Kleene algebras, adjunction and structural control
aa r X i v : . [ m a t h . L O ] M a y Kleene algebras, adjunction and structural control
Giuseppe Greco , Fei Liang , and Alessandra Palmigiano ∗ Utrecht University, the Netherlands Delft University of Technology, the Netherlands University of Johannesburg, South Africa
Abstract
In the present paper, we introduce a multi-type calculus for the logic of mea-surable Kleene algebras, for which we prove soundness, completeness, conserva-tivity, cut elimination and subformula property. Our proposal imports ideas andtechniques developed in formal linguistics around the notion of structural control[19].
Keywords : display calculus, measurable Kleene algebras, structural control.
Math . Subject Class 2010 : 03B45, 03G25, 03F05, 08A68.
A general pattern.
In this paper, we are going to explore the proof-theoretic ram-ifications of a pattern which recurs, with di ff erent motivations and guises, in variousbranches of logic, mathematics, theoretical computer science and formal linguistics.Since the most immediate application we intend to pursue is related to the issue of structural control in categorial grammar [19], we start by presenting this pattern in away that is amenable to make the connection with structural control. The pattern wefocus on features two types (of logical languages, of mathematical structures, of datastructures, of grammatical behaviour, etc.), a General one and a
Special one. Objectsof the
Special type can be regarded as objects of the
General type; moreover, each
General object can be approximated both “from above” and “from below” by
Special objects. That is, there exists a natural notion of order such that the collection of specialobjects order-embeds into that of general objects; moreover, for every general objectthe smallest special object exists which is greater than or equal to the given general one,and the greatest special object exists which is smaller than or equal to the given generalone. The situation just described can be captured category-theoretically by stipulatingthat a given faithful functor E : A → B between categories A (of the Special objects)and B (of the General objects) has both a left adjoint F : B → A and a right adjoint G : B → A , and moreover FE = GE = Id A . If we specialize this picture from categoriesto posets, the condition above can be reformulated by stating that the order-embedding e : A ֒ → B has both a left adjoint f : B ։ A and a right adjoint g : B ։ A such that ∗ This research is supported by the NWO Vidi grant 016.138.314, the NWO Aspasia grant 015.008.054,and a Delft Technology Fellowship awarded to the second author in 2013. e = ge = id A . From these conditions it also follows that the endomorphisms e f and eg on B are respectively a closure operator γ : B → B (mapping each general object to thesmallest special object which is greater than or equal to the given one) and an interioroperator ι : B → B (mapping each general object to the greatest special object whichis smaller than or equal to the given one). Examples.
A prime example of this situation is the natural embedding map e of theHeyting algebra A of the up-sets of a poset W , understood as an intuitionistic Kripkestructure, into the Boolean algebra B of the subsets of the domain of the same Kripkestructure. This embedding is a complete lattice homomorphism, and hence both itsright adjoint and its left adjoint exist. This adjunction situation is the mechanism se-mantically underlying the celebrated McKinsey-G¨odel-Tarski translation of intuition-istic logic into the classical normal modal logic S4 (cf. [2] for an extended discussion).Another example arises from the theory of quantales [21] (order-theoretic structuresarising as “noncommutative” generalizations of locales, or pointfree topologies). Forevery unital quantale, its two-sided elements form a locale, which is embedded in thequantale, and this embedding has both a left and a right adjoint, so that every elementof the quantale is approximated from above and from below by two-sided elements.A third example arises from the algebraic team semantics of inquisitive logic [13, 14],in which the embedding of the algebra interpreting flat formulas into the algebra in-terpreting general formulas has both a left adjoint and a right adjoint (cf. [6] for anexpanded discussion). Structural control.
These and other similar adjunction situations provide a promis-ing semantic environment for a line of research in formal linguistics, started in [19],and aimed at establishing systematic forms of communication between di ff erent gram-matical regimes. In [19], certain well known extensions of the Lambek calculus arestudied as logics for reasoning about the grammatical structure of linguistic resources,in such a way that the requirement of grammatical correctness on the linguistic sideis matched by the requirement of derivability on the logical side. In this regard, thevarious axiomatic extensions of the Lambek calculus correspond to di ff erent gram-matical regimes which become progressively laxer (i.e. recognize progressively moreconstructions as grammatically correct) as their associated logics become progressivelystronger. In this context, the basic Lambek calculus incarnates the most general gram-matical regime, and the ‘special’ behaviour of its extensions is captured by additionalanalytic structural rules. A systematic two-way communication between these gram-matical regimes is captured by introducing extra pairs of adjoint modal operators (the structural control operators), which make it possible to import a degree of flexibilityfrom the special regime into the general regime, and conversely, to endow the specialregime with enhanced ‘structural discrimination’ coming from the general regime. Thecontrol operators are normal modal operators inspired by the exponentials of linearlogic [7] but are not assumed to satisfy the modal S4-type conditions that are satisfiedby the linear logic exponentials. Interestingly, in linear logic, precisely the S4-type ax-ioms guarantee that the ‘of course’ exponential ! is an interior operator and the ‘whynot’ exponential ? is a closure operator , and hence each of them can be reobtained asthe composition of adjoint pairs of maps between terms of the linear (or general) typeand terms of the classical (or special) type, which are section / (co-)retraction pairs. In-stead, in [19], the adjunction situation is taken as primitive, and the structural control I.e. those elements x such that x · ≤ x and 1 · x ≤ x . / (co-)retraction pairs. In [10], a multi-type environ-ment for linear logic is introduced in which the Linear type encodes the behaviour ofgeneral resources, and the
Classical/Intuitionistic type encodes the behaviour of special(renewable) resources. The special behaviour is captured by additional analytic rules(weakening and contraction), and is exported in a controlled form into the general typevia the pairs of adjoint connectives which account for the well known controlled appli-cation of weakening and contraction in linear logic. This approach has made it possibleto design the first calculus for linear logic in which all rules are closed under uniformsubstitution (within each type), so that its cut elimination result becomes straightfor-ward. In [10] it is also observed that the same underlying mechanisms can be used toaccount for the controlled application of other structural rules, such as associativity andexchange. Since these are precisely the structural analytic rules capturing the specialgrammatical regimes in the setting of [19], this observation strengthens the connectionbetween linear logic and the structural control approach of [19].
Kleene algebras: similarities and di ff erences. In this paper, we focus on the casestudy of Kleene algebras in close relationship with the ideas of structural control andthe multi-type approach illustrated above. Kleene algebras have been introduced toformally capture the behaviour of programs modelled as relations [17, 18]. While gen-eral programs are encoded as arbitrary elements of a Kleene algebra, the Kleene starmakes it possible to access the special behaviour of reflexive and transitive programsand to import it in a controlled way within the general environment. Hence, the roleplayed by the Kleene star is similar to the one played by the exponential ? in linearlogic, which makes it possible to access the special behaviour of renewable resources,captured proof-theoretically by the analytic structural rules of weakening and contrac-tion, and to import it, in a controlled way, into the environment of general resources.Another similarity between the Kleene star and ? is that their axiomatizations guaran-tee that their algebraic interpretations are closure operators, and hence can be obtainedas the composition of adjoint maps in a way which provides the approximation “fromabove” which is necessary to instantiate the general pattern described above, and useit to justify the soundness of the controlled application of the structural rules capturingthe special behaviour. However, in the general setting of Kleene algebras there is noapproximation “from below”, as e.g. it is easy to find examples in the context of Kleenealgebras of relations in which more than one reflexive transitive relation can be maxi-mally contained in a given general relation. Our analysis (cf. Section 4) identifies thelack of such an approximation “from below” as the main hurdle preventing the devel-opment of a smooth proof-theoretic treatment of the logic of general Kleene algebras,which to date remains very challenging.
Extant approaches to the logic of Kleene algebras and PDL.
The di ffi culties in theproof-theoretic treatment of the logic of Kleene algebras propagate into the di ffi cultiesin the proof-theoretic treatment of Propositional Dynamic Logic (PDL) [23, 12, 4].Indeed, PDL can be understood (cf. [4]) as an expansion of the logic of Kleene alge-bras with a Formula type. Heterogeneous binary operators account for the connectionbetween the action / program types and the Formula type. The properties of these bi-nary operators are such that their proof-theoretic treatment is per se unproblematic.However, the PDL axioms encoding the behaviour of the Kleene star are non ana-lytic , and in the literature several approaches have been proposed to tackle this hurdle,which always involve some trade-o ff : from sequent calculi with finitary rules but with3 non-eliminable analytic cut [12, 15], to cut-free sequent calculi with infinitary rules[23, 22]. Measurable Kleene algebras.
In this paper, we introduce a subclass of Kleene alge-bras, referred to as measurable
Kleene algebras, which are Kleene algebras endowedwith a dual Kleene star operation, associating any element with its reflexive transitive interior . Similar definitions have been introduced in the context of dioids (cf. e.g. [11]and [20]; in the latter, however, the order-theoretic behaviour of the dual Kleene star isthat of a second closure operator rather than that of an interior operator). In measurableKleene algebras, the defining properties of the dual Kleene star are those of an interior operator, which then provides the approximation “from below” which is missing in thesetting of general Kleene algebras. Hence measurable Kleene algebras are designed toprovide yet another instance of the pattern described in the beginning of the present in-troduction. In this paper, this pattern is used as a semantic support of a proper displaycalculus for the logic of measurable Kleene algebras, and for establishing a concep-tual and technical connection between Kleene algebras and structural control which ispotentially beneficial for both areas.
Structure of the paper.
In Section 2, we collect preliminaries on (continuous) Kleenealgebras and their logics, introduce the notion of measurable Kleene algebra, and pro-pose an axiomatization for the logic corresponding to this class. In Section 3, we in-troduce the heterogeneous algebras corresponding to (continuous, measurable) Kleenealgebras and prove that each class of Kleene algebras can be equivalently presentedin terms of its heterogeneous counterpart. In Section 4, we introduce multi-type lan-guages corresponding to the semantic environments of heterogeneous Kleene algebras,define a translation from the single -type languages to the multi-type languages, andanalyze the proof-theoretic hurdles posed by Kleene logic with the lenses of the multi-type environment. This analysis leads to our proposal, introduced in Section 5, of aproper display calculus for the logic of measurable Kleene algebras. In Section 6 weverify that this calculus is sound, complete, conservative and has cut elimination andsubformula property.
Definition 1. A Kleene algebra [16] is a structure K = ( K , ∪ , · , () ∗ , , such that:K1 ( K , ∪ , is a join-semilattice with bottom element ;K2 ( K , · , is a monoid with unit , moreover · preserves ∪ in each coordinate, and is an annihilator for · ;K3 ∪ α · α ∗ ≤ α ∗ , ∪ α ∗ · α ≤ α ∗ , and ∪ α ∗ · α ∗ ≤ α ∗ ;K4 α · β ≤ β implies α ∗ · β ≤ β ;K5 β · α ≤ β implies β · α ∗ ≤ β . The name is chosen by analogy with measurable sets in analysis, which are defined in terms of theexistence of approximations “from above” and “from below”. Kleene algebra is continuous [16] if: K1’ ( K , ∪ , is a complete join-semilattice;K2’ · is completely join-preserving in each coordinate;K6 α ∗ = S α n for n ≥ . Lemma 1. [17, Section 2.1] For any Kleene algebra K and any α, β ∈ K,1. α ≤ α ∗ ;2. α ∗ = α ∗∗ ;3. if α ≤ β , then α ∗ ≤ β ∗ . By Lemma 1, the operation ∗ : K → K is a closure operator on K seen as a poset. Lemma 2.
For any continuous Kleene algebra K and any α, β ∈ K,If α ≤ β and ≤ β and β · β ≤ β then α ∗ ≤ β. Next, we introduce a subclass of Kleene algebras endowed with both a Kleene starand a dual Kleene star. To our knowledge, this definition has not appeared as suchin the literature, although similar definitions have been proposed in di ff erent settings(cf. [20, 1]). Definition 2. A measurable Kleene algebra is a structure K = ( K , ∪ , · , () ∗ , () ⋆ , , suchthat:MK1 ( K , ∪ , · , () ∗ , , is a continuous Kleene algebra;MK2 () ⋆ is a monotone unary operation;MK3 ≤ α ⋆ , and α ⋆ · α ⋆ ≤ α ⋆ ;MK4 α ⋆ ≤ α and α ⋆ ≤ α ⋆⋆ ;MK5 β ≤ α and ≤ β and β · β ≤ β implies β ≤ α ⋆ . Lemma 3.
For any measurable Kleene algebra K and any α ∈ K, if ≤ α and α · α ≤ α ,then α ∗ = α = α ⋆ . Hence,
Range ( ∗ ) = Range ( ⋆ ) = { β ∈ K | ≤ β and β · β ≤ β } . Proof.
By MK4 α ⋆ ≤ α ; the converse direction follows by MK5 with β : = α . ByLemma 1, α ≤ α ∗ ; the converse direction follows from Lemma 2. This completes theproof of the first part of the statement, and of the inclusion of the set of the β s with thespecial behaviour into Range ( ∗ ) and Range ( ⋆ ). The converse inclusions immediatelyfollow from K3 and MK3. (cid:3) For any n ∈ N let α n be defined by induction as follows: α : = α n + : = α n · α . .2 The logics of Kleene algebras Fix a denumerable set
Atprop of propositional variables, the elements of which aredenoted a , b possibly with sub- or superscripts. The language KL over Atprop is definedrecursively as follows: α :: = a | | | α ∪ α | α · α | α ∗ In what follows, we use α, β, γ (with or without subscripts) to denote formulas in KL . Definition 3.
Kleene logic, denoted S . KL , is presented in terms of the following axioms ⊢ α, α ⊢ α, α ⊢ α ∨ β, β ⊢ α ∨ β, · α ⊣⊢ α · , · α ⊣⊢ , · α ⊣⊢ α · , · α ⊣⊢ α, α · ( β ∪ γ ) ⊣⊢ ( α · β ) ∪ ( α · γ ) , ( β ∪ γ ) · α ⊣⊢ ( β · α ) ∪ ( γ · α )( α · β ) · γ ⊣⊢ α · ( β · γ ) , ∪ α · α ∗ ⊢ α ∗ , ∪ α ∗ · α ⊢ α ∗ , ∪ α ∗ · α ∗ ⊢ α ∗ and the following rules: α ⊢ β β ⊢ γα ⊢ γ α ⊢ γ β ⊢ γα ∨ β ⊢ γ α ⊢ β α ⊢ β α · α ⊢ β · β α · β ⊢ β K4 α ∗ · β ⊢ β β · α ⊢ β K5 β · α ∗ ⊢ β Continuous Kleene logic, denoted S . KL ω , is the axiomatic extension of S . KL deter-mined by the following axioms: α · ( [ i ∈ ω β i ) ⊣⊢ [ i ∈ ω ( α · β i ) , [ i ∈ ω β i · α ⊣⊢ [ i ∈ ω ( β i · α ) , [ n ≥ β · α n · γ ⊣⊢ β · α ∗ · γ Theorem 1. [17] (S . KL ω ) S . KL is complete with respect to (continuous) Kleene alge-bras. The language
MKL over
Atprop is defined recursively as follows: α :: = a | | | α ∪ α | α · α | α ∗ | α ⋆ . Definition 4.
Measurable Kleene logic, denoted S . MKL , is presented in terms of theaxioms and rules of S . KL plus the following axioms: ⊢ α ⋆ α ⋆ · α ⋆ ⊢ α ⋆ α ⋆ ⊢ α α ⋆ ⊢ ( α ⋆ ) ⋆ and the following rules: α ⊢ βα ⋆ ⊢ β ⋆ β ⊢ α ⊢ β β · β ⊢ ββ ⊢ α ⋆ In the present section, we introduce the algebraic environment which justifies seman-tically the multi-type approach to the logic of measurable Kleene algebras which wedevelop in Section 2.2. In the next subsection, we take Kleene algebras as startingpoint, and expand on the properties of the image of the algebraic interpretation of theKleene star, leading to the notion of ‘kernel’. In the remaining subsections, we showthat (continuous, measurable) Kleene algebras can be equivalently presented in termsof their corresponding heterogeneous algebras.6 .1 Kleene algebras and their kernels
By Lemma 1, for any Kleene algebra K , the operation () ∗ : K → K is a closure operatoron K seen as a poset. By general order-theoretic facts (cf. [3, Chapter 7]) this meansthat () ∗ = e γ, where γ : K ։ Range ( ∗ ), defined by γ ( α ) = α ∗ for every a ∈ K , is the left adjoint of thenatural embedding e : Range ( ∗ ) ֒ → K , i.e. for every α ∈ K , and ξ ∈ Range ( ∗ ), γ ( α ) ≤ ξ i ff α ≤ e ( ξ ) . In what follows, we let S be the subposet of K identified by Range ( ∗ ) = Range ( γ ).We will also use the variables α, β , possibly with sub- or superscripts, to denote ele-ments of K , and π, ξ, χ , possibly with sub- or superscripts, to denote elements of S . Lemma 4.
For every Kleene algebra K and every ξ ∈ S , γ ( e ( ξ )) = ξ. (1) Proof.
By adjunction, γ ( e ( ξ )) ≤ ξ i ff e ( ξ ) ≤ e ( ξ ), which always holds. As to the con-verse inequality ξ ≤ γ ( e ( ξ )), since e is an order-embedding, it is enough to show that e ( ξ ) ≤ e ( γ ( e ( ξ ))), which by adjunction is equivalent to γ ( e ( ξ )) ≤ γ ( e ( ξ )), which alwaysholds. (cid:3) Definition 5.
For any Kleene algebra K = ( K , ∪ , · , () ∗ , , , let the kernel of K be thestructure S = ( S , ⊔ , s ) defined as follows:KK1. S : = Range ( ∗ ) = Range ( γ ) , where γ : K ։ S is defined by letting γ ( α ) = α ∗ forany α ∈ K;KK2. ξ ⊔ χ : = γ ( e ( ξ ) ∪ e ( χ )) ;KK3. s : = γ (0) . Proposition 1. If K is a (continuous) Kleene algebra, then its kernel S defined as aboveis a (complete) join-semilattice with bottom element.Proof. By KK1, S is a subposet of K . Let ξ, χ ∈ S . Using KK2 and Lemma 1, oneshows that ξ ⊔ χ is a common upper bound of ξ and χ w.r.t. the order S inherits from K .Since e and γ are monotone, ξ ≤ π and χ ≤ π imply that ξ ⊔ χ = γ ( e ( ξ ) ∪ e ( χ )) ≤ γ ( e ( π )) = π , the last equality due to Lemma 1. This shows that ξ ⊔ χ is the least upper bound of ξ and χ w.r.t. the inherited order. Analogously one shows that, if K is continuous and Y ⊆ S , F Y : = γ ( S e [ Y ]) is the least upper bound of Y . Finally, γ (0) being the bottomelement of S follows from 0 being the bottom element of K and the monotonicity andsurjectivity of γ . (cid:3) Remark 1.
We have proved a little more than what is stated in Proposition 1. Namely,we have proved that all (finite) joins exist w.r.t. the order that S inherits from K, andhence the join-semilattice structure of S is also in a sense inherited from K. However,this does not mean or imply that S is a sub-join-semilattice of K, since joins in S are‘closures’ of joins in K, and hence ⊔ is certainly not the restriction of ∪ to S . .2 Measurable Kleene algebras and their kernels The results of Section 3.1 apply in particular to measurable Kleene algebras, where inaddition, by definition, the operation () ⋆ : K → K is an interior operator on K seen as aposet. By general order-theoretic facts (cf. [3, Chapter 7]) this means that() ⋆ = e ′ ι, where ι : K ։ Range ( ⋆ ), defined by ι ( α ) = α ⋆ for every a ∈ K , is the right adjoint ofthe natural embedding e ′ : Range ( ⋆ ) ֒ → K , i.e. for every α ∈ K and ξ ∈ Range ( ⋆ ), e ′ ( ξ ) ≤ α i ff ξ ≤ ι ( α ) . Moreover, Lemma 3 guarantees that
Range ( ∗ ) = Range ( ⋆ ) = { β ∈ K | ≤ β and β · β ≤ β } . Hence, e ′ coincides with the natural embedding e : Range ( ∗ ) ֒ → K , which is then en-dowed with both the left adjoint and the right adjoint.In what follows, we let S be the subposet of K identified by Range ( ∗ ) = Range ( γ ) = Range ( ι ) = Range ( ⋆ ) . We will use the variables α, β , possibly with sub- or superscripts, to denote elements of K , and π, ξ, χ , possibly with sub- or superscripts, to denote elements of S . Lemma 5.
For every measurable Kleene algebra K and every ξ ∈ S , γ ( e ( ξ )) = ξ = ι ( e ( ξ )) . (2) Proof.
The first identity is shown in Lemma 1. As to the second one, by adjunction, ξ ≤ ι ( e ( ξ )) i ff e ( ξ ) ≤ e ( ξ ), which always holds. As to the converse inequality ι ( e ( ξ )) ≤ ξ ,since e is an order-embedding, it is enough to show that e ( ι ( e ( ξ ))) ≤ e ( ξ ), which byadjunction is equivalent to ι ( e ( ξ )) ≤ ι ( e ( ξ )), which always holds. (cid:3) Definition 6.
For any measurable Kleene algebra K = ( K , ∪ , · , () ∗ , () ⋆ , , , let the ker-nel of K be the structure S = ( S , ⊔ , s ) defined as follows:KK1. S : = Range ( ∗ ) = Range ( γ ) = Range ( ι ) = Range ( ⋆ ) ;KK2. ξ ⊔ χ : = γ ( e ( ξ ) ∪ e ( χ )) ;KK3. s : = γ (0) . Definition 7. A heterogeneous Kleene algebra is a tuple H = ( A , ∼ , ⊗ , ⊗ , γ, e ) verify-ing the following conditions:H1 A = ( A , ⊔ , · , s , is such that ( A , ⊔ , a join-semilattice with bottom element and ( A , · , s ) a monoid with unit , moreover · preserves finite joins in eachcoordinate, and is an annihilator for · ;H2 ∼ = ( S , ⊔ , s ) is a join-semilattice with bottom element s ; ⊗ : ∼ × A → A preserves finite joins in its second coordinate, is monotone inits first coordinate, and has unit in its second coordinate, and ⊗ : A × ∼ → A preserves finite joins in its first coordinate, is monotone in its second coordinate,and has unit in its first coordinate. Moreover, for all α ∈ A and ξ ∈ S , ξ ⊗ α = e ( ξ ) · α and α ⊗ ξ = α · e ( ξ ); (3) H4 γ : A ։ ∼ and e : ∼ ֒ → A are such that γ ⊣ e and γ ( e ( ξ )) = ξ for all ξ ∈ S ;H5 ≤ e ( ξ ) , and e ( ξ ) · e ( ξ ) ≤ e ( ξ ) for any ξ ∈ ∼ ;H6 α · β ≤ β implies γ ( α ) ⊗ β ≤ β , and β · α ≤ β implies β ⊗ γ ( α ) ≤ β for all α, β ∈ A .A heterogeneous Kleene algebra is continuous ifH1’ ( A , ⊔ , is a complete join-semilattice and · preserves arbitrary joins in eachcoordinate;H2’ ∼ = ( S , ⊔ , s ) is a complete join-semilattice;H7 e ( γ ( α )) = S α n for any n ∈ N . Definition 8.
For any Kleene algebra K = ( K , ∪ , · , () ∗ , , , let K + = ( A , ∼ , ⊗ , ⊗ , γ, e ) be the structure defined as follows:1. A : = ( K , ∪ , · , , is the () ∗ -free reduct of K ;2. ∼ is the kernel of K (cf. Definition 5);3. γ : A ։ ∼ and e : ∼ ֒ → A are defined as the maps into which the closure operator () ∗ decomposes (cf. discussion before Lemma 1);4. ⊗ (resp. ⊗ ) is the restriction of · to S in the first (resp. second) coordinate. Proposition 2.
For any (continuous) Kleene algebra K , the structure K + defined aboveis a (continuous) heterogeneous Kleene algebra.Proof. Since K verifies by assumption K1 and K2, K + verifies H1. Condition H2(resp. H2’) is verified by Proposition 1. Condition H3 immediately follows from thedefinition of ⊗ and ⊗ in K + . Condition H4 holds by Lemma 1 and 1. ConditionH5 follows from K verifying assumption K3. Condition H6 follows from K verifyingassumption K4 and K5. If K is continuous, then K verifies conditions K1’, K2’ andK6, which guarantee that K + verifies H1’ and H7. (cid:3) Definition 9.
For any heterogeneous Kleene algebra H = ( A , ∼ , ⊗ , ⊗ , γ, e ) , let H + : = ( A , () ∗ ) , where () ∗ : A → A is defined by α ∗ : = e ( γ ( α )) for every α ∈ A . Proposition 3.
For any (continuous) heterogeneous Kleene algebra H = ( A , ∼ , ⊗ , ⊗ , γ, e ) , the structure H + defined above is a (continuous) Kleene algebra. Moreover,the kernel of H + is join-semilattice-isomorphic to ∼ .Proof. As to the first part of the statement, we only need to show that () ∗ satisfiesconditions K3-K5 (resp. K1’, K2’ and K6) of Definition 1. Condition K3 easily followsfrom assumption H5 and the proof is omitted. As to K4, let α, β ∈ A such that α · β ≤ β .9 · β ≤ β ⇒ γ ( α ) ⊗ β ≤ β (H6) ⇒ e ( γ ( α )) · β ≤ β (H3) ⇒ α ∗ · β ≤ β (definition of () ∗ )The proof of K5 is analogous. Conditions K1’, K2’ and K6 readily follow from as-sumptions H1’ and H7.This completes the proof of the first part of the statement. As to the second part, letus show preliminarily that the following identities hold:AK2. ξ ⊔ χ : = γ ( e ( ξ ) ∪ e ( χ )) for all ξ, χ ∈ S ;AK3. 0 s : = γ (0).Being a left adjoint, γ preserves existing joins. Hence, γ (0) = s , which proves (AK2),and, using H4, γ ( e ( ξ ) ∪ e ( χ )) = γ ( e ( ξ )) ⊔ γ ( e ( χ )) = ξ ⊔ χ , which proves (AK3). Toshow that the kernel of H + and S are isomorphic as (complete) join-semilattices, noticethat the domain of the kernel of H + is defined as K ∗ : = Range (() ∗ ) = Range ( e ◦ γ ) = Range ( e ). Since e is an order-embedding (which is easily shown using H4), this im-plies that K ∗ , regarded as a sub-poset of A , is order-isomorphic to the domain of S with its join-semilattice order. Let i : S → K ∗ denote the order-isomorphism between S and K ∗ . To show that ∼ = ( S , ⊔ ∼ , s ) and K ∗ = ( K ∗ , ⊔ K ∗ , s ∗ ) are isomorphic as join-semilattices, we need to show that for all ξ, χ ∈ S , i ( ξ ⊔ S χ ) = i ( ξ ) ⊔ K ∗ i ( χ ) and i (0 s ) = s ∗ . Let e ′ : K ∗ ֒ → A and γ ′ : A ։ K ∗ be the pair of adjoint maps arising from ∗ . Thus, e = e ′ i and γ ′ = i γ , and so, i ( ξ ) ⊔ K ∗ i ( χ ) = γ ′ ( e ′ ( i ( ξ )) ∪ e ′ ( i ( χ ))) (definition of ⊔ K ∗ ) = γ ′ ( e ( ξ ) ∪ e ( χ )) ( e = e ′ i ) = i ( γ ( e ( ξ ) ∪ e ( χ ))) ( γ ′ = i γ ) = i ( ξ ⊔ S χ ). (AK2)0 s ∗ = γ ′ (0) (KK3) = i ( γ (0)) ( γ ′ = i γ ) = i (0 s ) (AK3) (cid:3) The following proposition immediately follows from Propositions 2 and 3:
Proposition 4.
For any Kleene algebra K and heterogeneous Kleene algebra H , K (cid:27) ( K + ) + and H (cid:27) ( H + ) + . Moreover, these correspondences restrict to continuous Kleene algebras and continu-ous heterogeneous Kleene algebras.
The extra conditions of measurable Kleene algebras allow for their ‘heterogeneouspresentation’ (encoded in the definition below) being much simpler than the one forKleene algebras:
Definition 10. A heterogeneous measurable Kleene algebra is a tuple H = ( A , ∼ , ι, γ, e ) verifying the following conditions: M1 A = ( A , ⊔ , · , s , is such that ( A , ⊔ , a complete join-semilattice with bottomelement and ( A , · , s ) a monoid with unit , moreover · preserves arbitraryjoins in each coordinate, and is an annihilator for · ;HM2 ∼ = ( S , ⊔ , s ) is a complete join-semilattice with bottom element s ;HM3 e ( γ ( α )) = S α n for any n ∈ N .HM4 γ : A ։ ∼ and ι : A ։ ∼ and e : ∼ ֒ → A are such that γ ⊣ e ⊣ ι and γ ( e ( ξ )) = ξ = ι ( e ( ξ )) for all ξ ∈ S ;HM5 ≤ e ( ξ ) , and e ( ξ ) · e ( ξ ) ≤ e ( ξ ) for any ξ ∈ ∼ ;HM6 For any β ∈ A , if ≤ β and β · β ≤ β , then γ ( β ) ≤ ι ( β ) . Definition 11.
For any measurable Kleene algebra K = ( K , ∪ , · , () ∗ , () ⋆ , , , let K + = ( A , ∼ , ι, γ, e ) be the structure defined as follows:1. A : = ( K , ∪ , · , , is the { () ∗ , () ⋆ } -free reduct of K ;2. ∼ is the kernel of K (cf. Definition 6);3. γ : A ։ ∼ and e : ∼ ֒ → A are defined as the maps into which the closure operator () ∗ decomposes, and ι : A ։ ∼ and e : ∼ ֒ → A are defined as the maps into whichthe interior operator () ⋆ decomposes (cf. discussion before Lemma 5). Proposition 5.
For any measurable Kleene algebra K , the structure K + defined aboveis a heterogeneous measurable Kleene algebra.Proof. Since K verifies by assumption K1’, K2, and K6, K + verifies HM1. ConditionHM2 is verified by Proposition 1. Condition HM3 immediately follows from the defi-nition of () ∗ and assumption K6. Condition HM4 holds by Lemma 5. Condition HM5follows from K verifying assumption K3. As to condition HM6, if 1 ≤ β and β · β ≤ β ,then by Lemma 3, e ( γ ( β )) = β ∗ = β ⋆ = e ( ι ( β )), which implies, since e is injective, that γ ( β ) ≤ ι ( β ), as required. (cid:3) Definition 12.
For any heterogeneous measurable Kleene algebra H = ( A , ∼ , ι, γ, e ) ,let H + : = ( A , () ∗ , () ⋆ ) , where () ∗ : A → A and () ⋆ : A → A are respectively defined by α ∗ : = e ( γ ( α )) and α ⋆ : = e ( ι ( α )) for every α ∈ A . Proposition 6.
For any heterogeneous measurable Kleene algebra H = ( A , ∼ , ι, γ, e ) ,the structure H + defined above is a measurable Kleene algebra. Moreover, the kernelof H + is join-semilattice-isomorphic to ∼ .Proof. The part of the statement which concerns the verification of axioms K1’, K2’,K3-K6 is accounted for as in the proof of Proposition 3. Let us verify that () ⋆ satisfiesconditions MK2-MK5 of Definition 2. Conditions MK2 and MK4 easily follow fromthe assumption that e ⊣ ι (HM4). Condition MK3 follows from the surjectivity of ι andassumption HM5. As to MK5, it is enough to show that if α, β ∈ K such that β ≤ α and 1 ≤ β and β · β ≤ β , then β ≤ e ( ι ( α )). Since β ≤ α by assumption and e and ι aremonotone, it is enough to show that β ≤ e ( ι ( β )). By adjunction, this is equivalent to γ ( β ) ≤ ι ( β ), which holds by assumption HM6. This completes the proof of the first partof the statement. The proof of the second part is analogous to the corresponding partof the proof of Proposition 3, and is omitted. (cid:3) Proposition 7.
For any measurable Kleene algebra K and heterogeneous measurableKleene algebra H , K (cid:27) ( K + ) + and H (cid:27) ( H + ) + . In Section 3.3, (continuous) heterogeneous (measurable) Kleene algebras have beenintroduced (cf. Definitions 7 and 10) and shown to be equivalent presentations of (con-tinuous, measurable) Kleene algebras. These constructions motivate the multi-typepresentations of Kleene logics we introduce in the present section. Indeed, heteroge-neous Kleene algebras are natural models for the following multi-type language L MT ,defined by simultaneous induction from a set AtAct of atomic actions (the elements ofwhich are denoted by letters a , b ): Special ∋ ξ :: = (cid:7) α General ∋ α :: = a | | | (cid:3) ξ | α ∪ α while heterogeneous measurable Kleene algebras are natural models for the follow-ing multi-type language L MT , defined by simultaneous induction from AtAct : Special ∋ ξ :: = (cid:7) α | (cid:4) α General ∋ α :: = a | | | (cid:3) ξ | α ∪ α where, in any heterogeneous (measurable) Kleene algebra, the maps γ and e (and ι )interpret the heterogeneous connectives (cid:7) , (cid:3) (and (cid:4) ) respectively. The interpretation of L MT -terms into heterogeneous algebras is defined as the straightforward generalizationof the interpretation of propositional languages in algebras of compatible signature, andis omitted.The toggle between Kleene algebras and heterogeneous Kleene algebras is reflectedsyntactically by the following translation ( · ) t : L → L MT between the original language L of Kleene logic and the language L MT defined above: a t = a t = t = α ∪ β ) t = α t ∪ β t ( α · β ) t = α t · β t ( α ∗ ) t = (cid:3)(cid:7) α t ( α ⋆ ) t = (cid:3) (cid:4) α t The following proposition is proved by a routine induction on L -formulas. Proposition 8.
For all L -formulas A and B and every Kleene algebra K , K | = α ≤ β i ff K + | = α t ≤ β t . The general definition of analytic inductive inequalities can be instantiated to in-equalities in the L MT -signature according to the order-theoretic properties of the al-gebraic interpretation of the L MT -connectives in heterogeneous (measurable) Kleene12lgebras. In particular, all connectives but ⊗ and ⊗ are normal. Hence, we are nowin a position to translate the axioms and rules describing the behaviour of () ∗ and () ⋆ from the single-type languages into L MT using ( · ) t , and verify whether the resultingtranslations are analytic inductive.1 ∪ α ≤ α ∗ ∪ α t ≤ (cid:3)(cid:7) α t ( i ) (cid:3)(cid:7) α t ≤ ∪ α t ( ii )1 ∪ α ∗ = α ∗ ∪ (cid:3)(cid:7) α t ≤ (cid:3)(cid:7) α t ( iii ) (cid:3)(cid:7) α t ≤ ∪ (cid:3)(cid:7) α t ( iv ) α · β ≤ β implies α ∗ · β ≤ β n α t · β t ≤ β t implies (cid:3)(cid:7) α t · β t ≤ β t ( v ) β · α ≤ β implies β · α ∗ ≤ β n β t · α t ≤ β t implies β t · (cid:3)(cid:7) α t ≤ β t ( vi )Notice that, relative to the order-theoretic properties of their interpretations on hetero-geneous Kleene algebras, · , 1, (cid:7) are F -connectives, while (cid:3) is a G -connective. How-ever, relative to the order-theoretic properties of their interpretations on heterogeneousmeasurable Kleene algebras, · , 1, (cid:7) are F -connectives, while (cid:3) is both an F -connectiveand a G -connective. Hence, it is easy to see that, relative to the first interpretation, ( i )is the only analytic inductive inequality of the list above, due to the occurrences ofthe McKinsey-type nesting (cid:3)(cid:7) α t in antecedent position. However, relative to the sec-ond interpretation, the same nesting becomes harmless, since the occurrences of (cid:3) inantecedent position are part of the Skeleton.Likewise, it is very easy to see that the conditions HM1-HM6 in the definition ofheterogeneous measurable Kleene algebras do not violate the conditions on nesting ofanalytic inductive inequalities. However, some of these conditions do not consist ofinequalities taken in isolation but are given in the form of quasi-inequalities. Whenembedded into a quasi-inequality, the proof-theoretic treatment of an inequality suchas β · β ≤ β (which in isolation would be unproblematic) becomes problematic, sincethe translation of the quasi-inequality into a logically equivalent rule would not allowto ‘disentangle’ the occurrences of β in precedent position from the occurrences of β insuccedent position, thus making it impossible to translate the quasi-inequality directlyas an analytic structural rule. This is why the calculus defined in the following sectionfeatures an infinitary rule, introduced to circumvent this problem. In the present section, we define a multi-type language for the proper multi-type displaycalculus for measurable Kleene logic. As usual, this language includes constructors forboth logical (operational) and structural terms. • Structural and operational terms: 13 eneral α :: = a | | | (cid:3) ξ | α ∪ α | α · α Γ :: = Φ | ◦ Π | Γ ⊙ Γ | Γ < Γ | Γ > Γ Special ξ :: = (cid:7) α | (cid:4) α Π :: = • Γ In what follows, we reserve α, β, γ (with or without subscripts) to denote
General -type operational terms, and ξ, χ, π (with or without subscripts) to denote formulas in
Special -type operational terms. Moreover, we reserve Γ , ∆ , Θ (with or without sub-scripts) to denote General -type structural terms, and Π , Ξ , Λ (with or without sub-scripts) to denote Special -type structural terms. • Structural and operational terms:
General S → G G → S Φ ⊙ < > ◦ • · ( / ) ( \ ) (cid:3) (cid:3) (cid:7) (cid:4) Notice that, for the sake of minimizing the number of structural symbols, we areassigning the same structural connective • to (cid:7) and (cid:4) although these modal operatorsare not dual to one another, but are respectively interpreted as the left adjoint andthe right adjoint of (cid:3) , which is hence both an F -operator and a G -operator, and cantherefore correspond to the structural connective ◦ both in antecedent and in succedentposition. In the rules below, the symbols Γ , ∆ and Θ denote structural variables of general type,and Σ , Π and Ξ structural variables of special type. The calculus D.MKL consists thefollowing rules: • Identity and cut rules: Id a ⊢ a Γ ⊢ α α ⊢ ∆ Cut g Γ ⊢ ∆ Π ⊢ ξ ξ ⊢ Ξ Cut s Π ⊢ Ξ • General type display rules: Γ ⊙ ∆ ⊢ Θ res ∆ ⊢ Γ > Θ Γ ⊙ ∆ ⊢ Θ res Γ ⊢ Θ < ∆ • Multi-type display rules: Γ ⊢ ◦ Ξ adj • Γ ⊢ Ξ ◦ Ξ ⊢ Γ adj Ξ ⊢ • Γ • General type structural rules: 14 ⊢ ∆ Φ L Φ ⊙ Γ ⊢ ∆ Γ ⊢ ∆ Φ R Γ ⊙ Φ ⊢ ∆ ( Γ ⊙ Γ ) ⊙ Γ ⊢ ∆ assoc Γ ⊙ ( Γ ⊙ Γ ) ⊢ ∆ Γ ⊢ Φ Φ -W Γ ⊢ ∆ • Multi-type structural rules: one Φ ⊢ ◦ Π Γ ⊢ ◦
Π ∆ ⊢ ◦ Π abs Γ ⊙ ∆ ⊢ ◦ ΠΠ ⊢ Σ b-bal • ◦ Π ⊢ • ◦ Σ Π ⊢ Ξ w-bal ◦ Π ⊢ ◦ Ξ ( Γ ( n ) ⊢ ∆ | n ≥ ω ◦ • Γ ⊢ ∆ ◦ Π ⊙ ◦ Π ⊢ ∆ ◦ -C ◦ Π ⊢ ∆ • General type operational rules: in what follows, i ∈ { , } , Φ ⊢ ∆ ⊢ ∆ Φ ⊢ ⊢ Φ Γ ⊢ Φ Γ ⊢ α ⊢ ∆ α ⊢ ∆ ∪ α ∪ α ⊢ ∆ Γ ⊢ α i ∪ Γ ⊢ α ∪ α α ⊙ β ⊢ ∆ · α · β ⊢ ∆ Γ ⊢ α ∆ ⊢ β · Γ ⊙ ∆ ⊢ α · β • Multi-type operational rules: • α ⊢ Π (cid:7) (cid:7) α ⊢ Π Γ ⊢ α (cid:7) • Γ ⊢ (cid:7) αα ⊢ Γ (cid:4) (cid:4) α ⊢ • Γ Π ⊢ • α (cid:4) Π ⊢ (cid:4) α ◦ ξ ⊢ Γ (cid:3) (cid:3) ξ ⊢ Γ Γ ⊢ ◦ ξ (cid:3) Γ ⊢ (cid:3) ξ The following fact is proven by a straightforward induction on α and ξ . We omitthe details. Proposition 9.
For every α ∈ General and ξ ∈ Special , the sequents α ⊢ α and ξ ⊢ ξ are derivable in D.MKL. In the present subsection, we outline the verification of the soundness of the rules ofD . MKL w.r.t. heterogenous measurable Kleene algebras (cf. Definition 10). The first Let Γ ( n ) be defined by setting Γ (1) : = Γ and Γ ( n + : = Γ ⊙ Γ ( n ) . ?? . Thismakes it possible to interpret sequents as inequalities, and rules as quasi-inequalities.For example, (modulo standard manipulations) the rules on the left-hand side belowcorrespond to the (quasi-)inequalities on the right-hand side: Φ ⊢ ◦ Π ∀ ξ [1 ≤ (cid:3) ξ ] Γ ⊢ ◦ Π ∆ ⊢ ◦ Π abs Γ ⊙ ∆ ⊢ ◦ Π ∀ α ∀ β [ (cid:7) ( α · β ) ≤ (cid:7) α ⊔ (cid:7) β ] Π ⊢ Σ b-bal • ◦ Π ⊢ • ◦ Σ ∀ ξ [ (cid:7)(cid:3) ξ ≤ (cid:4) (cid:3) ξ ] Π ⊢ Ξ w-bal ◦ Π ⊢ ◦ Ξ ∀ ξ ∀ π [ π ≤ ξ ⇔ (cid:3) π ≤ (cid:3) ξ ]( Γ ( n ) ⊢ ∆ | n ≥ ω ◦ • Γ ⊢ ∆ ∀ α [ (cid:3)(cid:7) α ≤ S n ∈ ω α n ] ◦ Π ⊙ ◦ Π ⊢ ∆ ◦ -C ◦ Π ⊢ ∆ ∀ ξ [ (cid:3) ξ ≤ (cid:3) ξ · (cid:3) ξ ]Then, the verification of the soundness of the rules of D . MKL boils down to check-ing the validity of their corresponding quasi-inequalities in heterogenous measurableKleene algebras. This verification is routine and is omitted.
In the present section, we show that the translations – by means of the map () t definedin Section 4 – of the axioms and rules of S . MKL (cf. Section 2.2) are derivable in thecalculus D . MKL. For the reader’s convenience, here below we report the recursivedefinition of () t : a t :: = a t :: = t :: = α · β ) t :: = α t · β t ( α ∪ β ) t :: = α t ∪ β t ( α ∗ ) t :: = (cid:3)(cid:7) α t ( α ⋆ ) t :: = (cid:3) (cid:4) α t Proposition 10.
For every α ∈ S . KL , the sequent α t ⊢ α t is derivable in D.MKL. Let α ( n ) be defined by setting α (1) : = α and α ( n + : = α ⊙ α ( n ) . Lemma 6 (Omega) . If α ⊙ β ⊢ β (resp. β ⊙ α ⊢ β ) is derivable, then α ( n ) ⊙ β ⊢ β (resp. β ⊙ α ( n ) ⊢ β ) is derivable for every n ≥ .Proof. Let us show that for any n ≥
1, if α ( n ) ⊙ β ⊢ β is derivable, then α ( n + ⊙ β ⊢ β isderivable (the proof that β ⊙ α ( n ) ⊢ β is derivable from β ⊙ α ( n ) ⊢ β is analogous and it isomitted). Indeed: 16 ⊢ α hyp α ( n ) ⊙ β ⊢ βα ⊙ ( α ( n ) ⊙ β ) ⊢ α · β ( α ⊙ α ( n ) ) ⊙ β ⊢ α · βα ( n + ⊙ β ⊢ α · β assump α · β ⊢ β cut α ( n + ⊙ β ⊢ β Hence, the sequent α ( n ) ⊙ β ⊢ β for any n is obtained from a proof of α ⊙ β ⊢ β byconcatenating n derivations of the shape shown above. (cid:3) As to the rule K4 (cf. Definition 3), if α · β ⊢ β is derivable in D.MKL, then α ⊙ β ⊢ β is derivable in D.MKL , hence by Lemma 6 so are the sequents α ( n ) ⊙ β ⊢ β for any n ≥
1. By applying the appropriate display postulate to each such sequent, we obtainderivations of α ( n ) ⊢ β < β for any n ≥
1. Hence:( α ( n ) ⊢ β < β | ≤ n ) ω ◦ • α ⊢ β < β • α ⊢ • ( β < β ) (cid:7) α ⊢ • ( β < β ) ◦ (cid:7) α ⊢ β < β (cid:3)(cid:7) α ⊢ β < β (cid:3)(cid:7) α ⊙ β ⊢ β (cid:3)(cid:7) α · β ⊢ β The proof that the rule K5 is derivable is analogous and we omit it. As to theaxioms of Definition 3 in which () ∗ -terms occur, one Φ ⊢ ◦ (cid:7) α Φ ⊢ (cid:3)(cid:7) α ⊢ (cid:3)(cid:7) α α ⊢ α • α ⊢ (cid:7) αα ⊢ ◦ (cid:7) α α ⊢ α • α ⊢ (cid:7) α (cid:7) α ⊢ (cid:7) α w-bal ◦ (cid:7) α ⊢ ◦ (cid:7) α (cid:3)(cid:7) α ⊢ ◦ (cid:7) α abs α ⊙ (cid:3)(cid:7) α ⊢ ◦ (cid:7) αα ⊙ (cid:3)(cid:7) α ⊢ (cid:3)(cid:7) αα · (cid:3)(cid:7) α ⊢ (cid:3)(cid:7) α ∪ α · (cid:3)(cid:7) α ⊢ (cid:3)(cid:7) α one Φ ⊢ ◦ (cid:7) α Φ ⊢ (cid:3)(cid:7) α ⊢ (cid:3)(cid:7) α α ⊢ α • α ⊢ (cid:7) α (cid:7) α ⊢ (cid:7) α w-bal ◦ (cid:7) α ⊢ ◦ (cid:7) α (cid:3)(cid:7) α ⊢ ◦ (cid:7) α α ⊢ α • α ⊢ (cid:7) α (cid:7) α ⊢ (cid:7) α w-bal ◦ (cid:7) α ⊢ ◦ (cid:7) α (cid:3)(cid:7) α ⊢ ◦ (cid:7) α abs (cid:3)(cid:7) α ⊙ (cid:3)(cid:7) α ⊢ ◦ (cid:7) α (cid:3)(cid:7) α ⊙ (cid:3)(cid:7) α ⊢ (cid:3)(cid:7) α (cid:3)(cid:7) α · (cid:3)(cid:7) α ⊢ (cid:3)(cid:7) α ∪ (cid:3)(cid:7) α · (cid:3)(cid:7) α ⊢ (cid:3)(cid:7) α The translations of 0 · α ⊣⊢ This is due to the fact that · is a normal F -operator, and in proper display calculi the left introductionrules of F -operators are invertible. ⊢ Φ Φ -W ⊢ Φ < α ⊙ α ⊢ Φ · α ⊢ Φ · α ⊢ ⊢ Φ Φ -W ⊢ α · ∗ ⊢ one Φ ⊢ ◦ • Φ Φ ⊙ ⊢ ◦ • Φ ⊢ ◦ • • ⊢ • (cid:7) ⊢ • ◦ (cid:7) ⊢ (cid:3)(cid:7) ⊢ ⊢ ∗ is derivable as follows: one Φ ⊢ ◦ (cid:7) Φ ⊢ (cid:3)(cid:7) ⊢ (cid:3)(cid:7) ∗ ⊢ ◦ • Φ (that is derivable usingthe ω -rule): Φ ⊢ ◦ • Φ ◦ • Φ ⊢ • Φ ⊢ • Φ ⊢ ◦ • ⊢ ◦ • • ⊢ • (cid:7) ⊢ • ◦ (cid:7) ⊢ (cid:3)(cid:7) ⊢ one Φ ⊢ ◦ (cid:4) α Φ ⊢ (cid:3) (cid:4) α ⊢ (cid:3) (cid:4) α α ⊢ α (cid:4) α ⊢ • α (cid:4) α ⊢ (cid:4) α w-bal ◦ (cid:4) α ⊢ ◦ (cid:4) α (cid:3) (cid:4) α ⊢ ◦ (cid:4) α α ⊢ α (cid:4) α ⊢ • α (cid:4) α ⊢ (cid:4) α w-bal ◦ (cid:4) α ⊢ ◦ (cid:4) α (cid:3) (cid:4) α ⊢ ◦ (cid:4) α abs (cid:3) (cid:4) α ⊙ (cid:3) (cid:4) α ⊢ ◦ (cid:4) α (cid:3) (cid:4) α ⊙ (cid:3) (cid:4) α ⊢ (cid:3) (cid:4) αα ⊢ α (cid:4) α ⊢ • α ◦ (cid:4) α ⊢ α (cid:3) (cid:4) α ⊢ α α ⊢ α (cid:4) α ⊢ • α (cid:4) α ⊢ (cid:4) α b-bal • ◦ (cid:4) α ⊢ • ◦ (cid:4) α ◦ • ◦ (cid:4) α ⊢ ◦ (cid:4) α ◦ • ◦ (cid:4) α ⊢ (cid:3) (cid:4) α • ◦ (cid:4) α ⊢ • (cid:3) (cid:4) α • ◦ (cid:4) α ⊢ (cid:4) (cid:3) (cid:4) α ◦ (cid:4) α ⊢ ◦ (cid:4) (cid:3) (cid:4) α ◦ (cid:4) α ⊢ (cid:3) (cid:4) (cid:3) (cid:4) α (cid:3) (cid:4) α ⊢ (cid:3) (cid:4) (cid:3) (cid:4) α α ⊢ α (cid:4) α ⊢ • α (cid:4) α ⊢ (cid:4) α w-bal ◦ (cid:4) α ⊢ ◦ (cid:4) α (cid:3) (cid:4) α ⊢ ◦ (cid:4) α (cid:3) (cid:4) α ⊢ (cid:3) (cid:4) α β ⊢ α , and 1 ⊢ β and β · β ⊢ β are derivable. Hence, by the invertibilityof the introduction rules of F -connectives in proper display calculi, Φ ⊢ β and β ⊙ β ⊢ β are derivable. By Lemma 6, β ( n ) ⊙ β ⊢ β is derivable. Therefore, we can derive thefollowing sequents for any n ≥
1: ( β ( n ) ⊢ β < β ( n ) | ≤ n ) ω ◦ • β ⊢ β < β ( n ) ◦ • β ⊙ β ( n ) ⊢ ββ ( n ) ⊢ ◦ • β > β Hence: ( β ( n ) ⊢ ◦ • β > β | ≤ n ) ω ◦ • β ⊢ ◦ • β > β ◦ • β ⊙ ◦ • β ⊢ β ◦ -C ◦ • β ⊢ β β ⊢ α Cut ◦ • β ⊢ α • β ⊢ • α • β ⊢ (cid:4) αβ ⊢ ◦ (cid:4) αβ ⊢ (cid:3) (cid:4) α For any heterogeneous measurable Kleene algebra H = ( A , ∼ , γ, ι, e ), the algebra A is acomplete join-semilattice, and · distributes over arbitrary joins in each coordinate. Thisimplies that the right residuals exist of · in each coordinate, which we denote / and \ : α \ β : = [ { α ′ : α · α ′ ≤ β } , β/α : = [ { α ′ : α ′ · α ≤ β } . From here on, the proof of conservativity proceeds in the usual way as detailed in [9].
The cut elimination of D.MKL follows from the Belnap-style meta-theorem proven in[5], of which a restriction to proper multi-type display calculi is stated in [10]. Theproof boils down to verifying the conditions C - C of [10, Section 6.4]. Most ofthese conditions are easily verified by inspection on rules; the most interesting one iscondition C ′ , concerning the principal stage in the cut elimination, on which we expandin the lemma below. Lemma 7. D . MKL satisfies C ′ .Proof. By induction on the shape of the cut formula.
Atomic propositions: a ⊢ a a ⊢ aa ⊢ a a ⊢ a onstants: Φ ⊢ ... π Φ ⊢ ∆ ⊢ ∆Φ ⊢ ∆ ... π Φ ⊢ ∆ The cases for 0, 0 s are standard and similar to the one above. Unary connectives:
As to (cid:7) α , ... π Γ ⊢ α • Γ ⊢ (cid:7) α ... π • α ⊢ Ξ (cid:7) α ⊢ Ξ • Γ ⊢ Ξ ... π Γ ⊢ α ... π • α ⊢ Ξ α ⊢ ◦ ΞΓ ⊢ ◦ Ξ • Γ ⊢ Ξ As to (cid:3) α , ... π Γ ⊢ ◦ ξ Γ ⊢ (cid:3) ξ ... π ξ ⊢ Ξ (cid:3) ξ ⊢ ◦ ΞΓ ⊢ ◦ Ξ ... π Γ ⊢ ◦ ξ • Γ ⊢ ξ ... π ξ ⊢ Ξ • Γ ⊢ ΞΓ ⊢ ◦ Ξ Binary connectives:
As to α ∪ α , ... π Γ ⊢ α Γ ⊢ α ∪ α ... π α ⊢ ∆ ... π α ⊢ ∆ α ∪ α ⊢ ∆Γ ⊢ ∆ ... π Γ ⊢ α ... π α ⊢ ∆Γ ⊢ ∆ (cid:3) References [1] Thomas Brunsch, Laurent Hardouin, and Jrg Raisch. Modelling manufacturingsystems in a dioid framework. In
Formal Methods in Manufacturing , pages 29–74, 2017.[2] Willem Conradie, Alessandra Palmigiano, and Zhiguang Zhao. Sahlqvist viatranslation. Submitted. ArXiv preprint 1603.08220.[3] Brian A. Davey and Hilary A. Priestley.
Lattices and Order . Cambridge UniverityPress, 2002.[4] Sabine Frittella, Giuseppe Greco, Alexander Kurz, and Alessandra Palmigiano.Multi-type display calculus for propositional dynamic logic.
Journal of Logicand Computation , 26 (6):2067–2104, 2016.205] Sabine Frittella, Giuseppe Greco, Alexander Kurz, Alessandra Palmigiano, andVlasta Sikimi´c. Multi-type sequent calculi.
Proceedings Trends in Logic XIII, A.Indrzejczak, J. Kaczmarek, M. Zawidski eds , 13:81–93, 2014.[6] Sabine Frittella, Giuseppe Greco, Alessandra Palmigiano, and Fan Yang. A multi-type calculus for inquisitive logic. In Jouko V¨a¨an¨anen, Åsa Hirvonen, and Ruyde Queiroz, editors,
Logic, Language, Information, and Computation: 23rd Inter-national Workshop, WoLLIC 2016, Puebla, Mexico, August 16-19th, 2016. Pro-ceedings , LNCS 9803, pages 215–233. Springer, 2016.[7] Jean-Yves Girard. Linear logic: its syntax and semantics.
London MathematicalSociety Lecture Note Series , pages 1–42, 1995.[8] Giuseppe Greco, Fei Liang, and Alessandra Palmigiano. Multi-type display cal-culus for measurable Kleene algebras. in preparation , 2017.[9] Giuseppe Greco, Minghui Ma, Alessandra Palmigiano, Apostolos Tzimoulis, andZhiguang Zhao. Unified correspondence as a proof-theoretic tool.
Journal ofLogic and Computation , 2016. doi: 10.1093 / logcom / exw022.[10] Giuseppe Greco and Alessandra Palmigiano. Linear logic properly displayed.Submitted. ArXiv preprint:1611.04181.[11] Laurent Hardouin, Olivier Boutin, Bertrand Cottenceau, Thomas Brunsch, andJ¨org Raisch. Discrete-event systems in a dioid framework: Control theory. Con-trol of Discrete-Event Systems , 433:451–469, 2013.[12] Chrysafis Hartonas. Analytic cut for propositional dynamic logic. unpublishedmanuscript.[13] Wilfrid Hodges. Compositional semantics for a language of imperfect informa-tion.
Logic Journal of IGPL , 5(4):539–563, 1997.[14] Wilfrid Hodges. Some strange quantifiers. In
Structures in logic and computerscience , pages 51–65. Springer, 1997.[15] Peter Jipsen. From semirings to residuated Kleene lattices.
Studia Logica ,76(2):291–303, 2004.[16] Dexter Kozen. On Kleene algebras and closed semirings. In
Mathematical Foun-dations of Computer Science 1990 , pages 26–47. Springer, 1990.[17] Dexter Kozen. A completeness theorem for Kleene algebras and the algebra ofregular events.
Information and computation , 110(2):366–390, 1994.[18] Dexter Kozen. Kleene algebra with tests.
ACM Transactions on ProgrammingLanguages and Systems (TOPLAS) , 19(3):427–443, 1997.[19] Natasha Kurtonina and Michael Moortgat. Structural control. In P. Blackburn andM. de Rijke, editors,
Specifying syntactic structures (Amsterdam, 1994) , Studiesin Logic, Language and Information, pages 75–113, Stanford, CA, 1997. CSLI.[20] Michael R. Laurence and Georg Struth. Completeness theorems for bi-Kleenealgebras and series-parallel rational pomset languages. In
Formal Methods inManufacturing , pages 65–82, 2014. 2121] Christopher J. Mulvey and Mohammad Nawaz. Quantales: quantal sets.
Non-Classical Logics and Their Applications to Fuzzy Subsets (Linz, 1992) , 32:159–217, 1995.[22] Ewa Palka. An infinitary sequent system for the equational theory of *-continuousaction lattices.
Fundamenta Informaticae , 78(2):295–309, 2007.[23] Francesca Poggiolesi.