S ),the space of finitely additive S -valued measures on X . An element of M ( X, S ) is afunction µ : B → S satisfying µ (0) = 0 and µ ( a ∨ b ) + µ ( a ∧ b ) = µ ( a ) + µ ( b ) ∀ a, b ∈ B, (1)and the set M ( X, S ) is equipped with the subspace topology induced by the producttopology on S B . The equations in (1), encoding finite additivity, translate into Perhaps more natural would be to first identify V ( X ) with the space of filters on the dual Booleanalgebra of X , as explained towards the end of Section 2.2 in the case of the Smyth powerspace, and thenobserve that filters can be seen as two-valued finitely additive measures. qualiser diagrams in the category of Boolean spaces. Hence, the resulting space M ( X, S ) is again Boolean. Explicitly, the topology of M ( X, S ) is generated by the(clopen) subsets of the form[ a, k ] = { µ ∈ M ( X, S ) | µ ( a ) = k } , for a ∈ B and k ∈ S. In order to describe the dual of the construction B B i ∃ S , we perform two steps.First, given a finite model with assignment ( A, α ) ∈ Fin n \ i , let δ ( A,α ) : Fin n → S (2)be the ‘ S -valued characteristic function’ of π − i ( A, α ), where π i : Fin n → Fin n \ i isthe map which forgets the assignment of the i th variable. That is, δ i ( A,α ) ( A ′ , α ′ ) is1 if A = A ′ and α agrees with α ′ on the variables v , . . . , v i − , v i +1 , . . . , v n , and 0otherwise. Since A is finite, δ i ( A,α ) belongs to the set S (Fin n ) of finitely supported S -valued functions on Fin n . In the second step, in order to construct a measure, weextend the function δ i ( A,α ) to subsets of Fin n by adding up all the non-zero values ina given subset. More generally, if T is a set and g : T → S is a finitely supportedfunction, the map Z g : P ( T ) → S, P Z P g computed as X x ∈ P g ( x )is a finitely additive S -valued measure on β ( T ). We obtain an integration map Z : S ( T ) → M ( β ( T ) , S ) . Now, let f : β (Fin n ) → X be the dual of the embedding j : B ֒ → P (Fin n ). Con-sider the compositeFin n \ i δ i (-) −−−−→ S (Fin n ) R −−−→ M ( β (Fin n ) , S ) f ∗ −−−→ M ( X, S ) (3)where f ∗ sends a measure to its pushforward along f , i.e. f ∗ ( µ )( a ) = µ ( f − ( b a )) forevery µ ∈ M ( β (Fin n ) , S ) and a ∈ B . The space M ( X, S ) is compact and Hausdorff,whence the above composition extends to a (unique) continuous function R : β (Fin n \ i ) → M ( X, S ) . (4)The following result generalises Proposition 3.1 and can be proved in a similar manner(we omit the details here). Theorem 3.3.
The image of the continuous map R : β (Mod n \ i ) → M ( X, S ) is thedual space of B i ∃ S . In fact, the construction X
7→ M ( X, S ) yields a monad on
BStone and the integration map canbe upgraded to a monad morphism R : S ◦ U → U ◦ M ( − , S ), where S is the semiring monad on Set and U : BStone → Set is the forgetful functor. Cf. (Gehrke et al., 2017). While, for the purpose ofthis section, we may assume S is any pointed monoid, the monadic treatment requires the full semiringstructure. he connection between semiring quantifiers and spaces of finitely additive meas-ures was first explored, in the context of logic on words, in (Gehrke et al., 2017).The treatment in this section could be adapted to deal with any profinite semiring,such as the tropical semiring ( N ∪ {∞} , min , + , ∞ , Topological methods are also employed in the study of structural limits in finite modeltheory. A systematic investigation of limits of finite structures has been developedby Neˇsetˇril and Ossona de Mendez and is based on an embedding, called the
Stonepairing , of the collection of finite structures into a space of probability measures(Neˇsetˇril and Ossona de Mendez, 2012, 2020). The latter space is complete, thus itprovides the limit objects for those sequences of finite structures which embed asCauchy sequences. Although this space of measures and the Stone pairing embed-ding did not originate from duality, in recent work we showed that a closely relatedversion of the Stone pairing can be understood — via duality — as the embeddingof finite structures into a space of types. Namely, the space of 0-types of an exten-sion of first-order logic obtained by adding a layer of certain probabilistic quantifiers(Gehrke et al., 2020). In the following, we highlight the similarities between theStone pairing embedding and the space-of-measures construction introduced abovein the context of existential and semiring quantification.For every first-order formula ϕ with free variables contained in v , . . . , v n , andfinite structure A , the Stone pairing of ϕ and A is defined as h ϕ, A i = |{ a ∈ A n | A | = ϕ ( a ) }|| A | n . In other words, h ϕ, A i is the probability that a random assignment of the variables v , . . . , v n in A satisfies the formula ϕ . Upon fixing the second coordinate, the map h - , A i is a finitely additive measure on the dual space of the Lindenbaum-Tarskialgebra of all first-order formulas FO ω , with values in the unit interval [0 , h⊥ , A i = 0 and h ϕ ∨ ψ, A i + h ϕ ∧ ψ, A i = h ϕ, A i + h ψ, A i ∀ ϕ, ψ ∈ FO ω . Since the Boolean algebra FO ω is dual to the space of models and valuations Mod ω ,we obtain an embedding h - , - i : Fin −→ M (Mod ω , [0 , , A
7→ h - , A i where Fin is the collection of finite structures, up to isomorphism (with the notationof Section 3.2, Fin = Fin ). This is the Stone pairing embedding introduced byNeˇsetˇril and Ossona de Mendez.By restricting h - , A i to suitable fragments of first-order logic, Neˇsetˇril and Ossonade Mendez obtained a unifying framework that captures various notions of conver-gence of finite structures, such as Lovasz–Szegedy convergence, Benjamini–Schrammconvergence, elementary convergence, etc. Their insight was that each of these Note that the restriction of the Stone pairing embedding to a fragment of FO may fail to be injective. otions of convergence corresponds to a fragment of first-order logic. Further, sincethe ensuing spaces of finitely additive measures are complete, they admit a limit forevery sequence of finite structures which embeds as a Cauchy sequence.In section 3.2, we defined a map from a set of finite structures with evaluationsinto a space of finitely additive measures, see equation (3), and showed that it duallycaptures the adding of a layer of semiring quantifiers. By analogy, we may ask ifthe Stone pairing also corresponds to applying a layer of quantifiers. One immediateobstacle is that the spaces [0 ,
1] and M (Mod ω , [0 , ,
1] with a profinite version of theunit interval obtained from a codirected system of finitary approximations of realnumbers in [0 , Γ is naturally equipped with a Priestley spacestructure and can therefore be studied using Stone-Priestley duality for distributivelattices. To define Γ , we divide the unit interval into n segments of equal length, i.e. Γ n = { < n < n < . . . < } . The chain Γ n provides a finite approximation of [0 , n ∈ N ,the better the approximation is. Whenever n | m , we consider the flooring function Γ m → Γ n sending am to the largest bn ∈ Γ n such that bn ≤ am . Note that the finitechains Γ n with flooring functions between them form a codirected diagram in thecategory Pos f of finite posets with monotone maps. The limit of this diagram isan object Γ of the pro-completion of Pos f , which is the category of Priestley spaceswith continuous monotone maps. See e.g. Corollary VI.3.3 in (Johnstone, 1986).Concretely, the elements of Γ are the sequences of approximations ( x n ) n ∈ Q n ∈ N Γ n which are compatible with the flooring functions. Every q ∈ (0 ,
1] determines an ele-ment q − ∈ Γ , namely the sequence q − = ( q − , q − , q − , . . . ) where q − n = max { an ∈ Γ n | an < q } which approximates q from below while never reaching it. Further, if q is rational,we also get a lower approximating sequence q ◦ ∈ Γ which eventually stabilises at q : q ◦ = ( q ◦ , q ◦ , q ◦ , . . . ) where q ◦ n = max { an ∈ Γ n | an ≤ q } . In fact, any point of Γ is of one of these two types. We can thus think of Γ as a copyof the unit interval where all the non-zero rationals are doubled (in the picture, q isrational while r is irrational): r − q ◦ q − ◦ − ◦ Γ =Equivalently, Γ is a copy of the Cantor space with an extra top element which istopologically isolated (corresponding to 1 ◦ ). The natural order of Γ , illustrated inthe previous picture, is the total order defined by the two conditions A Priestley space is a pair ( X, ≤ ) where X is a compact space and ≤ is a partial order such that,whenever x y , there is a clopen subset C ⊆ X which is upward closed and satisfies x ∈ C and y / ∈ C . r ◦ < s − if and only if r < s in [0 , • q − < q ◦ for every q ∈ (0 , Γ retracts onto [0 , γ : Γ → [0 , , q − , q ◦ q has a (lower semicontinuous) section ι : [0 , → Γ , ι ( q ) = ( q ◦ if q is rational q − otherwise . The additive structure of [0 ,
1] lifts to Γ (as can be derived by duality for ad-ditional operators) so that it makes sense to consider the set M ( X, Γ ) of finitelyadditive probability measures on a Boolean space X with values in Γ . This construc-tion can be generalised to any Priestley space X , and it turns out that the assignment X
7→ M ( X, Γ ) is an endofunctor on the category of Priestley spaces. In particular,a continuous monotone map of Priestley spaces f : X → Y is sent to the map f ∗ : M ( X, Γ ) → M ( Y, Γ )taking a measure to its pushforward along f . Furthermore, the retraction-sectionpair γ : Γ ⇆ [0 ,
1] : ι lifts to a retraction-section pair γ : M ( X, Γ ) ⇆ M ( X, [0 , ι , where γ ( µ ) = γ ◦ µ and ι ( µ ) = ι ◦ µ. Now we define a Γ -valued variant of the Stone pairing by following the strategy setout in Section 3.2 in the case of semiring quantifiers. Fix n ∈ N , and let F (Fin n , Γ )be the set of finitely supported functions Fin n → Γ with total value 1 ◦ . We get amap δ (-) : Fin → F (Fin n , Γ ) sending a finite structure A to δ A : Fin n → Γ , where δ A ( A ′ , α ′ ) = (cid:16) | A | n (cid:17) ◦ if A ′ = A ◦ otherwise . The map δ (-) is the (normalized) Γ -valued version of the function introduced in (2)for semiring quantifiers. In a similar way, to move from finitely supported functionsto measures, for every set T we consider the integration map Z : F ( T, Γ ) → M ( β ( T ) , Γ ) , f Z f. Lastly, define the following composition R n : Fin F (Fin n , Γ ) M ( β (Fin n ) , Γ ) M (Mod n , Γ ) δ (-) R f ∗ where f : β (Fin n ) → Mod n is the dual map of the Boolean algebra homomorphismFO n → P (Fin n ) , ϕ J ϕ K ∩ Fin n . he map R n can be extended to a continuous function e R n : β (Fin) → M (Mod n , Γ ),corresponding to the map in (4). Using the fact that the space Mod ω is the codirectedlimit of the Mod n ’s for n ∈ N , and the functor M (- , Γ ) preserves codirected limits,we can ‘glue’ the maps e R n to get a continuous function e R : β (Fin) → M (Mod ω , Γ ).The restriction R : Fin → M (Mod ω , Γ ) of e R is an equivalent Γ -valued version of theStone pairing, as expressed by the commutativity of the following diagram. M (Mod ω , Γ )Fin M (Mod ω , [0 , γ R h - , - i ι The map R , and more precisely the way it is constructed, provides an interestinglink between the theory of structural limits and the inductive study of semiringquantifiers. Further, the duality approach allows us to see (the Γ -valued version of)the Stone pairing as an embedding of the finite structures into a space of types. Thisis the content of the following theorem, which is a special case of more general resultsin (Gehrke et al., 2020). Theorem 3.4.
The Boolean space M (Mod ω , Γ ) is dual to the Lindenbaum-Tarskialgebra of the propositional logic having as atoms p ≥ q ϕ and p We saw in Section 3.1 that adding a layer of existential quantifier ∃ to a Booleanalgebra B of first-order formulas (with free variables in v , . . . , v n ) dually correspondsto taking the image of a continuous map β (Mod n ) → V ( X ) × X , where X is the dualStone space of B . A similar statement holds for semiring quantifiers, cf. Section 3.2.This continuous map is defined in a canonical way, and ensures the soundness ofthe construction. But we do not know, so far, how to characterise the continuousmaps β (Mod n ) → V ( X ) × X arising in this manner, which would establish the completeness of the construction. This is a notable obstacle to a full duality theoreticunderstanding of step-by-step quantification in predicate logics. On the other hand,such a completeness result is available for semiring quantifiers in logic on words, andmakes use of the richer structure of the spaces of models (in the form of monoidactions). See Proposition VI.7 and Theorem VI.8 of (Gehrke et al., 2017), wherethis is called a ‘Reutenauer-type theorem’. A question arises, whose answer wouldsignificantly further the use of topological methods in logic: Is there a Reutenauer-type result for first-order logic over arbitrary structures? In this paper we have discussed several examples of topological methods in logicand computer science, highlighting their duality theoretic nature. However, there aretopological methods in logic which have been successfully developed and applied, butfor which no duality theoretic explanation is available so far. An appealing example isthe theory of limits of schema mappings as developed in database theory by Kolaitisand his collaborators (Kolaitis et al., 2018). Understanding these tools and resultsfrom a duality theoretic perspective may yield new useful insights and is an excitingvenue for future investigations. Another example are 0–1 laws in finite model theory,illustrating the limits of the expressive power of first-order logic over finite structures,see e.g. (Fagin, 1976). These are only some of the many opportunities for furtherdevelopment of the duality approach, which would contribute to unify the ‘structure’and ‘power’ strands in theoretical computer science.One of the main themes of our present contribution has been the analysis of step-by-step constructions in logic, which yield free objects on the algebra side and co-free objects on the space side. Note that, even though the step-wise process of adding alayer of connectives yields a monad in the (co)limit, the one-step functor is typicallya comonad . For instance, the functor on Boolean algebras which adds one layer ofmodality ♦ is a comonad, whose dual is the Vietoris monad on Boolean spaces.The recent work of Samson Abramsky and his coauthors on comonads for model-theoretic games (Abramsky et al., 2017; Abramsky and Shah, 2018) is tightly relatedto this viewpoint. The connection between the comonadic approach and the dualityone remains to be explored, and is an interesting avenue of research. In this direction,one may point out that the Ehrenfeucht-Fra¨ıss´e comonad introduced by Abramskyand Shah arises as the density comonad for a certain (contravariant) realizationfunctor from a category of primitive positive sentences into the category of structures.Besides the inductive treatment of quantifiers, another important theme of this aper has been the lack of freeness of Lindenbaum-Tarski algebras of first-ordertheories. Indeed, we pointed out that this is one of the main obstacles to a satisfactoryalgebraic and duality theoretic approach to predicate logics.Another place where the lack of freeness plays an important role is quantum in-formation and computation, to which Samson Abramsky has greatly contributed.There, as recently observed by Abramsky, the lack of freeness (of certain Booleansubalgebras of partial Boolean algebras) can be regarded as an obstruction to classic-ality. In fact, in the presence of freeness, the Kochen-Specker theorem does not apply.See (Abramsky and Barbosa, 2020). Interestingly, in this context, this obstructionrepresents a (quantum) advantage.We conclude with a question concerning a wider issue, which is instrumental inaddressing the divide between structure and power, one of the main focuses of SamsonAbramsky’s recent research. 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