Abstract
This paper presents a simple decidable logic of functional dependence LFD, based on an extension of classical propositional logic with dependence atoms plus dependence quantifiers treated as modalities, within the setting of generalized assignment semantics for first order logic. The expressive strength, complete proof calculus and metaproperties of LFD are explored. Various language extensions are presented as well, up to undecidable modalstyle logics for independence and dynamic logics of changing dependence models. Finally, more concrete settings for dependence are discussed: continuous dependence in topological models, linear dependence in vector spaces, and temporal dependence in dynamical systems and games.
Notes
Note the similarity of the unraveled model in this proof and the tree construction in the proof of Theorem 4.8. Indeed, the two decidability proofs are based on similar ideas, but there are also notable differences. The proof in Section 4 is based on a syntactic construction (“type models”) and is very elaborate; the proof of the key Truth Lemma is a very syntactic complex induction on formulas. In contrast, the proof in this section is purely semantic, and it offers a shortcut, by relying on known results and techniques in Modal Logic.
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Acknowledgements
We thank various audiences in Amsterdam, Bayreuth, Beijing, Leiden, Moscow, Nice, Rennes, Pittsburgh, Stanford and Tbilisi for their feedback on this work, and in particular, Fausto Barbero, Adam Bjorndahl, Denis Bonnay, Jan van Eijck, Malvin Gattinger, David Gonzalez, Helle Hansen, Andreas Herzig, Kevin Kelly, Raoul Koudijs, Dazhu Li, Graham Priest, Phil Pützstück, Valentin Shehtman, Gabriel Sandu, Chenwei Shi, Sonja Smets, Yde Venema, Jouko Väänänen, Yanjing Wang, and Dag Westerståhl.
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Appendices
Appendix A: Modal proofs of LFD decidability and completeness
To study LFD as a modal logic, we need to generalize the ’standard’ relational models introduced in Section 3.4 to a wider class of relational models. Viewing LFD as a modal language in the usual sense, with modalities \(\mathbb {D}_{X}\varphi \) and atomic formulas Px and D_{X}y, our general relational models will be just ordinary Kripke models for this language. This move allows us to apply to them wellknown notions and methods in modal logic, such as pmorphisms, unraveling, and filtration. In the following we will assume familiarity with these standard modal techniques. See [22] for definitions and explanations.
So there are two main differences between general relational models and the standard models introduced earlier: (a) each relation =_{X} for sets \(X\subseteq V\) is taken as primitive, without being reduced to an intersection of basic relations =_{x}, and (b) D_{X}y is treated as just another atom, whose semantics is given by a valuation (although one subject to restrictions).
1.1 A1. Relational Semantics
Definition A.1
A relational model is a structure M = (A, =_{X}, D_{X}y, Px), where: A is a set of possible worlds (“abstract assignments”); \(=_{X}\subseteq A\times A\) are binary relations on worlds, one for each set \(X\subseteq V\) of variables; \({D^{s}_{X}} y\subseteq {\mathcal P}(V)\times V\) are relations between sets of variables X and variables y, one for each world s ∈ A; and P^{s} are nary relations on variables, one for each nary predicate P and each world s. These ingredients are required to satisfy four conditions:
 (1):

all relations =_{X} are equivalence relations on A;
 (2):

all relations D^{s} satisfy Projection and Transitivity;
 (3):

if s =_{Y}t and \({D^{s}_{X}} y\) for all y ∈ Y, then s =_{Y}t and \({D^{t}_{X}} y~\text {for all}~ y\in Y\);
 (4):

if s =_{X}t and P^{s}y for some \(\{y_{1}, \ldots , y_{m}\}\subseteq X\), then P^{t}y;
 (5):

=_{∅} is the global relation on A (relating every two worlds)
The semantics of LFD on relational models is just as on dependence models, except that the abstract relations s =_{X}t, \({D^{s}_{X}} y\) and P^{s}x are used instead of their concrete counterparts.
Fact A.2
Standard relational models in the sense of Section 3.4 are exactly those relational models satisfying the following two additional conditions:
 (5):

if s =_{X}t and s =_{Y}t, then s =_{X∪Y}t.
 (6):

if s =_{X}t implies s =_{y}t holds for all t ∈ A, then \({D^{s}_{X}} y\).
1.2 A2. Equivalence Between Relational Models and Dependence Models
We now show that the logic of relational models is the same as the logic of dependence models.
To go from dependence models to relational models: we can just use the equivalence between dependence models and standard relational models (cf. Fact 3.15 and Fact 3.17).
But to go the other way, from relational models to dependence models, we need a representation of relational models in terms of standard ones:
Proposition A.3
Every relational model is a pmorphic image of some standard relational model (in the sense of Section 3.4).
Proof
The proof is essentially a variation of modal unravelling, making infinitely many copies of each world.^{Footnote 1}
Let M = (A, =_{X}, D, P) be a relational model, and let s_{0} ∈ A be any designated world. To construct a standard relational model M^{st}, take as worlds the set A^{st} of all ‘histories’, i.e. all finite sequences h = (s_{0}, X^{1}, s_{1},…, X^{n}, s_{n}), with n ≥ 0 and s_{o},…, s_{n} ∈ A satisfying \(s_{k1} =_{X^{k}} s_{k}\) for all k = 1, n. We denote by last(h) := s_{n} the last state in history h, and by →_{X} the natural onestep relation on histories, given by h →_{X}h^{′} iff \(h^{\prime }=(h, X, s^{\prime })\) (with \(last(h) =_{X} s^{\prime }=last(h^{\prime })\)). The onestep relations structure A^{st} can be viewed as a tree with root (s_{0}) (where s_{0} is the designated world), in which any two nodes \(h, h^{\prime }\) are connected by a unique nonredundant path.
To structure this as a relational model, we define a new onestep relation→ =_{X}, incorporating all the onestep relations labelled by sets that locally determine X:
Then the required equivalence relations =_{X} on worlds/histories in A^{st} can be taken to be the reflexivetransitivesymmetric closure of the relations→ =_{X}. To check the claims below, it may be useful to note that \(h=_{X} h^{\prime }\) holds iff the unique nonredundant path from h to \(h^{\prime }\) consists only of steps of the form \(h_{n} \stackrel {=}{\to }_{Y^{n}} h_{n+1}\), or \(h_{n} \overset {=}{\leftarrow }_{Y^{n}} h_{n+1}\), with last(h_{n})⊧D_{Y}X.
Finally, the valuation on atoms is given by truth at the last world in the history (in the original model):
The fact that this definition yields a standard relational model M^{st} is an easy verification.
To finish the proof, we define a map f : A^{st} → A, by putting f(h) := last(h) for all h ∈ A^{st}. It is easy to check that f is a surjective pmorphism f from M^{st} to M. (Surjectivity follows from the fact that every world s ∈ A satisfies s_{0} =_{∅}s, by condition 5 on relational models, hence h = (s_{0}, ∅, s) is a history with f(h) = last(h) = s.) □
Combining Fact 3.15, Propositions 3.17 and A.3, plus the preservation of modal formulas under surjective pmorphisms (and so under surjective homomorphisms), yields the following:
Corollary A.4
(Modal equivalence of relational and dependence models) The same LFD formulas are valid on dependence models, relational models and standard relational models.
1.3 A3. Decidability via Relational Models
The preceding detour into abstract relational models and the above Corollary A.4 on modal equivalence can be used to give a second, more general proof of decidability using the Modal Logic concept of filtration [22].
Proposition A.5
The language LFD has the Strong Finite Relational Model Property: if φ is satisfied in some relational model M, then it is satisfied in a finite relational model, whose size is bounded by a computable function of φ. As a consequence, the logic LFD is decidable.
Proof
Start with the singleton F = {φ}, and construct the finite set of formulas Φ =Φ _{F} as in Section 4.1 (whose size was bounded by a computable function of φ).
The filtrated model M^{f} has as worlds the equivalence classes [s] of original worlds s ∈ A modulo Φequivalence ≡_{Φ} (with respect to all formulas in Φ). Note that there are only finitely many such classes (their number is bounded by a computable function F(φ)).
To define the relations =_{X} in the filtrated model, we take the following ‘dependent filtration’:
This is well defined (independent from the choice of representatives), and the definition implies that {y ∈ V : s⊧D_{X}y} = {y ∈ V : t⊧D_{X}y} whenever [s] =_{X}[t].
As for the valuation: the truth values at [s] for atoms D_{X}y, Px ∈Φ are inherited from the original truth values at s in M. The resulting finite relational model M^{f} is a filtration of M in the usual sense. By the standard Filtration Lemma, [s] will satisfy φ in M^{f}.
As usual, the Strong Finite Relational Model Property provides an obvious algorithm for deciding satisfiability on relational models (and thus by Corollary A.4 also on dependence models). Given formula φ generate all the relational models (up to isomorphism) of size ≤ F(φ); check whether φ is satisfied in any of these models. If so, φ is satisfiable; else, it is not. □
It should be noted that in general the filtrated model is typically a nonstandard relational model, not a dependence model.
1.4 A4. Completeness via Relational Models
Completeness of LFD with respect to dependence models follows from Corollary A.4 together with the following result:
Lemma A.6
The calculus LFD is sound and strongly complete wrt general relational models.
Proof
Soundness is immediate: the conditions on relational models were chosen to validate the matching axioms. For completeness, take the usual Henkinstyle ‘canonical model’ for LFD, considered as a basic modal logic. This canonical model is a relational model, and the calculus is strongly complete for this model.□
Appendix B: Restricted Cut Elimination and Subformula Property
As announced, it is convenient to absorb Weakening into the logical rules (cf. [70] for this technique), while simultaneously restricting Projection and Transitivity to variables that actually occur in the conclusion, and also restricting Cut to dependence atoms between actually occurring variables. This can be done by first modifying the axioms to
(a) \({\varGamma }, \varphi \vdash \varphi , \varDelta \quad \quad (b) {\varGamma } \vdash D_{X} x, \varDelta ~\text {where} x\in X\subseteq Var({\varGamma }\cup \varDelta )\),
while introduction rules are made ‘cumulative’, by repeating principal formulas in the premises.
For instance, the leftintroduction rule \((\mathbb {D}_{L})\) becomes
Transitivity needs special treatment: in addition to being made cumulative, it has to be restricted to relevant formulas, becoming the rule of ‘Restricted Transitivity’:
Likewise, the rightintroduction rule \((\mathbb {D}_{R})\) needs to be modified to:
Finally, we replace Cut by a restricted version (in which we also absorbed Weakening):
A restrictedcut proof is a proof that uses only these modified rules. The following observation shows how LFD allows for a tighter management of variables than FOL:
Lemma B.7
(Elimination of irrelevant variables)

If Γ ⊩ D_{X}Y, Δ has a restrictedcut proof, and Z = X ∩ (V ar(Γ) ∪ Y ∪ V ar(Δ)), then Γ ⊩ D_{Z}Y, Δ has a restrictedcut proof.

If \({\varGamma }\vdash \mathbb {D}_{X} \varphi ,\varDelta \) has a restrictedcut proof, and Z = X ∩ (V ar(Γ) ∪ V ar(φ) ∪ V ar(Δ)), then \({\varGamma }\vdash \mathbb {D}_{Z} \varphi ,\varDelta \) has a restrictedcut proof.
Using this lemma and a cursory inspection of the above modified rules, we obtain:
Lemma B.8
(Subformula/Subterm Property) Let \({\mathcal P}\) be a restrictedcut proof of the sequent Γ ⊩Δ. Each formula 𝜃 in \({\mathcal P}\) is either of the form D_{X}Y with \(X\cup Y\subseteq Var({\varGamma }\cup \varDelta )\), or it is a subformula of some formula in Γ ∪Δ. In particular, only variables x ∈ V ar(Γ∪Δ) occur in \({\mathcal P}\).
Finally, we can prove our Restricted Cut Elimination theorem:
Every provable sequent has a restrictedcut proof (which thus involves only subformulas of the sequent formulas, or dependence atoms for variables in the sequent)
Proof
To show this, first gradually eliminate Transitivity and Projection in favor of their modified versions, using the above lemma when necessary. Similarly replace all other rules except Cut by their cumulative versions. Finally, eliminate unrestricted cuts in the usual way, by successively removing topmost maximalrank cuts from a given proof of a sequent Γ ∪Δ. Here, since DA Cut is permitted, one need not worry about cutformulas of the form D_{X}Y, with all variables occurring in the original sequent Γ ∪Δ. As a result, the additional axioms and rules for dependency are innocuous: the cutformula to be removed was never introduced by such rules. The only case that presents any novelty is that of a dualquantifier cutformula \(\mathbb {D}_{X}\varphi \) that is principal in both antecedent and succedent: having been freshly introduced on both sides.□
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Baltag, A., van Benthem, J. A Simple Logic of Functional Dependence. J Philos Logic 50, 939–1005 (2021). https://doi.org/10.1007/s1099202009588z
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DOI: https://doi.org/10.1007/s1099202009588z
Keywords
 Functional dependence
 Generalized assignment semantics
 Modal logic
 Epistemic logic
 Logics of dependence