A Sequent Calculus for Dynamic Topological Logic - Semantic Scholar
A Sequent Calculus for Dynamic Topological Logic Samuel Reid∗ July 24, 2014
Abstract ◦∗/2
We introduce a sequent calculus for the temporal-over-topological fragment DTL0 of dynamic topological logic DTL, prove soundness semantically, and prove completeness syntactically using the ◦∗/2 ◦∗/2 axiomatization of DTL0 given in [3]. A cut-free sequent calculus for DTL0 is obtained as the union of the propositional fragment of Gentzen’s classical sequent calculus, two 2 structural rules for the modal extension, and nine ◦ (next) and ∗ (henceforth) structural rules for the temporal extension. Future research will focus on the construction of a hypersequent calculus for dynamic topological S5 logic in order to prove Kremer’s Next Removal Conjecture for the logic of homeomorphisms on almost discrete spaces S5H.
1
Introduction
Within recent years there has been an outburst of research activity in spatial-temporal reasoning, leading to important advances in computer science and logic. The Handbook of Spatial Logics [1], and references therein, summarize the main recent achievements such as mereotopology, spatial constraint calculi, modal logics of space, connections between topology and epistemic logic, logics for fragments of elementary geometry, mathematical morphology, logics of space-time and relativity theory, and dynamic topological logic. The present paper focuses on extending the research area of dynamic topological logic by means of a Gentzen-style ◦∗/2 proof calculus for the temporal-over-topological fragment of dynamic topological logic DTL0 . We achieve this by extending the sequent calculi developed for the fragments of dynamic topological logic defined in [2] as the logic of control action S4F and the logic of continuous control action S4C. The main motivation for this work is to combine structural proof theory and dynamic topological logic. Thus laying the groundwork for defining hypersequent calculi for dynamic topological S5 logic and the conservative axiomatizable extensions S5C, S5H, S5Ct, and S5Ht known as the logics of continuous functions on almost discrete spaces, homeomorphisms on almost discrete spaces, functions on trivial spaces, and homeomorphisms on trivial spaces, respectively [4]. These logics have been defined quite recently and the development of a sequent calculus for them would lead to applications such as proof search and automated theorem proving in dynamic topological logic. We identify a main conjecture of this recent research area and propose that future research regarding hypersequent calculi for dynamic topological S5 will provide a positive solution to this conjecture. Conjecture 1 (Kremer’s Next Removal Conjecture). The logic of homeomorphisms on almost discrete spaces can be axiomatized without next removal by S5H = S5 + LTL + (◦2A ⇔ 2 ◦ A).
Dynamic topological logic provides the foundation for breakthroughs in topics ranging from control theory and robot-motion planning to dynamical systems and eschatological cosmology, as statements regarding the possibility and necessity of spatio-temporal properties can be understood with systematic logical precision. ∗ University
of Calgary, Department of Mathematics and Statistics, Calgary, AB, Canada. e − mail : [email protected].
1
2
Dynamic Topological Logic
We now present an introduction to dynamic topological logic, freely citing from the seminal papers on dynamic topological logic [2], [3], [4], [5]. The main idea of dynamic topological logic is to combine of topological semantics in logic, temporal logic, and topological dynamics (asymptotic properties of continuous maps on topological spaces). Interestingly predating the well-known Kripke semantics for modal logic, the McKinsey-Tarki topological semantics interprets the purely temporal modal propositional language L2 = | ∧ | ¬ | 2 | p1 | · · · | pn | · ··
in terms of topological spaces with the interptation of 2 given by topological interior. Then the propositional letters denote subsets of X; ∨, ∧, and ¬ express union, intersection, and complement, respectively, with 3 := ¬2¬ interpreted as closure. We then have n ^
pi →
i=1
k _
qj ⇔
j=1
n \
pi ⊆
i=1
k [
qj
j=1
thus providing us with a semantic interpretation for sequents and language to prove the soundess of structural rules. A topological model is an ordered pair M = hX, V i, where X is a topological space and V : Var → P(X). The function V is extended to formulas of L2 by V (¬A) = X − V (A) V (A ∧ B) = V (A) ∩ V (B) V (2A) = int(V (A)) with four validity relations, where T is a class of topological spaces. M A iff V (A) = X X A iff hX, V i A, ∀V : Var → X T A iff X A, ∀X ∈ T A iff X A, ∀X A dynamic topological system is an ordered pair hX, f i, where X is a topological space and f is a continuous function on X. We interpret the temporal connectives of the modal-temporal language L by means of the function f : • ◦A is true at x iff A is true at f x. V∞ • ∗A is true at x iff A is true at i=1 f i x, ∀i ∈ N. A dynamic topological model is an ordered triple M = hX, f, V i where hX, f i is a dynamic topological system and V : Var → P(X) is extended to all formulas of L by V (◦A) = f −1 (V (A)) ∞ \ V (∗A) = f −i (V (A)) i=1
with five validity relations, where FX is a class of continuous functions associated with the topological space
2
X and F is an arbitrary class of continuous functions. M A iff V (A) = X hX, f i A iff hX, f, V i A, ∀V : Var → P(X) X A iff hX, f i A, ∀f ∈ FX T A iff X A, ∀X ∈ T T , F A iff hX, f i A, ∀X ∈ T , ∀f ∈ F A iff X A, ∀X From this we define DTL0 = {A : A} to be the logic of all dynamic topological systems. We now provide the axiomatization of linear time logic (LTL). S4 axioms for *: ∗ (A ⊃ B) ⊃ (∗A ⊃ ∗B) ∗A⊃A ∗A⊃∗∗A ◦ commutes with ¬, ∨, ∗ : ◦ ¬A ⇔ ¬ ◦ A ◦ (A ∨ B) ⇔ ◦A ∨ ◦B ◦ ∗A ⇔ ∗ ◦ A henceforth implies next: ∗ A ⊃ ◦A induction axiom: A ∧ ∗(A ⊃ ◦A) ⊃ ∗A We can then provide the axiomatization given in [3] for the temporal-over-topological fragment of dynamic topological logic as follows: ◦∗/2 DTL0 = S4 L2 + LTL L◦∗/2 where S4 L2 is the S4 axioms for 2 with A, B ∈ L2 and LTL L◦∗/2 is LTL where the scope of 2 in any ◦∗/2
subformula is ◦-free and ∗-free. This defines DTL0 with the temporal-over-topological language L◦∗/2 . ◦∗/2 We now define the sequent calculus for DTL0 by means of six weak structural rules: weakening in the antecedent (WA), weakening in the succedent (WS), contraction in the antecedent (CA), contraction in the succedent (CS), interchange in the antecedent (IA), and interchange in the succedent (IS); eight logical rules: ∧R, ∧L, ∨R, ∨L, ⊃ R, ⊃ L, ¬R, ¬L; the strong structural rule of cut; two modal rules for the 2-modality: 2L, 2R; and nine temporal rules: ∗L, ∗R, ◦ ∗ R, ◦ ∗ L, ◦¬R, ◦¬L, ◦ ∗ CA, IN D, L ◦ R. The 22 unary inference rules and 4 binary inference rules are as follows: A, A, Γ → Θ Γ → Θ, A, A ∆, A, B, Γ → Θ Γ→Θ Γ→Θ WA WS CA CS IA A, Γ → Θ Γ → Θ, A A, Γ → Θ Γ → Θ, A ∆, B, A, Γ → Θ Γ → Θ, A, B, Λ Γ → Θ, A Γ → Θ, B A/B, Γ → Θ Γ → Θ, A/B IS ∧R ∧L ∨R Γ → Θ, B, A, Λ Γ → Θ, A ∧ B A ∧ B, Γ → Θ Γ → Θ, A ∨ B A, Γ → Θ B, Γ → Θ A, Γ → Θ, B Γ → Θ, A B, ∆ → Λ ⊃R ⊃L ∨L A ∨ B, Γ → Θ Γ → Θ, A ⊃ B A ⊃ B, Γ, ∆ → Θ, Λ Γ, A → Θ Γ → A, Θ Γ → Θ, A A, ∆ → Λ ¬R ¬L CU T Γ → ¬A, Θ Γ, ¬A → Θ Γ, ∆ → Θ, Λ Γ, A → Θ Γ, A → Θ 2Γ → A ∗Γ → A 2R ∗R 2L ∗L ∗Γ → ∗A 2Γ → 2A Γ, 2A → Θ Γ, ∗A → Θ Γ → ∗ ◦ A, Θ Γ, ∗ ◦ A → Θ Γ → ¬ ◦ A, Θ Γ, ¬ ◦ A → Θ ◦∗R ◦∗L ◦¬R ◦¬L Γ → ◦ ∗ A, Θ Γ, ◦ ∗ A → Θ Γ → ◦¬A, Θ Γ, ◦¬A → Θ ∗A, ∗(A ⊃ ◦A) → ◦ ∗ A A, ◦ ∗ A, Γ → Θ Γ→Θ L◦R ◦ ∗ CA IN D ◦Γ → ◦Θ ∗A, Γ → Θ A, ∗(A ⊃ ◦A) → ∗A
3
3
◦∗/2
Soundness of DTL0
We show soundness for the temporal-over-topological fragment of dynamic topological logic by giving semantic proofs of the soundness of the rules of the temporal-over-topological sequent calculus. Proposition 1. The 2L rule is sound.
Γ, A → Θ 2L Γ, 2A → Θ
Proof. Assume Γ ∩ A ⊆ Θ. Then Γ ∩ int(A) ⊆ Θ. Proposition 2. The 2R rule is sound.
2Γ → A 2R 2Γ → 2A
Proof. Assume int(Γ) ⊆ A. Then int(Γ) ⊆ int(A). Proposition 3. The ∗L rule is sound.
Γ, A → Θ ∗L Γ, ∗A → Θ Proof. Assume Γ ∩ A ⊆ Θ. Then Γ∩
∞ \
f −i (A) ⊆ Θ.
i=1
Proposition 4. The ∗R rule is sound. ∗Γ → A ∗R ∗Γ → ∗A Proof. Assume
∞ \
f −i (Γ) ⊆ A.
i=1
Then
∞ \
f −i (Γ) ⊆
i=1
∞ \
f −i (A).
i=1
Proposition 5. The ◦ ∗ R rule is sound. Γ → ∗ ◦ A, Θ ◦∗R Γ → ◦ ∗ A, Θ Proof. Assume Γ⊆
∞ \
f −i (f −1 (A)) ∪ Θ.
i=1
Then Γ ⊆ f −1
∞ \
! f −i (A)
i=1
4
∪ Θ.
Proposition 6. The ◦ ∗ L rule is sound. Γ, ∗ ◦ A → Θ ◦∗L Γ, ◦ ∗ A → Θ Proof. Assume Γ∩
∞ \
f −i (f −1 (A)) ⊆ Θ.
i=1
Then Γ ∩ f −1
∞ \
! f −i (A))
⊆ Θ.
i=1
Proposition 7. The ◦¬R rule is sound. Γ → ¬ ◦ A, Θ ◦¬R Γ → ◦¬A, Θ Proof. Assume Γ ⊆ (X − f −1 (A)) ∪ Θ. Then Γ ⊆ f −1 (X − A) ∪ Θ. Proposition 8. The ◦¬L rule is sound. Γ, ¬ ◦ A → Θ ◦¬L Γ, ◦¬A → Θ Proof. Assume Γ ∩ (X − f −1 (A)) ⊆ Θ. Then Γ ∩ f −1 (X − A) ⊆ Θ. Proposition 9. The ◦ ∗ CA rule is sound. A, ◦ ∗ A, Γ → Θ ◦ ∗ CA ∗A, Γ → Θ Proof. Assume A∩f
−1
∞ \
! f
−i
(A))
∩ Γ ⊆ Θ.
i=1
Then
∞ \
f −i (A)) ∩ Γ ⊆ Θ.
i=1
Proposition 10. The IN D rule is sound. ∗A, ∗(A ⊃ ◦A) → ◦ ∗ A IN D A, ∗(A ⊃ ◦A) → ∗A Proof. Assume ∞ \
f −i (A) ∩
i=1
∞ \
f −i ((X − A) ∪ f −1 (A)) ⊆ f −1
i=1
Then A∩
∞ \
∞ \
! f −i (A) .
i=1
f
−i
((X − A) ∪ f
i=1
−1
(A)) ⊆
∞ \ i=1
5
f −i (A).
Proposition 11. The L ◦ R rule is sound. Γ→Θ L◦R ◦Γ → ◦Θ Proof. Assume Γ ⊆ Θ. Then f −1 (Γ) ⊆ f −1 (Θ).
4
◦∗/2
Completeness for DTL0
We show completeness for the temporal-over-topological fragment of dynamic topological logic by giving ◦∗/2 sequent calculus derivations of the axioms of DTL0 using the soundness of the sequent calculus rules. ◦∗/2
Proposition 12. The sequent calculus for DTL0 Proof.
A 2A
→ A 2L →A → 2A ⊃ A
◦∗/2
Proposition 13. The sequent calculus for DTL0 Proof.
2A 2A
◦∗/2
Proof.
→A
A ⊃ B, A A ⊃ B, 2A 2A, A ⊃ B 2A, 2(A ⊃ B) 2A, 2(A ⊃ B) 2(A ⊃ B)
◦∗/2
A ∗A
⊃R
proves 2(A ⊃ B) ⊃ (2A ⊃ 2B).
◦∗/2
∗A ∗A
⊃R
proves ∗A ⊃ ∗ ∗ A.
→ ∗A → ∗ ∗ A ∗R → ∗A ⊃ ∗ ∗ A ◦∗/2
Proposition 17. The sequent calculus for DTL0
⊃R
proves ∗A ⊃ A.
→ A ∗L →A → ∗A ⊃ A
Proposition 16. The sequent calculus for DTL0 Proof.
proves 2A ⊃ 22A.
B→B →B ⊃L → B 2L → B IA → B 2L → 2B 2R → 2A ⊃ 2B ⊃ R → 2(A ⊃ B) ⊃ (2A ⊃ 2B)
Proposition 15. The sequent calculus for DTL0 Proof.
⊃R
→ 2A 2R → 22A → 2A ⊃ 22A
Proposition 14. The sequent calculus for DTL0 A
proves 2A ⊃ A.
⊃R
proves ∗(A ⊃ B) ⊃ (∗A ⊃ ∗B).
6
A
→A
A ⊃ B, A A ⊃ B, ∗A ∗A, A ⊃ B ∗A, ∗(A ⊃ B) ∗A, ∗(A ⊃ B) ∗(A ⊃ B)
Proof.
B→B →B ⊃L → B ∗L → B IA → B ∗L → ∗B ∗R → ∗A ⊃ ∗B ⊃ R → ∗(A ⊃ B) ⊃ (∗A ⊃ ∗B) ◦∗/2
Proposition 18. The sequent calculus for DTL0
Proof.
A ◦A A, ◦A A, ∗ ◦ A ∗A
◦∗/2
Proof.
◦∗/2
¬ ◦ ¬A ◦¬A
◦∗/2
→A → A, B
WS
A∨B ◦(A ∨ B) ◦(A ∨ B) ◦(A ∨ B) ◦(A ∨ B) ◦(A ∨ B)
→ → → → → → →
⊃R ∧R
proves ◦¬A ⇔ ¬ ◦ A.
→ ¬ ◦ A ◦¬L ¬◦A → ¬◦A ◦¬R → ¬◦A ¬ ◦ A → ◦¬A ⊃R → ◦¬A ⊃ ¬ ◦ A → ¬ ◦ A ⊃ ◦¬A → (◦¬A ⊃ ¬ ◦ A) ∧ (¬ ◦ A ⊃ ◦¬A)
Proposition 21. The sequent calculus for DTL0
A A
proves ◦ ∗ A ⇔ ∗ ◦ A.
→ ∗◦A ◦∗L ∗◦A → ∗◦A ◦∗R → ∗◦A ∗◦A → ◦∗A ⊃R → ◦∗A⊃∗◦A → ∗◦A⊃◦∗A → (◦ ∗ A ⊃ ∗ ◦ A) ∧ (∗ ◦ A ⊃ ◦ ∗ A)
Proposition 20. The sequent calculus for DTL0
Proof.
proves ∗A ⊃ ◦A.
→ A ◦R → ◦A LWA → ◦A → ◦A ∗L → ◦A ◦ ∗ CA⊃ R → ∗A ⊃ ◦A
Proposition 19. The sequent calculus for DTL0 ∗◦A ◦∗A
⊃R
⊃R ∧R
proves ◦(A ∨ B) ⇔ (◦A ∨ ◦B).
Proof. B→B WS B → B, A IS B → A, B ∨L A, B L◦R ◦A, ◦B ∨R ◦A, ◦A ∨ ◦B A→A B→B IS ∨R ∨R ◦A ∨ ◦B, ◦A A → A∨B B → A∨B ∨R L◦R L◦R ◦A ∨ ◦B, ◦A ∨ ◦B ◦A → ◦(A ∨ B) ◦B → ◦(A ∨ B) CS ∨L ◦A ∨ ◦B ◦A ∨ ◦B → ◦(A ∨ B) ⊃R ◦(A ∨ B) ⊃ (◦A ∨ ◦B) → (◦A ∨ ◦B) ⊃ ◦(A ∨ B) ⊃ R ∧R → (◦(A ∨ B) ⊃ (◦A ∨ ◦B)) ∧ ((◦A ∨ ◦B) ⊃ ◦(A ∨ B))
7
◦∗/2
Proposition 22. The sequent calculus for DTL0
A→A L◦R ◦A→ ◦A ⊃L A ⊃ ◦A, A → ◦A ∗L (A ⊃ ◦A), ∗A → ◦A IA ∗A, (A ⊃ ◦A) → ◦A ∗L ∗A, ∗(A ⊃ ◦A) → ◦A ∗R ∗A, ∗(A ⊃ ◦A) → ∗ ◦ A ◦∗R ∗A, ∗(A ⊃ ◦A) → ◦ ∗ A IND A, ∗(A ⊃ ◦A) → ∗A ∧L A ∧ ∗(A ⊃ ◦A), ∗(A ⊃ ◦A) → ∗A IA ∗(A ⊃ ◦A), A ∧ ∗(A ⊃ ◦A) → ∗A ∧L A ∧ ∗(A ⊃ ◦A), A ∧ ∗(A ⊃ ◦A) → ∗A CA A ∧ ∗(A ⊃ ◦A) → ∗A → (A ∧ ∗(A ⊃ ◦A)) ⊃ ∗A ⊃ R A
Proof.
proves (A ∧ ∗(A ⊃ ◦A)) ⊃ ∗A.
→A
◦∗/2
Lastly, we remark that all derivations in the sequent calculus for DTL0
5
◦∗/2
Admissibility of DTL0
are cut-free and harmonious.
Rules of Inference
We follow the strategy for cut-elimination and recall the definitions of descendents, ancestors, and depth of ◦∗/2 a formula from [6]. All of the inferences of DTL0 (with the exception of the cut rule) have a principal formula which is, by definition, the formula occurring in the lower sequent of the inference which is not in the cedents Γ or ∆ (or Θ or Λ). The exchange inferences have two principle formulas, as do ◦ ∗ CA, IN D, and L ◦ R. Every inference, except weakenings, has one or more auxiliary formulas which are the formulas A and B, occurring in the upper sequent(s) of the inference. The formulas which occur in the cedents Γ, ∆, Θ, Λ are called side formulas of the inference. The two auxiliary formulas of a cut inference are called the cut formulas. If C is a side formula in an upper sequent of an inference then the immediate descendent of C is the corresponding occurrence of the same formula in the same position in the same cedent in the lower sequent of the inference. If C is an auxiliary formula of any inference except an exchange or cut inference, then the principal formula of the inference is the immediate descendent of C. For an exchange inference, the immediate descendent of the A or B in the upper sequent is the A or B, respectively, in the lower sequent. The cut formulas of a cut inference do not have immediate descendents. We say that C is an immediate ancestor of D if and only if D is an immediate descendent of C. The ancestor relation is defined to be the reflexive, transitive closure of the immediate ancestor relation; thus, C is an ancestor of D if and only if there is a chain of zero or more immediate ancestors from D to C. A direct ancestor of D is an ancestor of C of D such that C is the same formula as D; descendent and direct descendent are defined conversely. The depth of a formula A is the height of the tree representation of the formula, with the depth of a cut inference the depth of its cut formula: • depth(A) = 0 for A atomic. • depth(A ∧ B) = depth(A ∨ B) = depth(A ⊃ B) = 1 + max{depth(A), depth(B)}. • depth(¬A) = depth(2A) = depth(◦A) = depth(∗A) = 1 + depth(A). x
The superexponentiation function 2xi , for i, x ≥ 0, is defined inductively by 2x0 = x and 2xi+1 = 22i .
8
◦∗/2
Theorem 1. Let Π be a DTL0 -proof and suppose every cut formula in Π has depth less than or equal ◦∗/2 to d. Then there is a cut-free DTL0 -proof Π∗ with the same endsequent as Π, with size kΠk
kΠ∗ k < 22d+2 . Proof. Lemma 1 shows how to replace a single cut by lower depth cut inference. Iteration of this construction removes all cuts of maximum depth d in a proof, which is stated as Lemma 2, from which the theorem is a consequence. ◦∗/2
Lemma 1. Let Π be a DTL0 -proof with final inference a cut of depth d such that every other cut in Π ◦∗/2 has depth strictly less than d. Then there is a DTL0 -proof Π∗ with the same endsequent as Π with all cuts in Π∗ of depth less than d with kΠ∗ k < kΠk2 . We closely follow the proof of Lemma 2.4.2.1 in [6] with case (e) and (f) omitted, and consider the additional cases of the cut formula as 2B, ◦B, and ∗B. Proof. The proof Π ends with a cut inference
β .. .
α .. .
Γ
→ Θ, A
A, Γ Θ, Θ
→ Γ→Θ
Γ, Γ
→Θ
CU T
where the depth of the cut formula A equals d and where all cuts in the subproofs α and β have depth strictly less than d. The lemma is proved by cases, based on the outermost logical connective of the cut formula A. We can assume without loss of generality that both α and β contain at least one strong inference; since otherwise, we must have A in Γ or Θ, or have a formula which occurs in both Γ and Θ, and in the former case, the sequent Γ → Θ is obtainable by weak inferences from one of the upper sequents and the cut can therefore be eliminated. In the latter case, Γ → Θ can be inferred with no cut inference at all. 1. Suppose A is a formula of the form ¬B. We shall form new proofs α∗ and β ∗ of the sequents B, Γ → Θ and Γ → Θ, B, which can then be combined with a cut inference of depth d − 1 to give the proof Π∗ of Γ → Θ. To form α∗ , first form α0 by replacing every sequent ∆ → Λ in α with the sequent ∆, B → Λ− , where Λ− is obtained from Λ by removing all direct ancestors of the cut formula A. Of B, ∆ → Λ course, α0 is not a valid proof; e.g., a ¬R inference in α of the form could become in ∆ → Λ, ¬B − B, ∆, B → Λ . α0 , This is not a valid inference, but can be modified to become a valid inference ∆, B → Λ− by inserting some exchanges and a contraction. In this manner, α0 can be modified so that it becomes a valid proof α∗ by removing some ¬L inferences and inserting some weak inferences. The proof β ∗ of Γ → Θ, B is formed in a similar manner from β. No new cuts are introduced by this process and, since we do not count weak inferences, kα∗ k ≤ kαk and kβ ∗ k ≤ kβk; thus Π∗ has only cuts of depth < d and has kΠ∗ k ≤ kΠk. 2. Now suppose the cut formula A is of the form B ∨ C. We defined α0 as a tree of sequents, with root labelled Γ → Θ, B, C, by replacing every sequent ∆ → Λ in α with the sequent ∆ → Λ− , B, C, where Λ− is Λ minus all occurrences of direct ancestors of the cut formula. By removing some formerly ∨R inferences from α0 and by adding some weak inferences, α0 can be transformed into a valid proof α∗ . Now construct βB from β by replacing every occurrence in β of B ∨ C as a direct ancestor of the cut formula with just the formula B. One way that βB can fail to be valid is that an ∨L inference B, ∆ → Λ C, ∆ → Λ B, ∆ → Λ C, ∆ → Λ in βB . This is no longer ∨L may become just B ∨ C, ∆ → Λ B, ∆ → Λ 9
a valid inference, but it can be fixed up by discarding the inference and its upper right hypothesis, including discarding the entire subproof of the upper right hypothesis. The only other changes needed to make βB valid are the addition of weak inferences, and in this way, a valid proof βB of B, Γ → Θ is formed. A similar process forms a valid proof βC of C, Γ → Θ. The proof Π∗ can now be defined to be βC α∗ .. .. . . Γ → Θ, B, C C, Γ → Θ CU T Γ, Γ → Θ, Θ, B Γ
→ Θ, B
βB .. . B, Γ → Θ
→ Θ, Θ Γ→Θ
Γ, Γ
CU T
The process of forming α∗ , βB , and βC did not introduce any new cuts or any new strong inferences. Thus, we clearly have that every cut in Π∗ has depth < d, and that kΠ∗ k ≤ kαk + 2kβk + 2. Since kΠk = kαk + kβk + 1 and kαk, kβk ≥ 1, this suffices to prove the lemma for this case. 3. The cases where A has outermost connective ∧ or ⊃ are very similar to the previous case and are omitted. 4. Now suppose A is of the form 2B; we remark that the case where A is of the form ∗B is identical. Then we transform the proof Π with a cut of depth d: .. .
B, Ω → Λ IA Ω, B → Λ 2L Ω, 2B → Λ
.. .
2∆ → B 2R 2∆ → 2B Γ → Θ, 2B 2B, Γ Γ, Γ → Θ, Θ Γ
→Θ
CU T
→Θ
into the proof Π∗ with a cut of depth d − 1: .. .
2∆
.. .
→B
B, Ω
→Λ Γ→Θ
→Λ
2∆, Ω
CU T
where clearly kΠ∗ k < kΠk2 . 5. Now suppose A is of the form ◦B. Then we transform the proof Π with a cut of depth d: .. .
.. .
∆→B B→Λ L◦R L◦R ◦∆ → ◦ B ◦B → ◦ Λ Γ → Θ, ◦B ◦B, Γ → Θ CU T Γ, Γ → Θ, Θ Γ
→Θ 10
into the proof Π∗ with a cut of depth d − 1: .. .
∆
.. .
→B
B→Λ CU T ∆→Λ L◦R ◦∆ → ◦ Λ Γ→Θ
where clearly kΠ∗ k < kΠk2 . 6. Finally, consider the case where A is atomic. Form β 0 from β by replacing every sequent ∆ → Λ in β with the sequent ∆− , Γ → Θ, Λ, where ∆− is ∆ minus all occurrences of direct ancestors of A. β 0 will end with the sequent Γ, Γ → Θ, Θ and will be valid as a proof, except for its initial sequents. The initial sequents B → B in β, with B not a direct ancestor of the cut formula A, become B, Γ → Θ, B in β 0 ; these are readily inferred from the initial sequent B → B with only weak inferences. On the other hand, the other initial sequents A → A in β become Γ → Θ, A which is just the endsequent of α. The desired proof Π∗ of Γ → Θ is thus formed from β 0 by adding some weak inferences and adding some copies of the subproof α to the leaves of β 0 , and by adding some exchanges and contractions to the end of β 0 . Since α and β have only cuts of degree < d (i.e., have no cuts, since d = 0), Π∗ likewise has only cuts of degree < d. Also, since the number of initial sequents in β 0 is bounded by kβk + 1, the size of Π∗ can be bounded by kΠ∗ k ≤ kβk + kαk(kβk + 1) < (kαk + 1)(kβk + 1) < kΠk2 . This completes the proof of Lemma 1. ◦∗/2
◦∗/2
Lemma 2. If Π is a DTL0 -proof with all cuts of depth at most d, there is a DTL0 same endsequent which has all cuts of depth strictly less than d and with size kΠk
kΠ∗ k < 22
-proof Π∗ with the
. ◦∗/2
We exactly follow Lemma 2.4.2.2 of [6] to prove the result for DTL0
as follows.
Proof. Lemma 2 will be proved by induction on the number of depth d cuts in Π. The base case with no kΠk depth d cuts is trivial as kΠk < 22 . For the induction, we prove the lemma in the case where Π ends with a cut inference β .. .
α .. .
Γ
→ Θ, A
A, Γ
→ Θ, Θ Γ→Θ
Γ, Γ
→Θ
CU T
where α and β are proofs of Γ → Θ, A and A, Γ → Θ, respectively, and the cut formula A is of depth d. First suppose that one of the subproofs, say β, does not have any strong inferences; i.e., kβk = 0. Therefore, β must contain either the axiom A → A, or must have direct ancestors of the cut formula A introduced only by weakenings. In the former case, A must appear in Θ, and the desired proof Π∗ can be obtained from α by adding some exchanges and a contraction to the end of α. In the second case, Π∗ can be obtained from β by removing all the WA inferences that introduce direct ancestors of the cut formula A (and possibly removing some exchanges and contractions involving these A’s). A similar argument works for the case kαk = 0. In kΠk both cases, kΠ∗ k < kΠk < 22 . Second, suppose that kαk and kβk are both nonzero. By the induction hypothesis, there are proofs α∗ and β ∗ of the same endsequents with all cuts of depth less than d, and with kαk kβk kα∗ k < 22 and kβ ∗ k < 22 . Applying Lemma 1 to the proof 11
β∗ α∗ .. .. . . Γ → Θ, A A, Γ → Θ CU T Γ, Γ → Θ, Θ Γ→Θ gives a proof Π∗ of Γ → Θ with all cuts of depth < d, so that kαk 2 kβk kαk+kβk+1 kΠk kΠ∗ k < (kα∗ k + kβ ∗ k + 1)2 ≤ 22 + 2 2 − 1 < 22 = 22 . The final inequality holds since kαk, kβk ≥ 1. This completes the proof of Lemma 2 and the Cut-Elimination ◦∗/2 Theorem for DTL0 . As a corollary, we cite a depth independent bound on kΠ∗ k from page 42 of [6]. ◦∗/2
Corollary 1. Suppose Π is a DTL0 the same sequent with size
-proof of the sequent Γ → Θ. Then there is a cut-free proof Π∗ of kΠk
kΠ∗ k < 22kΠk . ◦∗/2
◦∗/2
In conclusion, every theorem provable from the axiomatization of DTL0 in [3] has a DTL0 sequent calculus derivation, from which completeness follows. This holds due to cut elimination as the four rules of →A →A ◦∗/2 inference are admissible in the DTL0 -sequent calculus; namely the first three: → 2A , → ◦ A , →A ◦∗/2 → ∗ A , are easily proved to be admissible and the cut elimination theorem for DTL0 shows that
→ (A ⊃ B) →A →B ◦∗/2
is admissible in the sequent calculus for DTL0
.
References [1] M. Aiello, I. Pratt-Hartmann, J. van Benthem, Handbook of Spatial Logics. 2007, Springer Netherlands. [2] S. Artemov, J. Davoren, A. Nerode, Modal Logics and Topological Semantics for Hybrid Systems. 1997, Cornell University, Technical Report 97-05. [3] P. Kremer, G. Mints, Dynamic topological logic. 2005, Annals of Pure and Applied Logic, vol. 131, pg. 133 - 158. [4] P. Kremer, Dynamic topological S5. 2009, Annals of Pure and Applied Logic, vol. 160, pg. 96 - 116. [5] J. Davoren, Modal Logics for Continuous Dynamics. 1997, Cornell University, Technical Report 97-09. [6] S. Buss, Handbook of Proof Theory. 1998, Elsevier, vol. 137, pg. 1 - 78.
12
Abstract ◦∗/2
We introduce a sequent calculus for the temporal-over-topological fragment DTL0 of dynamic topological logic DTL, prove soundness semantically, and prove completeness syntactically using the ◦∗/2 ◦∗/2 axiomatization of DTL0 given in [3]. A cut-free sequent calculus for DTL0 is obtained as the union of the propositional fragment of Gentzen’s classical sequent calculus, two 2 structural rules for the modal extension, and nine ◦ (next) and ∗ (henceforth) structural rules for the temporal extension. Future research will focus on the construction of a hypersequent calculus for dynamic topological S5 logic in order to prove Kremer’s Next Removal Conjecture for the logic of homeomorphisms on almost discrete spaces S5H.
1
Introduction
Within recent years there has been an outburst of research activity in spatial-temporal reasoning, leading to important advances in computer science and logic. The Handbook of Spatial Logics [1], and references therein, summarize the main recent achievements such as mereotopology, spatial constraint calculi, modal logics of space, connections between topology and epistemic logic, logics for fragments of elementary geometry, mathematical morphology, logics of space-time and relativity theory, and dynamic topological logic. The present paper focuses on extending the research area of dynamic topological logic by means of a Gentzen-style ◦∗/2 proof calculus for the temporal-over-topological fragment of dynamic topological logic DTL0 . We achieve this by extending the sequent calculi developed for the fragments of dynamic topological logic defined in [2] as the logic of control action S4F and the logic of continuous control action S4C. The main motivation for this work is to combine structural proof theory and dynamic topological logic. Thus laying the groundwork for defining hypersequent calculi for dynamic topological S5 logic and the conservative axiomatizable extensions S5C, S5H, S5Ct, and S5Ht known as the logics of continuous functions on almost discrete spaces, homeomorphisms on almost discrete spaces, functions on trivial spaces, and homeomorphisms on trivial spaces, respectively [4]. These logics have been defined quite recently and the development of a sequent calculus for them would lead to applications such as proof search and automated theorem proving in dynamic topological logic. We identify a main conjecture of this recent research area and propose that future research regarding hypersequent calculi for dynamic topological S5 will provide a positive solution to this conjecture. Conjecture 1 (Kremer’s Next Removal Conjecture). The logic of homeomorphisms on almost discrete spaces can be axiomatized without next removal by S5H = S5 + LTL + (◦2A ⇔ 2 ◦ A).
Dynamic topological logic provides the foundation for breakthroughs in topics ranging from control theory and robot-motion planning to dynamical systems and eschatological cosmology, as statements regarding the possibility and necessity of spatio-temporal properties can be understood with systematic logical precision. ∗ University
of Calgary, Department of Mathematics and Statistics, Calgary, AB, Canada. e − mail : [email protected].
1
2
Dynamic Topological Logic
We now present an introduction to dynamic topological logic, freely citing from the seminal papers on dynamic topological logic [2], [3], [4], [5]. The main idea of dynamic topological logic is to combine of topological semantics in logic, temporal logic, and topological dynamics (asymptotic properties of continuous maps on topological spaces). Interestingly predating the well-known Kripke semantics for modal logic, the McKinsey-Tarki topological semantics interprets the purely temporal modal propositional language L2 = | ∧ | ¬ | 2 | p1 | · · · | pn | · ··
in terms of topological spaces with the interptation of 2 given by topological interior. Then the propositional letters denote subsets of X; ∨, ∧, and ¬ express union, intersection, and complement, respectively, with 3 := ¬2¬ interpreted as closure. We then have n ^
pi →
i=1
k _
qj ⇔
j=1
n \
pi ⊆
i=1
k [
qj
j=1
thus providing us with a semantic interpretation for sequents and language to prove the soundess of structural rules. A topological model is an ordered pair M = hX, V i, where X is a topological space and V : Var → P(X). The function V is extended to formulas of L2 by V (¬A) = X − V (A) V (A ∧ B) = V (A) ∩ V (B) V (2A) = int(V (A)) with four validity relations, where T is a class of topological spaces. M A iff V (A) = X X A iff hX, V i A, ∀V : Var → X T A iff X A, ∀X ∈ T A iff X A, ∀X A dynamic topological system is an ordered pair hX, f i, where X is a topological space and f is a continuous function on X. We interpret the temporal connectives of the modal-temporal language L by means of the function f : • ◦A is true at x iff A is true at f x. V∞ • ∗A is true at x iff A is true at i=1 f i x, ∀i ∈ N. A dynamic topological model is an ordered triple M = hX, f, V i where hX, f i is a dynamic topological system and V : Var → P(X) is extended to all formulas of L by V (◦A) = f −1 (V (A)) ∞ \ V (∗A) = f −i (V (A)) i=1
with five validity relations, where FX is a class of continuous functions associated with the topological space
2
X and F is an arbitrary class of continuous functions. M A iff V (A) = X hX, f i A iff hX, f, V i A, ∀V : Var → P(X) X A iff hX, f i A, ∀f ∈ FX T A iff X A, ∀X ∈ T T , F A iff hX, f i A, ∀X ∈ T , ∀f ∈ F A iff X A, ∀X From this we define DTL0 = {A : A} to be the logic of all dynamic topological systems. We now provide the axiomatization of linear time logic (LTL). S4 axioms for *: ∗ (A ⊃ B) ⊃ (∗A ⊃ ∗B) ∗A⊃A ∗A⊃∗∗A ◦ commutes with ¬, ∨, ∗ : ◦ ¬A ⇔ ¬ ◦ A ◦ (A ∨ B) ⇔ ◦A ∨ ◦B ◦ ∗A ⇔ ∗ ◦ A henceforth implies next: ∗ A ⊃ ◦A induction axiom: A ∧ ∗(A ⊃ ◦A) ⊃ ∗A We can then provide the axiomatization given in [3] for the temporal-over-topological fragment of dynamic topological logic as follows: ◦∗/2 DTL0 = S4 L2 + LTL L◦∗/2 where S4 L2 is the S4 axioms for 2 with A, B ∈ L2 and LTL L◦∗/2 is LTL where the scope of 2 in any ◦∗/2
subformula is ◦-free and ∗-free. This defines DTL0 with the temporal-over-topological language L◦∗/2 . ◦∗/2 We now define the sequent calculus for DTL0 by means of six weak structural rules: weakening in the antecedent (WA), weakening in the succedent (WS), contraction in the antecedent (CA), contraction in the succedent (CS), interchange in the antecedent (IA), and interchange in the succedent (IS); eight logical rules: ∧R, ∧L, ∨R, ∨L, ⊃ R, ⊃ L, ¬R, ¬L; the strong structural rule of cut; two modal rules for the 2-modality: 2L, 2R; and nine temporal rules: ∗L, ∗R, ◦ ∗ R, ◦ ∗ L, ◦¬R, ◦¬L, ◦ ∗ CA, IN D, L ◦ R. The 22 unary inference rules and 4 binary inference rules are as follows: A, A, Γ → Θ Γ → Θ, A, A ∆, A, B, Γ → Θ Γ→Θ Γ→Θ WA WS CA CS IA A, Γ → Θ Γ → Θ, A A, Γ → Θ Γ → Θ, A ∆, B, A, Γ → Θ Γ → Θ, A, B, Λ Γ → Θ, A Γ → Θ, B A/B, Γ → Θ Γ → Θ, A/B IS ∧R ∧L ∨R Γ → Θ, B, A, Λ Γ → Θ, A ∧ B A ∧ B, Γ → Θ Γ → Θ, A ∨ B A, Γ → Θ B, Γ → Θ A, Γ → Θ, B Γ → Θ, A B, ∆ → Λ ⊃R ⊃L ∨L A ∨ B, Γ → Θ Γ → Θ, A ⊃ B A ⊃ B, Γ, ∆ → Θ, Λ Γ, A → Θ Γ → A, Θ Γ → Θ, A A, ∆ → Λ ¬R ¬L CU T Γ → ¬A, Θ Γ, ¬A → Θ Γ, ∆ → Θ, Λ Γ, A → Θ Γ, A → Θ 2Γ → A ∗Γ → A 2R ∗R 2L ∗L ∗Γ → ∗A 2Γ → 2A Γ, 2A → Θ Γ, ∗A → Θ Γ → ∗ ◦ A, Θ Γ, ∗ ◦ A → Θ Γ → ¬ ◦ A, Θ Γ, ¬ ◦ A → Θ ◦∗R ◦∗L ◦¬R ◦¬L Γ → ◦ ∗ A, Θ Γ, ◦ ∗ A → Θ Γ → ◦¬A, Θ Γ, ◦¬A → Θ ∗A, ∗(A ⊃ ◦A) → ◦ ∗ A A, ◦ ∗ A, Γ → Θ Γ→Θ L◦R ◦ ∗ CA IN D ◦Γ → ◦Θ ∗A, Γ → Θ A, ∗(A ⊃ ◦A) → ∗A
3
3
◦∗/2
Soundness of DTL0
We show soundness for the temporal-over-topological fragment of dynamic topological logic by giving semantic proofs of the soundness of the rules of the temporal-over-topological sequent calculus. Proposition 1. The 2L rule is sound.
Γ, A → Θ 2L Γ, 2A → Θ
Proof. Assume Γ ∩ A ⊆ Θ. Then Γ ∩ int(A) ⊆ Θ. Proposition 2. The 2R rule is sound.
2Γ → A 2R 2Γ → 2A
Proof. Assume int(Γ) ⊆ A. Then int(Γ) ⊆ int(A). Proposition 3. The ∗L rule is sound.
Γ, A → Θ ∗L Γ, ∗A → Θ Proof. Assume Γ ∩ A ⊆ Θ. Then Γ∩
∞ \
f −i (A) ⊆ Θ.
i=1
Proposition 4. The ∗R rule is sound. ∗Γ → A ∗R ∗Γ → ∗A Proof. Assume
∞ \
f −i (Γ) ⊆ A.
i=1
Then
∞ \
f −i (Γ) ⊆
i=1
∞ \
f −i (A).
i=1
Proposition 5. The ◦ ∗ R rule is sound. Γ → ∗ ◦ A, Θ ◦∗R Γ → ◦ ∗ A, Θ Proof. Assume Γ⊆
∞ \
f −i (f −1 (A)) ∪ Θ.
i=1
Then Γ ⊆ f −1
∞ \
! f −i (A)
i=1
4
∪ Θ.
Proposition 6. The ◦ ∗ L rule is sound. Γ, ∗ ◦ A → Θ ◦∗L Γ, ◦ ∗ A → Θ Proof. Assume Γ∩
∞ \
f −i (f −1 (A)) ⊆ Θ.
i=1
Then Γ ∩ f −1
∞ \
! f −i (A))
⊆ Θ.
i=1
Proposition 7. The ◦¬R rule is sound. Γ → ¬ ◦ A, Θ ◦¬R Γ → ◦¬A, Θ Proof. Assume Γ ⊆ (X − f −1 (A)) ∪ Θ. Then Γ ⊆ f −1 (X − A) ∪ Θ. Proposition 8. The ◦¬L rule is sound. Γ, ¬ ◦ A → Θ ◦¬L Γ, ◦¬A → Θ Proof. Assume Γ ∩ (X − f −1 (A)) ⊆ Θ. Then Γ ∩ f −1 (X − A) ⊆ Θ. Proposition 9. The ◦ ∗ CA rule is sound. A, ◦ ∗ A, Γ → Θ ◦ ∗ CA ∗A, Γ → Θ Proof. Assume A∩f
−1
∞ \
! f
−i
(A))
∩ Γ ⊆ Θ.
i=1
Then
∞ \
f −i (A)) ∩ Γ ⊆ Θ.
i=1
Proposition 10. The IN D rule is sound. ∗A, ∗(A ⊃ ◦A) → ◦ ∗ A IN D A, ∗(A ⊃ ◦A) → ∗A Proof. Assume ∞ \
f −i (A) ∩
i=1
∞ \
f −i ((X − A) ∪ f −1 (A)) ⊆ f −1
i=1
Then A∩
∞ \
∞ \
! f −i (A) .
i=1
f
−i
((X − A) ∪ f
i=1
−1
(A)) ⊆
∞ \ i=1
5
f −i (A).
Proposition 11. The L ◦ R rule is sound. Γ→Θ L◦R ◦Γ → ◦Θ Proof. Assume Γ ⊆ Θ. Then f −1 (Γ) ⊆ f −1 (Θ).
4
◦∗/2
Completeness for DTL0
We show completeness for the temporal-over-topological fragment of dynamic topological logic by giving ◦∗/2 sequent calculus derivations of the axioms of DTL0 using the soundness of the sequent calculus rules. ◦∗/2
Proposition 12. The sequent calculus for DTL0 Proof.
A 2A
→ A 2L →A → 2A ⊃ A
◦∗/2
Proposition 13. The sequent calculus for DTL0 Proof.
2A 2A
◦∗/2
Proof.
→A
A ⊃ B, A A ⊃ B, 2A 2A, A ⊃ B 2A, 2(A ⊃ B) 2A, 2(A ⊃ B) 2(A ⊃ B)
◦∗/2
A ∗A
⊃R
proves 2(A ⊃ B) ⊃ (2A ⊃ 2B).
◦∗/2
∗A ∗A
⊃R
proves ∗A ⊃ ∗ ∗ A.
→ ∗A → ∗ ∗ A ∗R → ∗A ⊃ ∗ ∗ A ◦∗/2
Proposition 17. The sequent calculus for DTL0
⊃R
proves ∗A ⊃ A.
→ A ∗L →A → ∗A ⊃ A
Proposition 16. The sequent calculus for DTL0 Proof.
proves 2A ⊃ 22A.
B→B →B ⊃L → B 2L → B IA → B 2L → 2B 2R → 2A ⊃ 2B ⊃ R → 2(A ⊃ B) ⊃ (2A ⊃ 2B)
Proposition 15. The sequent calculus for DTL0 Proof.
⊃R
→ 2A 2R → 22A → 2A ⊃ 22A
Proposition 14. The sequent calculus for DTL0 A
proves 2A ⊃ A.
⊃R
proves ∗(A ⊃ B) ⊃ (∗A ⊃ ∗B).
6
A
→A
A ⊃ B, A A ⊃ B, ∗A ∗A, A ⊃ B ∗A, ∗(A ⊃ B) ∗A, ∗(A ⊃ B) ∗(A ⊃ B)
Proof.
B→B →B ⊃L → B ∗L → B IA → B ∗L → ∗B ∗R → ∗A ⊃ ∗B ⊃ R → ∗(A ⊃ B) ⊃ (∗A ⊃ ∗B) ◦∗/2
Proposition 18. The sequent calculus for DTL0
Proof.
A ◦A A, ◦A A, ∗ ◦ A ∗A
◦∗/2
Proof.
◦∗/2
¬ ◦ ¬A ◦¬A
◦∗/2
→A → A, B
WS
A∨B ◦(A ∨ B) ◦(A ∨ B) ◦(A ∨ B) ◦(A ∨ B) ◦(A ∨ B)
→ → → → → → →
⊃R ∧R
proves ◦¬A ⇔ ¬ ◦ A.
→ ¬ ◦ A ◦¬L ¬◦A → ¬◦A ◦¬R → ¬◦A ¬ ◦ A → ◦¬A ⊃R → ◦¬A ⊃ ¬ ◦ A → ¬ ◦ A ⊃ ◦¬A → (◦¬A ⊃ ¬ ◦ A) ∧ (¬ ◦ A ⊃ ◦¬A)
Proposition 21. The sequent calculus for DTL0
A A
proves ◦ ∗ A ⇔ ∗ ◦ A.
→ ∗◦A ◦∗L ∗◦A → ∗◦A ◦∗R → ∗◦A ∗◦A → ◦∗A ⊃R → ◦∗A⊃∗◦A → ∗◦A⊃◦∗A → (◦ ∗ A ⊃ ∗ ◦ A) ∧ (∗ ◦ A ⊃ ◦ ∗ A)
Proposition 20. The sequent calculus for DTL0
Proof.
proves ∗A ⊃ ◦A.
→ A ◦R → ◦A LWA → ◦A → ◦A ∗L → ◦A ◦ ∗ CA⊃ R → ∗A ⊃ ◦A
Proposition 19. The sequent calculus for DTL0 ∗◦A ◦∗A
⊃R
⊃R ∧R
proves ◦(A ∨ B) ⇔ (◦A ∨ ◦B).
Proof. B→B WS B → B, A IS B → A, B ∨L A, B L◦R ◦A, ◦B ∨R ◦A, ◦A ∨ ◦B A→A B→B IS ∨R ∨R ◦A ∨ ◦B, ◦A A → A∨B B → A∨B ∨R L◦R L◦R ◦A ∨ ◦B, ◦A ∨ ◦B ◦A → ◦(A ∨ B) ◦B → ◦(A ∨ B) CS ∨L ◦A ∨ ◦B ◦A ∨ ◦B → ◦(A ∨ B) ⊃R ◦(A ∨ B) ⊃ (◦A ∨ ◦B) → (◦A ∨ ◦B) ⊃ ◦(A ∨ B) ⊃ R ∧R → (◦(A ∨ B) ⊃ (◦A ∨ ◦B)) ∧ ((◦A ∨ ◦B) ⊃ ◦(A ∨ B))
7
◦∗/2
Proposition 22. The sequent calculus for DTL0
A→A L◦R ◦A→ ◦A ⊃L A ⊃ ◦A, A → ◦A ∗L (A ⊃ ◦A), ∗A → ◦A IA ∗A, (A ⊃ ◦A) → ◦A ∗L ∗A, ∗(A ⊃ ◦A) → ◦A ∗R ∗A, ∗(A ⊃ ◦A) → ∗ ◦ A ◦∗R ∗A, ∗(A ⊃ ◦A) → ◦ ∗ A IND A, ∗(A ⊃ ◦A) → ∗A ∧L A ∧ ∗(A ⊃ ◦A), ∗(A ⊃ ◦A) → ∗A IA ∗(A ⊃ ◦A), A ∧ ∗(A ⊃ ◦A) → ∗A ∧L A ∧ ∗(A ⊃ ◦A), A ∧ ∗(A ⊃ ◦A) → ∗A CA A ∧ ∗(A ⊃ ◦A) → ∗A → (A ∧ ∗(A ⊃ ◦A)) ⊃ ∗A ⊃ R A
Proof.
proves (A ∧ ∗(A ⊃ ◦A)) ⊃ ∗A.
→A
◦∗/2
Lastly, we remark that all derivations in the sequent calculus for DTL0
5
◦∗/2
Admissibility of DTL0
are cut-free and harmonious.
Rules of Inference
We follow the strategy for cut-elimination and recall the definitions of descendents, ancestors, and depth of ◦∗/2 a formula from [6]. All of the inferences of DTL0 (with the exception of the cut rule) have a principal formula which is, by definition, the formula occurring in the lower sequent of the inference which is not in the cedents Γ or ∆ (or Θ or Λ). The exchange inferences have two principle formulas, as do ◦ ∗ CA, IN D, and L ◦ R. Every inference, except weakenings, has one or more auxiliary formulas which are the formulas A and B, occurring in the upper sequent(s) of the inference. The formulas which occur in the cedents Γ, ∆, Θ, Λ are called side formulas of the inference. The two auxiliary formulas of a cut inference are called the cut formulas. If C is a side formula in an upper sequent of an inference then the immediate descendent of C is the corresponding occurrence of the same formula in the same position in the same cedent in the lower sequent of the inference. If C is an auxiliary formula of any inference except an exchange or cut inference, then the principal formula of the inference is the immediate descendent of C. For an exchange inference, the immediate descendent of the A or B in the upper sequent is the A or B, respectively, in the lower sequent. The cut formulas of a cut inference do not have immediate descendents. We say that C is an immediate ancestor of D if and only if D is an immediate descendent of C. The ancestor relation is defined to be the reflexive, transitive closure of the immediate ancestor relation; thus, C is an ancestor of D if and only if there is a chain of zero or more immediate ancestors from D to C. A direct ancestor of D is an ancestor of C of D such that C is the same formula as D; descendent and direct descendent are defined conversely. The depth of a formula A is the height of the tree representation of the formula, with the depth of a cut inference the depth of its cut formula: • depth(A) = 0 for A atomic. • depth(A ∧ B) = depth(A ∨ B) = depth(A ⊃ B) = 1 + max{depth(A), depth(B)}. • depth(¬A) = depth(2A) = depth(◦A) = depth(∗A) = 1 + depth(A). x
The superexponentiation function 2xi , for i, x ≥ 0, is defined inductively by 2x0 = x and 2xi+1 = 22i .
8
◦∗/2
Theorem 1. Let Π be a DTL0 -proof and suppose every cut formula in Π has depth less than or equal ◦∗/2 to d. Then there is a cut-free DTL0 -proof Π∗ with the same endsequent as Π, with size kΠk
kΠ∗ k < 22d+2 . Proof. Lemma 1 shows how to replace a single cut by lower depth cut inference. Iteration of this construction removes all cuts of maximum depth d in a proof, which is stated as Lemma 2, from which the theorem is a consequence. ◦∗/2
Lemma 1. Let Π be a DTL0 -proof with final inference a cut of depth d such that every other cut in Π ◦∗/2 has depth strictly less than d. Then there is a DTL0 -proof Π∗ with the same endsequent as Π with all cuts in Π∗ of depth less than d with kΠ∗ k < kΠk2 . We closely follow the proof of Lemma 2.4.2.1 in [6] with case (e) and (f) omitted, and consider the additional cases of the cut formula as 2B, ◦B, and ∗B. Proof. The proof Π ends with a cut inference
β .. .
α .. .
Γ
→ Θ, A
A, Γ Θ, Θ
→ Γ→Θ
Γ, Γ
→Θ
CU T
where the depth of the cut formula A equals d and where all cuts in the subproofs α and β have depth strictly less than d. The lemma is proved by cases, based on the outermost logical connective of the cut formula A. We can assume without loss of generality that both α and β contain at least one strong inference; since otherwise, we must have A in Γ or Θ, or have a formula which occurs in both Γ and Θ, and in the former case, the sequent Γ → Θ is obtainable by weak inferences from one of the upper sequents and the cut can therefore be eliminated. In the latter case, Γ → Θ can be inferred with no cut inference at all. 1. Suppose A is a formula of the form ¬B. We shall form new proofs α∗ and β ∗ of the sequents B, Γ → Θ and Γ → Θ, B, which can then be combined with a cut inference of depth d − 1 to give the proof Π∗ of Γ → Θ. To form α∗ , first form α0 by replacing every sequent ∆ → Λ in α with the sequent ∆, B → Λ− , where Λ− is obtained from Λ by removing all direct ancestors of the cut formula A. Of B, ∆ → Λ course, α0 is not a valid proof; e.g., a ¬R inference in α of the form could become in ∆ → Λ, ¬B − B, ∆, B → Λ . α0 , This is not a valid inference, but can be modified to become a valid inference ∆, B → Λ− by inserting some exchanges and a contraction. In this manner, α0 can be modified so that it becomes a valid proof α∗ by removing some ¬L inferences and inserting some weak inferences. The proof β ∗ of Γ → Θ, B is formed in a similar manner from β. No new cuts are introduced by this process and, since we do not count weak inferences, kα∗ k ≤ kαk and kβ ∗ k ≤ kβk; thus Π∗ has only cuts of depth < d and has kΠ∗ k ≤ kΠk. 2. Now suppose the cut formula A is of the form B ∨ C. We defined α0 as a tree of sequents, with root labelled Γ → Θ, B, C, by replacing every sequent ∆ → Λ in α with the sequent ∆ → Λ− , B, C, where Λ− is Λ minus all occurrences of direct ancestors of the cut formula. By removing some formerly ∨R inferences from α0 and by adding some weak inferences, α0 can be transformed into a valid proof α∗ . Now construct βB from β by replacing every occurrence in β of B ∨ C as a direct ancestor of the cut formula with just the formula B. One way that βB can fail to be valid is that an ∨L inference B, ∆ → Λ C, ∆ → Λ B, ∆ → Λ C, ∆ → Λ in βB . This is no longer ∨L may become just B ∨ C, ∆ → Λ B, ∆ → Λ 9
a valid inference, but it can be fixed up by discarding the inference and its upper right hypothesis, including discarding the entire subproof of the upper right hypothesis. The only other changes needed to make βB valid are the addition of weak inferences, and in this way, a valid proof βB of B, Γ → Θ is formed. A similar process forms a valid proof βC of C, Γ → Θ. The proof Π∗ can now be defined to be βC α∗ .. .. . . Γ → Θ, B, C C, Γ → Θ CU T Γ, Γ → Θ, Θ, B Γ
→ Θ, B
βB .. . B, Γ → Θ
→ Θ, Θ Γ→Θ
Γ, Γ
CU T
The process of forming α∗ , βB , and βC did not introduce any new cuts or any new strong inferences. Thus, we clearly have that every cut in Π∗ has depth < d, and that kΠ∗ k ≤ kαk + 2kβk + 2. Since kΠk = kαk + kβk + 1 and kαk, kβk ≥ 1, this suffices to prove the lemma for this case. 3. The cases where A has outermost connective ∧ or ⊃ are very similar to the previous case and are omitted. 4. Now suppose A is of the form 2B; we remark that the case where A is of the form ∗B is identical. Then we transform the proof Π with a cut of depth d: .. .
B, Ω → Λ IA Ω, B → Λ 2L Ω, 2B → Λ
.. .
2∆ → B 2R 2∆ → 2B Γ → Θ, 2B 2B, Γ Γ, Γ → Θ, Θ Γ
→Θ
CU T
→Θ
into the proof Π∗ with a cut of depth d − 1: .. .
2∆
.. .
→B
B, Ω
→Λ Γ→Θ
→Λ
2∆, Ω
CU T
where clearly kΠ∗ k < kΠk2 . 5. Now suppose A is of the form ◦B. Then we transform the proof Π with a cut of depth d: .. .
.. .
∆→B B→Λ L◦R L◦R ◦∆ → ◦ B ◦B → ◦ Λ Γ → Θ, ◦B ◦B, Γ → Θ CU T Γ, Γ → Θ, Θ Γ
→Θ 10
into the proof Π∗ with a cut of depth d − 1: .. .
∆
.. .
→B
B→Λ CU T ∆→Λ L◦R ◦∆ → ◦ Λ Γ→Θ
where clearly kΠ∗ k < kΠk2 . 6. Finally, consider the case where A is atomic. Form β 0 from β by replacing every sequent ∆ → Λ in β with the sequent ∆− , Γ → Θ, Λ, where ∆− is ∆ minus all occurrences of direct ancestors of A. β 0 will end with the sequent Γ, Γ → Θ, Θ and will be valid as a proof, except for its initial sequents. The initial sequents B → B in β, with B not a direct ancestor of the cut formula A, become B, Γ → Θ, B in β 0 ; these are readily inferred from the initial sequent B → B with only weak inferences. On the other hand, the other initial sequents A → A in β become Γ → Θ, A which is just the endsequent of α. The desired proof Π∗ of Γ → Θ is thus formed from β 0 by adding some weak inferences and adding some copies of the subproof α to the leaves of β 0 , and by adding some exchanges and contractions to the end of β 0 . Since α and β have only cuts of degree < d (i.e., have no cuts, since d = 0), Π∗ likewise has only cuts of degree < d. Also, since the number of initial sequents in β 0 is bounded by kβk + 1, the size of Π∗ can be bounded by kΠ∗ k ≤ kβk + kαk(kβk + 1) < (kαk + 1)(kβk + 1) < kΠk2 . This completes the proof of Lemma 1. ◦∗/2
◦∗/2
Lemma 2. If Π is a DTL0 -proof with all cuts of depth at most d, there is a DTL0 same endsequent which has all cuts of depth strictly less than d and with size kΠk
kΠ∗ k < 22
-proof Π∗ with the
. ◦∗/2
We exactly follow Lemma 2.4.2.2 of [6] to prove the result for DTL0
as follows.
Proof. Lemma 2 will be proved by induction on the number of depth d cuts in Π. The base case with no kΠk depth d cuts is trivial as kΠk < 22 . For the induction, we prove the lemma in the case where Π ends with a cut inference β .. .
α .. .
Γ
→ Θ, A
A, Γ
→ Θ, Θ Γ→Θ
Γ, Γ
→Θ
CU T
where α and β are proofs of Γ → Θ, A and A, Γ → Θ, respectively, and the cut formula A is of depth d. First suppose that one of the subproofs, say β, does not have any strong inferences; i.e., kβk = 0. Therefore, β must contain either the axiom A → A, or must have direct ancestors of the cut formula A introduced only by weakenings. In the former case, A must appear in Θ, and the desired proof Π∗ can be obtained from α by adding some exchanges and a contraction to the end of α. In the second case, Π∗ can be obtained from β by removing all the WA inferences that introduce direct ancestors of the cut formula A (and possibly removing some exchanges and contractions involving these A’s). A similar argument works for the case kαk = 0. In kΠk both cases, kΠ∗ k < kΠk < 22 . Second, suppose that kαk and kβk are both nonzero. By the induction hypothesis, there are proofs α∗ and β ∗ of the same endsequents with all cuts of depth less than d, and with kαk kβk kα∗ k < 22 and kβ ∗ k < 22 . Applying Lemma 1 to the proof 11
β∗ α∗ .. .. . . Γ → Θ, A A, Γ → Θ CU T Γ, Γ → Θ, Θ Γ→Θ gives a proof Π∗ of Γ → Θ with all cuts of depth < d, so that kαk 2 kβk kαk+kβk+1 kΠk kΠ∗ k < (kα∗ k + kβ ∗ k + 1)2 ≤ 22 + 2 2 − 1 < 22 = 22 . The final inequality holds since kαk, kβk ≥ 1. This completes the proof of Lemma 2 and the Cut-Elimination ◦∗/2 Theorem for DTL0 . As a corollary, we cite a depth independent bound on kΠ∗ k from page 42 of [6]. ◦∗/2
Corollary 1. Suppose Π is a DTL0 the same sequent with size
-proof of the sequent Γ → Θ. Then there is a cut-free proof Π∗ of kΠk
kΠ∗ k < 22kΠk . ◦∗/2
◦∗/2
In conclusion, every theorem provable from the axiomatization of DTL0 in [3] has a DTL0 sequent calculus derivation, from which completeness follows. This holds due to cut elimination as the four rules of →A →A ◦∗/2 inference are admissible in the DTL0 -sequent calculus; namely the first three: → 2A , → ◦ A , →A ◦∗/2 → ∗ A , are easily proved to be admissible and the cut elimination theorem for DTL0 shows that
→ (A ⊃ B) →A →B ◦∗/2
is admissible in the sequent calculus for DTL0
.
References [1] M. Aiello, I. Pratt-Hartmann, J. van Benthem, Handbook of Spatial Logics. 2007, Springer Netherlands. [2] S. Artemov, J. Davoren, A. Nerode, Modal Logics and Topological Semantics for Hybrid Systems. 1997, Cornell University, Technical Report 97-05. [3] P. Kremer, G. Mints, Dynamic topological logic. 2005, Annals of Pure and Applied Logic, vol. 131, pg. 133 - 158. [4] P. Kremer, Dynamic topological S5. 2009, Annals of Pure and Applied Logic, vol. 160, pg. 96 - 116. [5] J. Davoren, Modal Logics for Continuous Dynamics. 1997, Cornell University, Technical Report 97-09. [6] S. Buss, Handbook of Proof Theory. 1998, Elsevier, vol. 137, pg. 1 - 78.
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