Non-distributive positive logic as a fragment of first-order logic over semilattices

We characterise non-distributive positive logic as the fragment of a single-sorted first-order language that is preserved by a new notion of simulation called a meet-simulation. Meet-simulations distinguish themselves from simulations because they relate pairs of states from one model to single states from another. En route to this result we use a more traditional notion of simulations and prove a Hennessy-Milner style theorem for it, using an analogue of modal saturation called meet-compactness.


Introduction
The celebrated Van Benthem characterisation theorem characterises normal modal logic as the bisimulation-invariant fragment of first-order logic [3]. More precisely, it states that a first-order formula with one free variable is equivalent to the standard translation of a modal formula if and only if it is invariant under bisimulations. Similar theorems have been proven for non-normal modal logics, such as monotone modal logic [12], neighbourhood logic [13], and instantial neighbourhood logic [11]. For the latter three we only consider equivalence over particular classes of first-order structures, namely those corresponding to the modal logic semantics. Apart from modal extensions of classical propositional logics, other propositional logics and their modal extensions can be viewed as fragments of first-order logic as well. Examples include (bi-)intuitionistic logic [18,17,2] and positive modal logic [17,7].
Sometimes "invariant under bisimulations" is replaced with "preserved by simulations." This allows one to exclude negations from the characterisation. Indeed, if a formula is invariant under bisimulations then so is its negation, but preservation by simulations does not imply preservation of its negation.
The goal of the present paper is to characterise non-distributive positive logic as a fragment of firstorder logic. It has recently been noted that non-distributive positive logic logic can be given frame semantics by means of meet-semilattices with a valuation, with filters serving as denotations of formulae [9,5]. Conjunctions are interpreted as usual, while ϕ ∨ ψ holds at a state w if there are two states v and u satisfying ϕ and ψ, respectively, such that their meet lies below w. This non-standard interpretation of disjunctions prevents distributivity.
Since meet-semilattices are in particular partially ordered sets, they can be viewed as interpreting structures for the first-order language with one binary predicate. Indeed, the partial order underlying a meet-semilattice then serves as the interpretation of the binary relation. Guided by this observation, we translate non-distributive positive logic into the first-order logic with one binary predicate and a unary predicate for each proposition letter. This translation is similar to the one in [5].
To establish this characterisation result, we have to define an appropriate analogue of simulations. The main challenge is to exclude classical (locally evaluated) disjunctions from being preserved by this notion. Ordinary simulations do not suffice because they preserve all locally evaluated monotone connectives. Instead, we introduce meet-simulations. These differ from ordinary simulations because they relate pairs of states from one model to states of another. This modification, it turns out, allows us to prevent preservation of classical disjunctions. We prove the following characterisation theorem: A first-order formula α(x) is preserved by meet-simulations if and only if it is equivalent to ¬(x = x) or to the standard translation of a non-distributive positive formula over the class of meet-semilattices.
Related work. The idea of using (bi)simulations to relate pairs of states to pairs of states appeared before in [10], where it is used to characterise fragments of the calculus of binary relations, and in [1] in the context of paths on data trees.
Structure of the paper. In Section 2 we recall the language and semantics of non-distributive positive logic. In Section 3 we summarise some basic first-order logic required for the results in this paper, and we define the standard translation for non-distributive positive logic.
The contributions of this paper start in Section 4, where we define simulations between models and show that states related by a simulation are logically inclusive (that is, the theory of the former is included in the theory of the latter). In Section 5 we give a Hennessy-Milner style theorem for simulations. We first prove that the finite models form a Hennessy-Milner class. Taking stock of the proof, we define an analogue of modal saturation called meet-compactness. We then prove that ω-saturated models are meet-compact, and that the meet-compact models form a Hennessy-Milner class.
Subsequently, in Section 6 we work towards a Van Benthem style characterisation and point out why simulations do not admit such a result. We then introduce meet-simulations in Section 7 and use the results from Section 6 to prove the characterisation theorem announced above. We conclude in Section 8.

Non-distributive positive logic
Denote by L the language of positive logic, i.e. the language generated by the grammar where p ranges over some arbitrary but fixed set Prop of proposition letters. One can define a logic of consequence pairs of L-formula whose algebraic semantics is given by lattices, see [5,Section 3.1].
Formulae from L can be interpreted in meet-semilattices with a valuation [9]. The intuition behind this is that the collection of filters of a meet-semilattices is closed under (arbitrary) intersections. Therefore they form a complete lattice, but disjunctions are not given by unions. This gives rise to a non-standard interpretation of disjunctions which prevents distributivity. We call the resulting frames and models L 1 -frames, to distinguish them from the slightly different L-frames used in [5], see Remark 2.3.
In this paper, by a meet-semilattice we mean a partially ordered set in which every finite subsets has a greatest lower bound, called its meet. The meet of w and v is denoted by w v, reserving the symbol ∧ for conjunctions of formulae. The empty meet is the top element, denoted by 1. If (W, 1, ) is a meet-semilattice then we write for the partial order given by w v iff w v = w. A filter of a meet-semilattice (W, 1, ) is a subset F of W which is upward closed under and closed under finite meets. Filters are nonempty because they contain the empty meet, 1.

2.1
Definition. An L 1 -frame is a meet-semilattice (W, 1, ). An L 1 -model is an L 1 -frame (W, 1, ) together with a valuation V that assigns to each proposition letter p ∈ Prop a filter V (p) of (W, 1, ). The interpretation of ϕ ∈ L at a state w of an L 1 -model M = (W, 1, , V ) is defined recursively via The second condition can be depicted as follows: 2.3 Remark. In [5] a variation of the semantics presented above is used where meet-semilattices are only assumed to have binary meets. As a consequence, they need not have a top element, and filters are allowed to be empty. An advantage of this approach is that the truth set of ⊥ is empty, so that models contain no inconsistent state. The drawback is that it complicates interpretation of ϕ 1 ∨ ϕ 2 , cf. [5,Definition 3.6]. This, in turn, affects the definition of truth-preserving morphisms between models.
In this paper we choose to allow an inconsistent state 1 satisfying ⊥ because it simplifies the definitions of simulations and meet-simulations.

First-order translation
We define the first-order language we work with and the standard translation of L into this language. Then we characterise the class of first-order structures corresponding to L 1 -models and we recall the definition of ω-saturation.
3.1 Definition. Let FOL be the single-sorted first-order language which has a unary predicate P p for every proposition letter p ∈ Prop, and a binary predicate R. To avoid confusion with the interpretation of disjunctions from L, we denote classical disjunctions like the one in FOL by ∨ c .
Models for FOL are denoted by M, N (as oppose to M, N for L 1 -frames). Intuitively, the relation symbol of our first-order language accounts of the poset structure of L 1 -frames. If x, y and z are variables, then we can express that x is the meet of y and z in the ordering induced by the interpretation of R using a first-order sentence. To streamline notation we abbreviate this as follows: We are now ready to define the standard translation.

Definition. Let x be a variable. Define the standard translation st
Every L 1 -model M = (W, 1, , V ) gives rise to a first-order structure for FOL. Indeed, we define the interpretation of R as , and the interpretation of the unary predicates is given via the valuations of the proposition letters. We write M • for the L 1 -model M conceived of as a first-order structure for FOL. The following proposition is an adaptation of [5,Proposition 3.26]. Satisfaction of FOL-formulae in a first-order structure is defined as usual.

Proposition.
For every L 1 -model M = (W, 1, , V ), state w ∈ W and formula ϕ ∈ L we have Proof. We use induction on the structure of ϕ. If ϕ = p or ϕ = ⊤ then the statement is obvious.
. If ϕ is of the form ϕ 1 ∧ ϕ 2 then the inductive step is routine. If ϕ = ϕ 1 ∨ ϕ 2 and M, w ϕ 1 ∨ ϕ 2 then there exist u, v ∈ W such that u v w and M, u ϕ 1 and M, v ϕ 2 . Taking x ′ = u v, y = u and z = v, this witnesses . Conversely, validity of (1) entails the existence of suitable states u and v witnessing M, w ϕ 1 ∨ ϕ 2 .
Clearly not every structure for FOL is of the form M • . We can classify the ones that are.
3.4 Definition. Let FSL be the class of first-order structures for FOL that satisfy the following axioms: Here P ranges over all unary predicates of FOL.
Axioms (M 1 ) to (M 3 ) state that the interpretation of R should be a partial order. The fourth one adds that this partial order should have binary greatest lower bounds and (M 5 ) stipulates a top element. Finally, we have an axiom for each unary predicate stating that its interpretation should be a filter in the meet-semilattice induced by the domain and the interpretation of R.
Proof. The direction from left to right is easy. Conversely, let M = (W, I(R), {I(P p ) | p ∈ Prop}) be a first-order structure that satisfies all axioms from Definition 3.4. Then I(R) is a partial order on W with binary greatest lower bounds and a top element. Denote the latter by 1 ∈ W and the greatest lower bound of w, v ∈ W by w v. Finally, we recall basic properties of ω-extensions needed for the proof of the characterisation theorem.
Using e.g. ultraproducts, one can show that every FOL-model has an ω-saturated elementary extension [8]. We denote this extension of M by M * , and the image of a state w under the extension is denoted by w * . Moreover, if M ∈ FSL then its ω-saturated elementary extension is also in FSL, since validity of the axioms from Definition 3.4 is preserved under elementary extensions.

Simulations
In this section we define simulations between L 1 -models. Simulations only preserve truth of formulae in one direction, that is, if S is a simulation and (w, w ′ ) ∈ S then every formula satisfied at w is also satisfied at w ′ . This prevents preservation of negations, and hence has been used to characterise the negation-free part of classical normal modal logic [17]. Since the language of non-distributive positive logic does not have negations, this is a good starting point for our attempt to characterise it.
However, by its nature simulations preserve all monotone connectives, including classical (locally evaluated) disjunctions. So this this approach is bound to fail, since the collection of first-order formulae preserved by simulations is closed under classical disjunctions. Simulations are still worth investigating because they provide a stepping stone towards the Van Benthem style characterisation we are after. In fact, in Section 6 we will prove that a first-order formula is preserved by simulations if and only if it is the classical disjunction of standard translations of formulae in L. We extend this to a proper characterisation in Section 7 using the notion of a meet-simulation.
We call w ∈ W and w ′ ∈ W ′ L 1 -similar if there is an L 1 -simulation S between M and M ′ such that (w, w ′ ) ∈ S. This is denoted by M, w → M ′ , w ′ .
It is straightforward to see that the collection of L 1 -simulations between two models is closed under arbitrary unions. Therefore we have: Taking f = id : M → M we obtain S 1 and S 2 from Example 4.3 as Gr f and Gr ↑ f .
As expected, L 1 -similarity implies logical inclusion.
Proof. Let ϕ ∈ L be such that M, w ϕ. We show by induction on the structure of ϕ that M ′ , w ′ ϕ.

A Hennessy-Milner style theorem
While Proposition 4.5 guarantees that L 1 -similarity implies logical inclusion, the opposite (i.e. logical inclusion implies L 1 -similarity) need not be true. Classes on which L 1 -similarity and logical inclusion coincide are often called Hennessy-Milner classes, after the authors who proved an analogous result for classical normal modal logic [15]. This section is devoted to finding Hennessy-Milner classes.

Definition. A class of L 1 -models is called a Hennessy-Milner class if for all M, M ′ ∈ and states
We start by proving that the class of finite models is a Hennessy-Milner class. This will serve as inspiration for an analogue of modal saturation which we formulate in terms of compactness in a topology, and a more general Hennessy-Milner style result.
5.2 Proposition. The class of finite L 1 -models is a Hennessy-Milner class.
Proof. Let M = (W, 1, , V ) and M ′ = (W ′ , 1 ′ , ′ , V ′ ) be two finite L 1 -models. We claim that is an L 1 -simulation. Together this with Proposition 4.5 this proves the proposition. Item (S 1 ) holds by definition of S, and (S 2 ) follows from the fact that 1 and 1 ′ are the only elements satisfying ⊥. Let (w, w ′ ) ∈ S and let v, u ∈ W be such that v u w. Suppose towards a contradiction that (S 3 ) is not satisfied. Then for each ∈ S, so there exists a formula ϕ that is satisfied at v but not at v ′ ; or • (u, u ′ ) / ∈ S, so there exists a formula ψ that is satisfied at u but not at u ′ .
For each (v ′ , u ′ ) ∈ M (w ′ ) select a ϕ or ψ as specified. Let Φ be the set of all such ϕ and Ψ of all such ψ. Letφ := Φ andψ := Ψ (taking to empty conjunction to be ⊤). Then v and u satisfyφ and ψ, respectively. On the other hand, for each This contradicts the assumption that (w, w ′ ) ∈ S. So (S 3 ) must hold and S is an L 1 -simulation.
If we try to apply this proof to infinite models we may encounter infinite sets Φ and Ψ. So our analogue of modal saturation should remedy this. Indeed, we need a compactness property that allows us to reduce infinite Φ and Ψ to finite sets.

Definition.
Let M = (W, 1, , V ) be an L 1 -model. Then we denote by τ V the topology on W generated by the clopen subbase The L 1 -model M is called meet-compact if for all w ∈ W the set M (w) := {(v, u) ∈ W × W | v u w} is a compact subset of (X, τ V ) × (X, τ V ).

Example.
Every finite monotone L 1 -model is meet-compact. Evidently, the finiteness entails the compactness requirement.
As another example, we show that ω-saturated L 1 -models are meet-compact. An L 1 -model M is called ω-saturated if M • is ω-saturated (see Definition 3.6).

Lemma. If
. By the Alexander subbase theorem, it suffices to show that every open cover of subbasic opens has a finite subcover. A subbase for the topology on (X, τ V ) × (X, τ V ) can be given by the collection of open squares a × b, where a and b range over the subbase for τ V , that is, over the truth sets of formulae and their complements. So suppose where I, J, K, L are index sets. Let A = {w} and Then Γ(x) is not satisfiable in M • A , because if it were then there is a state x ∈ W such that xRw which is the meet of a pair (y, z) that is not in the open cover given in (2). Since M • is ω-saturated, this implies that Γ(x) is not finitely satisfiable in M • A , which by a reverse argument yields a finite subcover of (2).

5.6
Theorem. The meet-compact L 1 -models form a Hennessy-Milner class.
Proof. Let M = (W, 1, , V ) and M ′ = (W ′ , 1 ′ , ′ , V ′ ) be two meet-compact L 1 -models. We claim that is an L 1 -simulation. Together with Proposition 4.5 this proves the theorem. Item (S 1 ) is satisfied by definition, and (S 2 ) follows from the fact that 1 and 1 ′ are the only elements satisfying ⊥. To prove (S 3 ), assume (w, w ′ ) ∈ S and u, v ∈ W are such that u v w. Suppose there are no u ′ , v ′ ∈ W ′ such that u ′ ′ v ′ ′ w ′ and (u, u ′ ) ∈ S and (v, v ′ ) ∈ S. Proceeding as in the proof of Proposition 5.2, we obtain (potentially infinite) sets Φ and Ψ. Now observe that we have an open cover By assumption M (w ′ ) is compact, so we can find a finite subcover indexed by finite sets Φ ′ ⊆ Φ and Ψ ′ ⊆ Ψ. Defineφ := Φ ′ andψ := Ψ ′ . The remainder of the proof is the analogous to Proposition 5.2.

Towards characterisation
In this section we characterise the L 1 -simulation-invariant fragment of FOL over the class FSL of firstorder structures corresponding to L 1 -models.
whenever M, w → N, v, for all L 1 -models M, N.
We cannot yet characterise the L 1 -simulation-preserving formulae as the language L, because we can find first-order formulae α(x) that are preserved by L 1 -simulations but not equivalent to the standard translation of a formula in L. The next example gives such an α(x). Along the lines of [6, Theorem 2.68], we characterise the first-order formulae preserved by simulations as those formulae in FOL equivalent to the disjunction of standard translations of L-formulae. 6.3 Theorem. A first-order formula α(x) ∈ FOL with one free variable x is equivalent over FSL to a formula of the form st x (ϕ 1 ) ∨ c · · · ∨ c st x (ϕ n ), where ϕ i ∈ L, if and only if it is preserved by L 1 -simulations.
So let M ∈ FSL and assume M |= MOC(α) [w]. Let M • = (W, 1, , V ). For w ∈ W , let 7.1 Definition. Let M = (W, 1, , V ) and M ′ = (W ′ , 1 ′ , ′ , V ′ ) be two L 1 -models. A meet-simulation from M to M ′ is a relation T ⊆ (W × W ) × W ′ such that for all (w 1 , w 2 , w ′ ) ∈ T : (M 1 ) If w 1 ∈ V (p) and w 2 ∈ V (p) then w ′ ∈ V ′ (p), for all p ∈ Prop; (M 2 ) If w 1 = w 2 = 1 then w ′ = 1 ′ ; 7.2 Remark. Instead of relating pairs of states from one model to states of another, we can also define meet-simulations as relations between three (possibly distinct) models. The definition above is then obtained as the special case where the first two models are the same. The results in this section work for either definition of meet-simulation.
We give some examples of meet-simulations.
is a meet-simulation on M. Let us verify this. Condition (M 1 ) follows from the fact that proposition letters are interpreted as filters, so if both w 1 and w 2 satisfy p ∈ Prop, then so does w 1 w 2 and everything above it. Condition (M 2 ) follows immediately from the definition. For (M 3 ), suppose (w 1 , w 2 , w 3 ) ∈ T , u 1 v 1 w 1 and u 2 v 2 w 2 . Then u 3 := u 1 u 2 and v 3 := v 1 v 2 witness truth of (M 3 ), as We show that T 2 is a meet-simulation. The verification for the other two is similar. Let (w 1 , w 2 , w ′ ) ∈ T 2 .
(M 1 ) Suppose w 1 , w 2 ∈ V (p) for some proposition letter p. Then w 1 w 2 ∈ V (p) because V (p) is a filter of (W, 1, ). By definition of an L 1 -morphism this implies that f (w 1 w 2 ) ∈ V ′ (p). Using the fact that V ′ (p) is a filter of (W ′ , 1 ′ , ′ ) we find w ′ ∈ V ′ (p).
Proof. The proof by induction on the structure of ϕ is analogous to the proof of Proposition 4.5.
In the remainder of this section we work towards the desired characterisation theorem. The following proposition allows us to exploit the result from Section 6.
It can then be shown that for each (X, w ′ ) ∈ T and formula ϕ ∈ L: if M, w ϕ for all w ∈ X, then M ′ , w ′ ϕ.
A FOL-formula α(x) is said to be preserved by meet ′ -simulations if for every meet ′ -simulation T between models M and M ′ and all (X, w ′ ) ∈ T we have: We claim that β(x) := ¬(x = x) is not preserved by meet ′ -simulations. Consider any L 1 -model M = (W, 1, , V ). Then T = {(∅, 1)} is a meet ′ -simulation on M. We vacuously have M |= β(x)[w] for every w ∈ ∅, but not M |= β(x) [1]. So β(x) is not preserved by meet ′ -simulations. Thus, with this notion of simulation one can prove that α(x) is equivalent to the standard translation of a formula ϕ ∈ L if and only if it is preserved by meet ′ -simulations.