A Simple Logic of Functional Dependence

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 meta-properties of LFD are explored. Various language extensions are presented as well, up to undecidable modal-style 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.

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_Xy, our general relational models will be just ordinary Kripke models for this language. This move allows us to apply to them well-known notions and methods in modal logic, such as p-morphisms, 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_Xy 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_Xy, 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 n-ary relations on variables, one for each n-ary 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 =_Yt and ${D^{s}_{X}} y$ for all y ∈ Y, then s =_Yt and ${D^{t}_{X}} y~\text {for all}~ y\in Y$;

(4):

if s =_Xt and P^sy for some $\{y_{1}, \ldots , y_{m}\}\subseteq X$, then P^ty;

(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 =_Xt, ${D^{s}_{X}} y$ and P^sx 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 =_Xt and s =_Yt, then s =_X∪Yt.

(6):

if s =_Xt implies s =_yt 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 p-morphic 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.¹

Let M = (A, =_X, D, P) be a relational model, and let s₀ ∈ 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 = (s₀, X¹, s₁,…, Xⁿ, s_n), with n ≥ 0 and s_o,…, s_n ∈ A satisfying $s_{k-1} =_{X^{k}} s_{k}$ for all k = 1, n. We denote by last(h) := s_n the last state in history h, and by →_X the natural one-step relation on histories, given by h →_Xh^′ iff $h^{\prime }=(h, X, s^{\prime })$ (with $last(h) =_{X} s^{\prime }=last(h^{\prime })$). The one-step relations structure A^st can be viewed as a tree with root (s₀) (where s₀ is the designated world), in which any two nodes $h, h^{\prime }$ are connected by a unique non-redundant path.

To structure this as a relational model, we define a new one-step relation→ =_X, incorporating all the one-step relations labelled by sets that locally determine X:

$$h \stackrel{=}{\to}_{X} h^{\prime} ~\text{iff}~ h\to_{Y} h^{\prime} ~\text{for some}~ Y~ \text{with}~ last(h)\models D_{Y} X.$$

Then the required equivalence relations =_X on worlds/histories in A^st can be taken to be the reflexive-transitive-symmetric 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 non-redundant 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_YX.

Finally, the valuation on atoms is given by truth at the last world in the history (in the original model):

$${D^{h}_{X}} y ~\text{iff}~ last(h)\models D_{X} y, \quad P^{h} \mathbf{x} ~\text{iff}~ last(h)\models P\mathbf{x}.$$

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 p-morphism f from M^st to M. (Surjectivity follows from the fact that every world s ∈ A satisfies s₀ =_∅s, by condition 5 on relational models, hence h = (s₀, ∅, 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 p-morphisms (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 Φ-equivalence ≡_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’:

$$[s] =_{X} [t] ~\text{iff}~ (s\models \theta ~\text{iff}~ s\models \theta) ~\text{for all}~\theta\! \in\! {\varPhi}~\text{with}~ Free(\theta)\! \subseteq\! \{y\in V: s\models D_{X} y\}.$$

This is well defined (independent from the choice of representatives), and the definition implies that {y ∈ V : s⊧D_Xy} = {y ∈ V : t⊧D_Xy} whenever [s] =_X[t].

As for the valuation: the truth values at [s] for atoms D_Xy, 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 non-standard 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 Henkin-style ‘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 left-introduction rule $(\mathbb {D}_{L})$ becomes

$$ \frac{\mathbb{D}_{X}\varphi,\varphi,{\varGamma} ~\vdash ~{\varDelta}}{\mathbb{D}_{X}{\varphi}, {\varGamma}~ \vdash~ {\varDelta}} $$

Transitivity needs special treatment: in addition to being made cumulative, it has to be restricted to relevant formulas, becoming the rule of ‘Restricted Transitivity’:

$$ \frac{{\varGamma} \vdash {\varDelta}, D_{X} Y, D_{X} Z \qquad {\varGamma} \vdash {\varDelta}, D_{Y} Z, D_{X} Z}{\varGamma \vdash \varDelta, D_{X} Z} \quad\text{where}~ Y\subseteq Var({\varGamma}\cup\varDelta)\cup X\cup Z $$

Likewise, the right-introduction rule $(\mathbb {D}_{R})$ needs to be modified to:

$$ \frac{~~~~~~~~~~~~~~{\varGamma} \vdash {\varDelta},{\varphi}, \mathbb{D}_{X}{\varphi}}{D_{X} Y,{\varGamma}, {\varGamma}^{\prime} \vdash {\varDelta}, {\varDelta}^{\prime}, \mathbb{D}_{X} {\varphi}} \quad \text{where}~ Free({\varGamma}\cup\varDelta)\subseteq Y $$

Finally, we replace Cut by a restricted version (in which we also absorbed Weakening):

$$ \text{(DA Cut)}\frac{{\varGamma} \vdash {\varDelta}, D_{X} y \quad D_{X} y,{\varGamma} \vdash \varDelta}{{\varGamma} \vdash {\varDelta}} \quad \text{where}~ X\cup \{y\}\subseteq Var({\varGamma}\cup{\varDelta}) $$

A restricted-cut 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_XY, Δ has a restricted-cut proof, and Z = X ∩ (V ar(Γ) ∪ Y ∪ V ar(Δ)), then Γ ⊩ D_ZY, Δ has a restricted-cut proof.

• If ${\varGamma }\vdash \mathbb {D}_{X} \varphi ,\varDelta$ has a restricted-cut proof, and Z = X ∩ (V ar(Γ) ∪ V ar(φ) ∪ V ar(Δ)), then ${\varGamma }\vdash \mathbb {D}_{Z} \varphi ,\varDelta$ has a restricted-cut 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 restricted-cut proof of the sequent Γ ⊩Δ. Each formula 𝜃 in ${\mathcal P}$ is either of the form D_XY 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 restricted-cut 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 maximal-rank cuts from a given proof of a sequent Γ ∪Δ. Here, since DA Cut is permitted, one need not worry about cut-formulas of the form D_XY, with all variables occurring in the original sequent Γ ∪Δ. As a result, the additional axioms and rules for dependency are innocuous: the cut-formula to be removed was never introduced by such rules. The only case that presents any novelty is that of a dual-quantifier cut-formula $\mathbb {D}_{X}\varphi$ that is principal in both antecedent and succedent: having been freshly introduced on both sides.□