Tableau Calculi for the Logics of Finite k-Ary Trees
read more
References
Hypertableau and Path-Hypertableau Calculi for Some Families of Intermediate Logics
Space‐efficient Decision Procedures for Three Interpolable Propositional Intermediate Logics
Related Papers (5)
Frequently Asked Questions (12)
Q2. What is the main property of the logics Dk?
An intermediate propositional logic (see, e.g., [5]) is any set L of formulas satisfying the following conditions: (i) L is consistent; (ii) Int ⊆ L; (iii) L is closed under modus ponens; (iv) L is closed under propositional substitution (where a propositional substitution is any function mapping every propositional variable to a formula).
Q3. What is the definition of the structure KD(S)?
Then ∆ is a member of P and ∆ is an immediate successor of Γ in KD(S); 3. ≤ is the transitive and reflexive closure of the immediate successor relation; 4. For every atom p and for every Γ ∈ P , Γ p iff Tp ∈ Γ .
Q4. How many hypersets are there in the consequence of the rule?
Dk is intrinsically inefficient, indeed the number of hypersets in the consequence of the rule is exponential in the number of F-signed implicative formulas occurring in the premise.
Q5. What is the main set of swff’s of the rule?
The rule properly characterizing the calculus TDk is Dk that applies to the premiseS,F(A1 → B1), . . . ,F(An → Bn),T((C1 → D1) → E1), . . . ,T((Cm → Dm) → Em)Let U = {F(A1 → B1), . . . ,F(An → Bn)} V = {T((C1 → D1) → E1), . . . ,T((Cm → Dm) → Em)}and let U ∪ V be the main set of swff’s of the rule.
Q6. What is the force relation for every atomic symbol?
A (propositional) Kripke model (see, e.g., [5]) is a structure K = 〈P,≤, 〉, where 〈P,≤〉 is a poset (partially ordered set), and (the forcing relation) is a binary relation between elements of P and atomic symbols such that, for every atomic symbol p, α p implies β p for every β ∈
Q7. What is the meaning of the signs T and F?
The meaning of the signs T and F is as follows: given a Kripke model K = 〈P,≤, 〉 and a swff H, α ∈ P realizes H (in symbols α✄H) if H ≡ TX and α X, or H ≡ FX and α X. α H means that α✄H does not hold.
Q8. What is the definition of a constructive intermediate logic?
c© Springer-Verlag Berlin Heidelberg 2002we call constructive any intermediate logic L satisfying the disjunction property : A ∨ B ∈ L implies A ∈ L or B ∈ L.
Q9. What is the axiom schema of the Kripke models?
let Tk be the the family of all the Kripke models K = 〈P,≤, 〉 where: – 〈P,≤〉 is a finite tree; – Given α ∈ P , α has at most k immediate successors in 〈P,≤〉.
Q10. What is the construction of the structure KD(S)?
Given a finite and TDk-consistent set S of swff’s, the authors use the construction above to define the structure KD(S) = 〈P,≤, 〉 as follows: 1. S ∈ P , where S is a node set of S; 2.
Q11. What is the decidability of a tableau calculus?
In the following sections the authors introduce a tableau calculus for every TDk and then the authors use the properties of such a calculus to deduce the decidability and the disjunction property for Dk.3 The Sequence of Tableau Calculi TDk (k ≥ 2) A signed formula (swff for short) is an expression of the form TX or FX where X is any formula.
Q12. what is the swff of the type F(AB)?
(S2) If S does not contain swff’s of the kind F(A∨B) and contains at least one swff of the kind F(A→ B), then let:U = {F(A → B) | F(A → B) ∈ S} V = {T((C → D) → E) | T((C → D) → E) ∈