Modal Reasoning through Resolution Cláudia Nalon http://www.cic.unb.br/∼nalon nalon@{cic.unb.br, unb.br} Universidade de Brasília Instituto de Ciências Exatas Departamento de Ciência da Computação TRS, Natal, 2015
C. Nalon
TRS, Natal, 2015 – 1 / 43
⊲ Introduction Motivation Syntax Semantics Another example Syntax Semantics Usual Modal Logics Resolution Destructive Non-Clausal Resolution
Introduction
Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 2 / 43
Motivation ⊲ Introduction ⊲ Motivation Syntax Semantics Another example Syntax Semantics Usual Modal Logics Resolution
Destructive Non-Clausal Resolution
Modal logics have been used in Computers Science to represent properties of complex systems: temporal, epistemic, obligations, and so on. Given a representation of a system in a logical language, we also want to reason about the systems and their properties. There are different proof methods we could use in order to reason about those properties. –
Clausal Resolution for Modal Logics
–
C. Nalon
Some modal languages can be translated into first-order and we could then use readily available automated reasoners. Provide a proof method within the language of a particular modal logic.
There are different methods for modal logics. We will restrict to resolution-based methods.
TRS, Natal, 2015 – 3 / 43
Syntax ⊲ Introduction Motivation ⊲ Syntax
Semantics Another example Syntax Semantics Usual Modal Logics
a ’ Modal logics are extensions of propositional logic with operators ‘ a ’, where a ∈ A = {1, . . . , n}, n ∈ N. and ‘ ♦
–
a ϕ: necessarily ϕ.
c sun_rises_in_the_east
Resolution
p the_program_finishes
Destructive Non-Clausal Resolution
–
Clausal Resolution for Modal Logics
a ϕ: possibly ϕ. ♦ c it_rains ♦ p the_program_loops ♦
Several agents: i (i_send_m_j →
C. Nalon
j ¬j_receives_m_i) ♦
TRS, Natal, 2015 – 4 / 43
Semantics ⊲ Introduction Motivation Syntax Semantics Another example Syntax Semantics Usual Modal Logics
⊲
Evaluation of a formula depends on a set of worlds and on the accessibility relation on this set. 1,2 p
s
Resolution Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
~p
1
2
t
u
1,2
(M, s) |= p ∧
C. Nalon
p 1,2
2 p ∧ 1 ( 2 p ∨ 2 ¬p) ∧ ♦ 1 p 1 ¬p ∧ 2 ♦
TRS, Natal, 2015 – 5 / 43
Another example ⊲ Introduction Motivation Syntax Semantics Another example Syntax Semantics Usual Modal Logics
Evaluation of a formula depends on a set of worlds and on the accessibility relation on this set.
⊲
A
Resolution Destructive Non-Clausal Resolution
secret key msg
secret key ¬msg
A,C
Clausal Resolution for Modal Logics
A,C
A C
C
C. Nalon
C
¬secret key ¬msg TRS, Natal, 2015 – 6 / 43
Syntax ⊲ Introduction Motivation Syntax Semantics Another example Syntax Semantics Usual Modal Logics
⊲
Resolution
A modal propositional language extends the usual syntax of a , propositional logic (built over P, ¬, ∧) with a set of operators where a ∈ An = {1, . . . , n}, a fixed, non-empty, finite set of indexes. Abbreviations: false ≡ p ∧ ¬p (for p ∈ P), true ≡ ¬false ϕ ∨ ψ ≡ ¬(¬ϕ ∧ ¬ψ), ϕ → ψ ≡ ¬ϕ ∨ ψ, a ¬ϕ. a ϕ = ¬ ϕ ↔ ψ ≡ (ϕ → ψ) ∧ (ψ → ϕ), ♦
Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 7 / 43
Semantics ⊲ Introduction Motivation Syntax Semantics Another example Syntax Semantics Usual Modal Logics
Resolution Destructive Non-Clausal Resolution
M – – –
⊲
Clausal Resolution for Modal Logics
M
A Kripke Structure for P and An is a tuple = hW, R1 , . . . , Rn , πi, where: W is a non-empty set; For each a ∈ An , Ra ∈ W × W; π : W × P −→ {T , F }.
The satisfiability relation |= between a world w ∈ W in a Kripke structure and a formula is inductively defined by:
M
M, w |= p, p ∈ P, iff π(w, p) = T ; M, w |= ¬ϕ iff M, w 6|= ϕ; M, w |= ϕ ∧ ψ iff M, w |= ϕ and M, w |= ψ; M, w |= ϕ iff for all w such that wR w we have that M, w |= ϕ. A formula ϕ is satisfiable iff there is a Kripke structure M and w ∈ W such that M, w |= ϕ. In this case, we say that M satisfies ϕ. A formula ϕ is valid iff for all Kripke structures M, M satisfies ϕ.
– – – –
a
′
a
′
′
C. Nalon
TRS, Natal, 2015 – 8 / 43
Usual Modal Logics ⊲ Introduction Motivation Syntax Semantics Another example Syntax Semantics Usual Modal Logics
⊲
Resolution Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
C. Nalon
Different restrictions on the accessibility relations Ra define different modal logics: – – – – – – – – –
No restrictions: Kn ; Reflexive: KTn ; Transitive: K4n ; Euclidean: K5n ; Serial: KDn ; Symmetric: KBn ; Reflexive and Transitive: S4n ; Reflexive and Euclidean: S5n ; ...
TRS, Natal, 2015 – 9 / 43
Introduction
⊲ Resolution Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example Non-Clausal Resolution for Propositional Logic Example
Resolution
Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 10 / 43
Clausal Resolution for Propositional Logic Introduction
⊲ Resolution Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example
⊲
Non-Clausal Resolution for Propositional Logic Example Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
[RES]
C. Nalon
Clausal resolution is a refutational procedure: in order to prove ϕ, we transform the negation of ϕ into Conjunctive Normal Form before applying the inference rule. There is only one inference rule: (ϕ ∨ (ψ ∨ (ϕ ∨
l) ¬l) ψ)
Premises are called parent clauses (or the resolvends). The conclusion is called the resolvent. The literals l and ¬l are known as complementary literals. The parent clauses are resolved on the the complementary literals, generating the resolvent. The inference rule is applied until either a contradiction is found or no new clauses can be generated.
TRS, Natal, 2015 – 11 / 43
Conjunctive Normal Form Introduction
⊲ Resolution Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example
1. Removal of operators: ϕ→ψ ϕ↔ψ
⊲
Non-Clausal Resolution for Propositional Logic Example Destructive Non-Clausal Resolution
−→ (¬ϕ ∨ ψ) −→ (¬ϕ ∨ ψ) ∧ (ϕ ∨ ¬ψ)
2. Moving negation inwards (De Morgan’s): ¬(ϕ ∧ ψ) −→ (¬ϕ ∨ ¬ψ) ¬(ϕ ∨ ψ) −→ (¬ϕ ∧ ¬ψ) 3. Double negation elimination, i.e. ¬¬ϕ −→ ϕ 4. Distribution:
Clausal Resolution for Modal Logics
ϕ ∨ (ψ ∧ χ) ϕ ∧ (ψ ∨ χ)
C. Nalon
−→ (ϕ ∨ ψ) ∧ (ϕ ∨ ψ) ←− (ϕ ∧ ψ) ∨ (ϕ ∧ χ)
TRS, Natal, 2015 – 12 / 43
Example Introduction
If we want to show that
⊲ Resolution Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example
⊲
Non-Clausal Resolution for Propositional Logic Example Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
{p1 → p2 } ⊢ ¬p2 → ¬p1 or, equivalently, that ⊢ (p1 → p2 ) → (¬p2 → ¬p1 ) we take the Conjunctive Normal Form of ¬((p1 → p2 ) → (¬p2 → ¬p1 )) which is given by the following set of clauses 1. ¬p1 ∨ p2 2. ¬p2 3. p1 and apply the inference rule: 4. 5.
C. Nalon
p2 [3, 1] false [4, 2]
TRS, Natal, 2015 – 13 / 43
CNF? Introduction
⊲ Resolution Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example
(p ↔ q) ∨ (r ↔ s) 1. ((p → q) ∧ (q → p)) ∨ ((r → s) ∧ (s → r))
⊲
Non-Clausal Resolution for Propositional Logic Example Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 14 / 43
CNF? Introduction
⊲ Resolution Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example
⊲
(p ↔ q) ∨ (r ↔ s) 1. ((p → q) ∧ (q → p)) ∨ ((r → s) ∧ (s → r)) 2. ((¬p ∨ q) ∧ (¬q ∨ p)) ∨ ((¬r ∨ s) ∧ (¬s ∨ r))
Non-Clausal Resolution for Propositional Logic Example Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 14 / 43
CNF? Introduction
⊲ Resolution Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example
⊲
(p ↔ q) ∨ (r ↔ s) 1. ((p → q) ∧ (q → p)) ∨ ((r → s) ∧ (s → r)) 2. ((¬p ∨ q) ∧ (¬q ∨ p)) ∨ ((¬r ∨ s) ∧ (¬s ∨ r)) 3. (((¬p ∨ q) ∨ (¬r ∨ s)) ∧ ((¬p ∨ q) ∨ (¬s ∨ r)) ∧ ((¬q ∨ p) ∨ (¬r ∨ s)) ∧ ((¬q ∨ p) ∨ (¬s ∨ r)))
Non-Clausal Resolution for Propositional Logic Example Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 14 / 43
CNF? Introduction
⊲ Resolution Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example
⊲
Non-Clausal Resolution for Propositional Logic Example Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
C. Nalon
(p ↔ q) ∨ (r ↔ s) 1. ((p → q) ∧ (q → p)) ∨ ((r → s) ∧ (s → r)) 2. ((¬p ∨ q) ∧ (¬q ∨ p)) ∨ ((¬r ∨ s) ∧ (¬s ∨ r)) 3. (((¬p ∨ q) ∨ (¬r ∨ s)) ∧ ((¬p ∨ q) ∨ (¬s ∨ r)) ∧ ((¬q ∨ p) ∨ (¬r ∨ s)) ∧ ((¬q ∨ p) ∨ (¬s ∨ r)))
((¬p ∨ q ∨ ¬r ∨ s)∧ (¬p ∨ q ∨ r ∨ ¬s)∧ (p ∨ ¬q ∨ ¬r ∨ s)∧ (p ∨ ¬q ∨ r ∨ ¬s))
TRS, Natal, 2015 – 14 / 43
Another Example Introduction
⊲ Resolution Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example
(a ∧ b) ∨ (c ∧ d) ⇒ ((a ∨ c) ∧ (a ∨ d) ∧ (b ∨ c) ∧ (b ∨ d))
⊲
Non-Clausal Resolution for Propositional Logic Example Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 15 / 43
Another Example Introduction
⊲ Resolution Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example
(a ∧ b) ∨ (c ∧ d) ⇒ ((a ∨ c) ∧ (a ∨ d) ∧ (b ∨ c) ∧ (b ∨ d))
⊲
Non-Clausal Resolution for Propositional Logic Example
(a ∧ b) ∨ (c ∧ d ∧ e) ⇒ ((a ∨ c) ∧ (a ∨ d) ∧ (a ∨ e) ∧ (b ∨ c) ∧ (b ∨ d) ∧ (b ∨ e))
Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 15 / 43
Another Example Introduction
⊲ Resolution Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example
(a ∧ b) ∨ (c ∧ d) ⇒ ((a ∨ c) ∧ (a ∨ d) ∧ (b ∨ c) ∧ (b ∨ d))
⊲
Non-Clausal Resolution for Propositional Logic Example Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
(a ∧ b) ∨ (c ∧ d ∧ e) ⇒ ((a ∨ c) ∧ (a ∨ d) ∧ (a ∨ e) ∧ (b ∨ c) ∧ (b ∨ d) ∧ (b ∨ e)) (a ∧ b ∧ f ) ∨ (c ∧ d ∧ e)
⇒ ((a ∨ c) ∧ (a ∨ d) ∧ (a ∨ e) ∧ (b ∨ c) ∧ (b ∨ d) ∧ (b ∨ e) ∧ (f ∨ c) ∧ (f ∨ d) ∧ (f ∨ e))
C. Nalon
TRS, Natal, 2015 – 15 / 43
Efficiency Introduction
⊲ Resolution Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example
⊲
Non-Clausal Resolution for Propositional Logic Example Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
C. Nalon
Combinatorial explosion: (l1 ∧ . . . ∧ lm ) ∨ (k1 ∧ . . . ∧ kn ) ⇒ O(m × n)
In general: ψ ϕ1 ∧ . . . ∧ ϕn ϕ1 ∨ . . . ∨ ϕn ϕ1 → ϕ2 ϕ1 ↔ ϕ2 ¬ϕ atomic
Pnw(ψ) Qni=1 w(ϕi ) i=1 w(ϕi ) w(ϕ1 )w(ϕ2 ) w(ϕ1 )w(ϕ2 ) + w(ϕ1 )w(ϕ2 ) w(ϕ) 1
Qnw(ψ) w(ϕi ) Pi=1 n i=1 w(ϕi ) w(ϕ1 ) + w(ϕ2 ) w(ϕ1 )w(ϕ2 ) + w(ϕ1 )w(ϕ2 ) w(ϕ) 1
TRS, Natal, 2015 – 16 / 43
Renaming Introduction
⊲ Resolution Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example
⊲
Non-Clausal Resolution for Propositional Logic Example
We introduce new literals which replace subformulae; We also need to introduce the definition clauses for those literals. Let ϕ be the formula to be replaced: P ol(ϕ) > 0 ⇒ newϕ → ϕ P ol(ϕ) < 0 ⇒ ϕ → newϕ P ol(ϕ) = 0 ⇒ newϕ ↔ ϕ
Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 17 / 43
Example Introduction
⊲ Resolution
(a ∧ b ∧ f ) ∨ (c ∧ d ∧ e)
Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example
⊲
Non-Clausal Resolution for Propositional Logic Example Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 18 / 43
Example Introduction
⊲ Resolution Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example
(a ∧ b ∧ f ) ∨ (c ∧ d ∧ e) new(a∧b∧f ) ∨ new(c∧d∧e) new(a∧b∧f ) → (a ∧ b ∧ f )
new(c∧d∧e) → (c ∧ d ∧ e)
⊲
Non-Clausal Resolution for Propositional Logic Example Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 18 / 43
Example Introduction
⊲ Resolution Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example
⊲
Non-Clausal Resolution for Propositional Logic Example
(a ∧ b ∧ f ) ∨ (c ∧ d ∧ e) new(a∧b∧f ) ∨ new(c∧d∧e) new(a∧b∧f ) → (a ∧ b ∧ f ) ¬new(a∧b∧f ) ∨ a ¬new(a∧b∧f ) ∨ b ¬new(a∧b∧f ) ∨ f
new(c∧d∧e) → (c ∧ d ∧ e) ¬new(c∧d∧e) ∨ c ¬new(c∧d∧e) ∨ d ¬new(c∧d∧e) ∨ e
Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 18 / 43
Non-Clausal Resolution for Propositional Logic Introduction
⊲ Resolution
Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example Non-Clausal Resolution for Propositional Logic Example
⊲
Destructive Non-Clausal Resolution
The inference rule assumes the following form: NC-RES
ϕ(p) ψ(p) ϕ[p 7→ true] ∨ ψ[p 7→ false]
where ϕ(l) and ψ(l) denote two formulae which share the propositional symbol p. We can show that if Γ ∪ {ϕ(p), ψ(p)} is a consistent set of formulae, then Γ ∪ {ϕ(p), ψ(p), ϕ[p 7→ true] ∨ ψ[p 7→ false]} is also consistent. Non-clausal resolution, together with simplification, is sound and complete for propositional logic.
Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 19 / 43
Example Introduction
{p1 → p2 } ⊢ ¬p2 → ¬p1
⊲ Resolution Clausal Resolution for Propositional Logic CNF Example CNF? Another Example Efficiency Renaming Example Non-Clausal Resolution for Propositional Logic Example
⊲
Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics
C. Nalon
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
p1 → p2 ¬(¬p2 → ¬p1 ) (true → p2 ) ∨ ¬(¬p2 → ¬false) (true → p2 ) ∨ ¬(¬p2 → true) ¬(¬true → ¬p1 ) ∨ (true → false) ∨ ¬(¬false → true) ¬(false → ¬p1 ) ∨ false ∨ ¬(false → true) ¬(false → ¬p1 ) ¬(false → ¬true) ∨ ¬(false → ¬false) ¬true ∨ ¬true false
[1, 2, p1 ] [3, SIMP] [4, 2, p2 ] [5, SIMP] [6, SIMP] [7, 7, p1 ] [8, SIMP] [9, SIMP]
TRS, Natal, 2015 – 20 / 43
Introduction Resolution Destructive Non-Clausal Resolution Overview Notation Replacement Rules Addition Rule Modal Rule Example Example - I Example - II On Confluence Extensions Example Extensions - II
⊲
Destructive Non-Clausal Resolution
Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 21 / 43
Overview Introduction
Resolution Destructive Non-Clausal Resolution Overview Notation Replacement Rules Addition Rule Modal Rule Example Example - I Example - II On Confluence Extensions Example Extensions - II
⊲ ⊲
The method proposed by Fitting in [Fit90] uses the splitting inference rule (as in DPLL) and it is suitable for local reasoning in modal logics that do not require storing information from “previous” worlds. For formulae which are not in clausal form, it applies the usual tableaux inference rules for classical operators. Modal formulae are dealt with by means of replacement and addition inference rules. Completeness is easily achieved if one requires that all replacement inference rules are exhaustively applied before the resolution rule.
Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 22 / 43
Notation Introduction Resolution Destructive Non-Clausal Resolution Overview Notation Replacement Rules Addition Rule Modal Rule Example Example - I Example - II On Confluence Extensions Example Extensions - II
⊲ ⊲
The inference rules are applied to lists of lists of formulae. A clause is denoted by: [ϕ1 , . . . , ϕn ]
where ϕi are any formula. A clause represents the n-ary disjunction ϕ1 ∨ . . . ∨ ϕn . The empty disjunction [] is identified with false. If Ci is a clause, then < C1 , . . . , Cm >
Clausal Resolution for Modal Logics
denotes a clause list. A clause list represents the m-ary conjunction C1 ∧ . . . ∧ Cm . The empty conjunction <> is identified with true. If Li are clause lists, then [L1 , . . . , Lp ] denotes a block, representing the p-ary disjunction L1 ∨ . . . ∨ Lp .
C. Nalon
TRS, Natal, 2015 – 23 / 43
Replacement Rules Introduction Resolution Destructive Non-Clausal Resolution Overview Notation Replacement Rules Addition Rule Modal Rule Example Example - I Example - II On Confluence Extensions Example Extensions - II
⊲
Also called reduction rules. Those inference rules are applied to clauses. Double negation: [ϕ1 , . . . , ϕn , ¬¬ϕ] ⊢ [ϕ1 , . . . , ϕn , ϕ]
⊲
Conjunctions: [ϕ1 , . . . , ϕn , ϕ ∧ ψ] ⊢ [ϕ1 , . . . , ϕn , ϕ], [ϕ1 , . . . , ϕn , ψ]
Clausal Resolution for Modal Logics
Disjunctions: [ϕ1 , . . . , ϕn , ϕ ∨ ψ] ⊢ [ϕ1 , . . . , ϕn , ϕ, ψ]
Special Case (operates on lists of clauses): < [ϕ1 , . . . , ϕn , ψ1 , . . . , ψm ], C1 , . . . , Cp > < [ϕ1 , . . . , ϕn ], C1 , . . . , Cp >, < [ψ1 , . . . , ψm ], C1 , . . . , Cp >
C. Nalon
TRS, Natal, 2015 – 24 / 43
Addition Rule Introduction
Resolution:
Resolution Destructive Non-Clausal Resolution Overview Notation Replacement Rules Addition Rule Modal Rule Example Example - I Example - II On Confluence Extensions Example Extensions - II
⊲
[ϕ1 , . . . , ϕn , χ], [ψ1 , . . . , ψm , ¬χ] [ϕ1 , . . . , ϕn , χ], [ψ1 , . . . , ψm , ¬χ], [ϕ1 , . . . , ϕn , ψ1 , . . . , ψm ]
⊲
Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 25 / 43
Modal Rule Introduction
For ν-formulae:
For π-formulae:
Resolution Destructive Non-Clausal Resolution Overview Notation Replacement Rules Addition Rule Modal Rule Example Example - I Example - II On Confluence Extensions Example Extensions - II
ν ϕ ¬ ♦ϕ
⊲
⊲
Clausal Resolution for Modal Logics
π ♦ϕ ¬ ϕ
π0 ϕ ¬ϕ
If C is a clause, then: – –
ν0 ϕ ¬ϕ
C ♯ = {ν0i | ϕi ∈ C, ϕi is of type ν} C ♭ = {π0i | ϕi ∈ C, ϕi is of type π}
The K-π rule: < C1 , . . . , Cn , [π1 , . . . , πm ] > < C1♯ , . . . , Cn♯ , [π1 , . . . , πm ]♭ > where Ci♯ is added iff Ci is of the form [ν1 , . . . , νp ].
C. Nalon
TRS, Natal, 2015 – 26 / 43
Example Introduction Resolution Destructive Non-Clausal Resolution Overview Notation Replacement Rules Addition Rule Modal Rule Example Example - I Example - II On Confluence Extensions Example Extensions - II
⊲
h[p], [ q, r], [ p, ¬ ♦q], [¬ r, ♦s]i h
i
⊲
Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 27 / 43
Example Introduction Resolution Destructive Non-Clausal Resolution Overview Notation Replacement Rules Addition Rule Modal Rule Example Example - I Example - II On Confluence Extensions Example Extensions - II
⊲
h[p], [ q, r], [ p, ¬ ♦q], [¬ r, ♦s]i h
i
⊲
Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 27 / 43
Example Introduction Resolution Destructive Non-Clausal Resolution Overview Notation Replacement Rules Addition Rule Modal Rule Example Example - I Example - II On Confluence Extensions Example Extensions - II
⊲
h[p], [ q, r], [ p, ¬ ♦q], [¬ r, ♦s]i h[¬r, s]
i
⊲
Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 27 / 43
Example Introduction Resolution Destructive Non-Clausal Resolution Overview Notation Replacement Rules Addition Rule Modal Rule Example Example - I Example - II On Confluence Extensions Example Extensions - II
⊲
h[p], [ q, r], [ p, ¬ ♦q], [¬ r, ♦s]i h[¬r, s]
i
⊲
Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 27 / 43
Example Introduction Resolution Destructive Non-Clausal Resolution Overview Notation Replacement Rules Addition Rule Modal Rule Example Example - I Example - II On Confluence Extensions Example Extensions - II
⊲
h[p], [ q, r], [ p, ¬ ♦q], [¬ r, ♦s]i h[¬r, s], [p, ¬q]i
⊲
Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 27 / 43
Example - I Introduction Resolution Destructive Non-Clausal Resolution Overview Notation Replacement Rules Addition Rule Modal Rule Example Example - I Example - II On Confluence Extensions Example Extensions - II
⊲
⊲
1. 2. 3. 4. 5. 6. 7.
h[¬(( p ∧ q) → (p ∧ q))]i h[( p ∧ q)], [¬ (p ∧ q)]i h[ p], [ q)], [¬ (p ∧ q)]i h[p], [q], [¬(p ∧ q)]i h[p], [q], [¬p, ¬q]i h[p], [q], [¬p, ¬q], [¬q]i h[p], [q], [¬p, ¬q], [¬q], []i
[1, α] [2, α] [3, K-π] [4, β] [5, RES, p] [6, RES, q]
Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 28 / 43
Example - II Introduction Resolution Destructive Non-Clausal Resolution Overview Notation Replacement Rules Addition Rule Modal Rule Example Example - I Example - II On Confluence Extensions Example Extensions - II
⊲
⊲
1. 2. 3. 4. 5. 6. 7. 8. 9.
h[¬(( p ∨ q) → (¬ ♦¬p ∨ ¬ ♦¬q))]i h[ p ∨ q], [¬(¬ ♦¬p ∨ ¬ ♦¬q)]i h[ p ∨ q], [ ♦¬p], [ ♦¬q]i h[ p, q], [ ♦¬p], [ ♦¬q]i h[ p], [ ♦¬p], [ ♦¬q]i, h[ q], [ ♦¬p], [ ♦¬q]i h[p], [¬p]i, h[ q], [ ♦¬p], [ ♦¬q]i h[p], [¬p], []i, h[ q], [ ♦¬p], [ ♦¬q]i h[p], [¬p], []i, h[q], [¬q]i h[p], [¬p], []i, h[q], [¬q], []i
[1, α] [2, α(×3)] [3, β] [4, Special Case] [5, K-π] [6, RES] [7, K-π] [8, RES]
Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 29 / 43
On Confluence Introduction
The calculus is not confluent:
Resolution Destructive Non-Clausal Resolution Overview Notation Replacement Rules Addition Rule Modal Rule Example Example - I Example - II On Confluence Extensions Example Extensions - II
h[ p], [ ♦¬p], [ ♦¬q]i
⊲
leads to two different derivations: – –
⊲
Clausal Resolution for Modal Logics
C. Nalon
using [ ♦¬p] (as done in the previous example); using [ ♦¬q] (where a proof is not found and we need to backtrack to apply the above).
The special case inference rule also requires branching: if we can produce a refutation for all clause lists (as done in the previous example), then the original set is unsatisfiable.
TRS, Natal, 2015 – 30 / 43
Extensions Introduction Resolution Destructive Non-Clausal Resolution Overview Notation Replacement Rules Addition Rule Modal Rule Example Example - I Example - II On Confluence Extensions Example Extensions - II
⊲
⊲
Clausal Resolution for Modal Logics
C. Nalon
K-T-ν rule: [ν, ϕ1 , . . . , ϕn ] ⊢ [ν, ϕ1 , . . . , ϕn ], [ν0 , ϕ1 , . . . , ϕn ]. K-4-π rule: < C1 , . . . , Cn , [π1 , . . . , πm ] > < C1 , C1♯ , . . . , Cn , Cn♯ , [π1 , . . . , πm ]♭ >
where Ci and Ci♯ are added iff Ci is of the form [ν1 , . . . , νp ]. S4-π rule: < C1 , . . . , Cn , [π1 , . . . , πm ] > < C1 , . . . , Cn , [π1 , . . . , πm ]♭ > where Ci is added iff Ci is of the form [ν1 , . . . , νp ].
TRS, Natal, 2015 – 31 / 43
Example Introduction Resolution Destructive Non-Clausal Resolution Overview Notation Replacement Rules Addition Rule Modal Rule Example Example - I Example - II On Confluence Extensions Example Extensions - II
⊲
⊲
1. 2. 3. 4. 5. 6. 7. 8. 9. 10
h[¬( (p → q) → ( ♦p → ♦q))]i h[( (p → q)], [¬ ( ♦p → ♦q)]i h[( (p → q)], [¬( ♦p → ♦q)]i h[( (p → q)], [ ♦p], [¬ ♦q)]i h[( (p → q)], [p], [¬ ♦q)]i h[( (p → q)], [p], [¬ ♦q)], [p → q]i h[( (p → q)], [p], [¬ ♦q)], [p → q], [¬q]i h[( (p → q)], [p], [¬ ♦q)], [¬p, q], [¬q]i h[( (p → q)], [p], [¬ ♦q)], [¬p, q], [¬q], [q]i h[( (p → q)], [p], [¬ ♦q)], [¬p, q], [¬q], [q], []i
[1, α] [2, S4-π] [3, α] [4, S4-π] [5, T-ν] [6, T-ν] [7, β] [8, RES, p] [9, RES, q]
Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 32 / 43
Extensions - II Introduction
Resolution Destructive Non-Clausal Resolution Overview Notation Replacement Rules Addition Rule Modal Rule Example Example - I Example - II On Confluence Extensions Example Extensions - II
⊲
The calculus was extended to first-order modal logic: usual rules for quantifiers together with unification are added; skolemisation is applied as soon as the existential quantifier is the main operator of a formula (within a clause). The calculus is truly destructive: all the information from previous steps are discarded. This calculus cannot be extended easily to modal logics that require some kind of memory (e.g. where we have symmetry or the inverse) for completeness.
⊲
Clausal Resolution for Modal Logics
C. Nalon
TRS, Natal, 2015 – 33 / 43
Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution
⊲ for Modal Logics
Introduction Inference rules Example Another Example S4 Example Example - Continued Some notes References
C. Nalon
Clausal Resolution for Modal Logics
TRS, Natal, 2015 – 34 / 43
Introduction Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution
⊲ for Modal Logics ⊲ Introduction
Inference rules Example Another Example S4 Example Example - Continued Some notes References
C. Nalon
The calculus was proposed by Mints in [Min90]. It uses renaming to achieve a normal form to which the inference rules are applied: – – – – –
A literal is a propositional symbols or its negation; Propositional clauses are disjunctions of literals; A modal literal is a formula of the form l, l, or ♦l, where l is a literal; Modal clauses are disjunctions of modal literals. Initial clauses are modal clauses as well as expressions of the form D, where D is a modal clause.
TRS, Natal, 2015 – 35 / 43
Inference rules Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics Introduction Inference rules Example Another Example S4 Example Example - Continued Some notes References
⊲ ⊲
C. Nalon
[Rp] (L1 ∨ . . . ∨ Ln ); ¬L1 ∨ D1 ; . . . ; ¬Ln ∨ Dn D1 ∨ . . . ∨ Dn −
D D
The − rule can only be applied together with the resolution rule.
TRS, Natal, 2015 – 36 / 43
Example Introduction Resolution
¬( p ∧ q → (p ∧ q))
Destructive Non-Clausal Resolution Clausal Resolution
⊲ for Modal Logics
Introduction Inference rules Example Another Example S4 Example Example - Continued Some notes References
⊲
C. Nalon
TRS, Natal, 2015 – 37 / 43
Example Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics Introduction Inference rules Example Another Example S4 Example Example - Continued Some notes References
⊲ ⊲
C. Nalon
¬( p ∧ q → (p ∧ q)) 1. p 2. q 3. ¬ s1 4. (¬s1 ∨ p) 5. (¬s1 ∨ q) 6. (s1 ∨ ¬p ∨ ¬q)
TRS, Natal, 2015 – 37 / 43
Example Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics Introduction Inference rules Example Another Example S4 Example Example - Continued Some notes References
⊲ ⊲
C. Nalon
¬( p ∧ q → (p ∧ q)) 1. p 2. q 3. ¬ s1 4. (¬s1 ∨ p) 5. (¬s1 ∨ q) 6. (s1 ∨ ¬p ∨ ¬q) 7. (s1 ∨ ¬q)
[1, 6]
TRS, Natal, 2015 – 37 / 43
Example Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics Introduction Inference rules Example Another Example S4 Example Example - Continued Some notes References
⊲ ⊲
C. Nalon
¬( p ∧ q → (p ∧ q)) 1. p 2. q 3. ¬ s1 4. (¬s1 ∨ p) 5. (¬s1 ∨ q) 6. (s1 ∨ ¬p ∨ ¬q) 7. (s1 ∨ ¬q) 8. s1
[1, 6] [2, 6]
TRS, Natal, 2015 – 37 / 43
Example Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics Introduction Inference rules Example Another Example S4 Example Example - Continued Some notes References
⊲ ⊲
C. Nalon
¬( p ∧ q → (p ∧ q)) 1. 2. 3. 4. 5. 6. 7. 8. 9.
p q ¬ s1 (¬s1 ∨ p) (¬s1 ∨ q) (s1 ∨ ¬p ∨ ¬q) (s1 ∨ ¬q) s1 false
[1, 6] [2, 6] [3, 8]
TRS, Natal, 2015 – 37 / 43
Another Example Introduction Resolution
¬(( p ∨ q) → (¬ ♦¬p ∨ ¬ ♦¬q))
Destructive Non-Clausal Resolution Clausal Resolution
⊲ for Modal Logics
Introduction Inference rules Example Another Example S4 Example Example - Continued Some notes References
⊲
C. Nalon
TRS, Natal, 2015 – 38 / 43
Another Example Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics Introduction Inference rules Example Another Example S4 Example Example - Continued Some notes References
⊲
¬(( p ∨ q) → (¬ ♦¬p ∨ ¬ ♦¬q)) 1. 2. 3.
p ∨ q ♦¬p ♦¬q
⊲
C. Nalon
TRS, Natal, 2015 – 38 / 43
Another Example Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics Introduction Inference rules Example Another Example S4 Example Example - Continued Some notes References
⊲ ⊲
C. Nalon
¬(( p ∨ q) → (¬ ♦¬p ∨ ¬ ♦¬q)) 1. 2. 3. 4.
p ∨ q ♦¬p ♦¬q p
[1, 3]
TRS, Natal, 2015 – 38 / 43
Another Example Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution for Modal Logics Introduction Inference rules Example Another Example S4 Example Example - Continued Some notes References
⊲ ⊲
C. Nalon
¬(( p ∨ q) → (¬ ♦¬p ∨ ¬ ♦¬q)) 1. 2. 3. 4. 5.
p ∨ q ♦¬p ♦¬q p false
[1, 3] [4, 2]
TRS, Natal, 2015 – 38 / 43
S4 Introduction Resolution Destructive Non-Clausal Resolution
It also includes inference rules for T and 4. Reflexivity: [T]
Clausal Resolution
⊲ for Modal Logics
Introduction Inference rules Example Another Example S4 Example Example - Continued Some notes References
⊲
C. Nalon
l
∨ D
♦l ∨ D
Transitivity: [4]
l
∨
♦D
l
∨
♦D
TRS, Natal, 2015 – 39 / 43
Example Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution
⊲ for Modal Logics
Introduction Inference rules Example Another Example S4 Example Example - Continued Some notes References
⊲
C. Nalon
¬ ¬ ¬( a ∨ ¬ a) |{z} s1 | {z } s2 | {z } s3 {z } | s4 {z } | s5 {z } | s6
TRS, Natal, 2015 – 40 / 43
Example Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution
⊲ for Modal Logics
Introduction Inference rules Example Another Example S4 Example Example - Continued Some notes References
⊲
C. Nalon
¬ ¬ ¬( a ∨ ¬ a) |{z} s1 | {z } s2 | {z } s3 {z } | s4 {z } | s5 {z } | s6
1. 2. 3. 4. 5. 6. 7. 8. 9.
( a ∨ s1 ) (s1 ∨ s2 ) ( ♦¬s2 ∨ s3 ) ( ♦ a ∨ s1 ) (s4 ∨ ¬s3 ) (s4 ∨ ¬s1 ) (¬s4 ∨ ¬s5 ) ( s5 ∨ ¬s6 ) ♦ s6 TRS, Natal, 2015 – 40 / 43
Example - Continued Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution
⊲ for Modal Logics Introduction Inference rules Example Another Example S4 Example Example -
⊲ Continued
1. 2. 3. 4. 5. 6. 7. 8. 9.
( a ∨ s1 ) (s1 ∨ s2 ) ( ♦¬s2 ∨ s3 ) ( ♦ a ∨ s1 ) (s4 ∨ ¬s3 ) (s4 ∨ ¬s1 ) (¬s4 ∨ ¬s5 ) ( s5 ∨ ¬s6 ) ♦ s6
Some notes References
C. Nalon
TRS, Natal, 2015 – 41 / 43
Example - Continued Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution
⊲ for Modal Logics Introduction Inference rules Example Another Example S4 Example Example -
⊲ Continued Some notes References
C. Nalon
1. ( a ∨ s1 ) 2. (s1 ∨ s2 ) 3. ( ♦¬s2 ∨ s3 ) 4. ( ♦a ∨ s1 ) 5. (s4 ∨ ¬s3 ) 6. (s4 ∨ ¬s1 ) 7. (¬s4 ∨ ¬s5 ) 8. ( s5 ∨ ¬s6 ) 9. ♦s6 10. s4 ∨ s2
[6, 2]
TRS, Natal, 2015 – 41 / 43
Example - Continued Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution
⊲ for Modal Logics Introduction Inference rules Example Another Example S4 Example Example -
⊲ Continued Some notes References
C. Nalon
1. ( a ∨ s1 ) 2. (s1 ∨ s2 ) 3. ( ♦¬s2 ∨ s3 ) 4. ( ♦a ∨ s1 ) 5. (s4 ∨ ¬s3 ) 6. (s4 ∨ ¬s1 ) 7. (¬s4 ∨ ¬s5 ) 8. ( s5 ∨ ¬s6 ) 9. ♦s6 10. s4 ∨ s2 11. s2 ∨ ¬s5
[6, 2] [10, 7]
TRS, Natal, 2015 – 41 / 43
Example - Continued Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution
⊲ for Modal Logics Introduction Inference rules Example Another Example S4 Example Example -
⊲ Continued Some notes References
C. Nalon
1. ( a ∨ s1 ) 2. (s1 ∨ s2 ) 3. ( ♦¬s2 ∨ s3 ) 4. ( ♦a ∨ s1 ) 5. (s4 ∨ ¬s3 ) 6. (s4 ∨ ¬s1 ) 7. (¬s4 ∨ ¬s5 ) 8. ( s5 ∨ ¬s6 ) 9. ♦s6 10. s4 ∨ s2 11. s2 ∨ ¬s5 12. s4 ∨ ♦¬s2
[6, 2] [10, 7] [5, 3]
TRS, Natal, 2015 – 41 / 43
Example - Continued Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution
⊲ for Modal Logics Introduction Inference rules Example Another Example S4 Example Example -
⊲ Continued Some notes References
C. Nalon
1. ( a ∨ s1 ) 2. (s1 ∨ s2 ) 3. ( ♦¬s2 ∨ s3 ) 4. ( ♦a ∨ s1 ) 5. (s4 ∨ ¬s3 ) 6. (s4 ∨ ¬s1 ) 7. (¬s4 ∨ ¬s5 ) 8. ( s5 ∨ ¬s6 ) 9. ♦s6 10. s4 ∨ s2 11. s2 ∨ ¬s5 12. s4 ∨ ♦¬s2 13. ♦s2 ∨ ¬s5
[6, 2] [10, 7] [5, 3] [12, 7]
TRS, Natal, 2015 – 41 / 43
Example - Continued Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution
⊲ for Modal Logics Introduction Inference rules Example Another Example S4 Example Example -
⊲ Continued Some notes References
C. Nalon
1. ( a ∨ s1 ) 2. (s1 ∨ s2 ) 3. ( ♦¬s2 ∨ s3 ) 4. ( ♦a ∨ s1 ) 5. (s4 ∨ ¬s3 ) 6. (s4 ∨ ¬s1 ) 7. (¬s4 ∨ ¬s5 ) 8. ( s5 ∨ ¬s6 ) 9. ♦s6 10. s4 ∨ s2 11. s2 ∨ ¬s5 12. s4 ∨ ♦¬s2 13. ♦s2 ∨ ¬s5 14. s2 ∨ ¬ ♦s5
[6, 2] [10, 7] [5, 3] [12, 7] [11, T, 4]
TRS, Natal, 2015 – 41 / 43
Example - Continued Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution
⊲ for Modal Logics Introduction Inference rules Example Another Example S4 Example Example -
⊲ Continued Some notes References
C. Nalon
1. ( a ∨ s1 ) 2. (s1 ∨ s2 ) 3. ( ♦¬s2 ∨ s3 ) 4. ( ♦a ∨ s1 ) 5. (s4 ∨ ¬s3 ) 6. (s4 ∨ ¬s1 ) 7. (¬s4 ∨ ¬s5 ) 8. ( s5 ∨ ¬s6 ) 9. ♦s6 10. s4 ∨ s2 11. s2 ∨ ¬s5 12. s4 ∨ ♦¬s2 13. ♦s2 ∨ ¬s5 14. s2 ∨ ¬ ♦s5 15. ♦s5
[6, 2] [10, 7] [5, 3] [12, 7] [11, T, 4] [13, 14]
TRS, Natal, 2015 – 41 / 43
Example - Continued Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution
⊲ for Modal Logics Introduction Inference rules Example Another Example S4 Example Example -
⊲ Continued Some notes References
C. Nalon
1. ( a ∨ s1 ) 2. (s1 ∨ s2 ) 3. ( ♦¬s2 ∨ s3 ) 4. ( ♦a ∨ s1 ) 5. (s4 ∨ ¬s3 ) 6. (s4 ∨ ¬s1 ) 7. (¬s4 ∨ ¬s5 ) 8. ( s5 ∨ ¬s6 ) 9. ♦s6 10. s4 ∨ s2 11. s2 ∨ ¬s5 12. s4 ∨ ♦¬s2 13. ♦s2 ∨ ¬s5 14. s2 ∨ ¬ ♦s5 15. ♦s5 16. ¬s6
[6, 2] [10, 7] [5, 3] [12, 7] [11, T, 4] [13, 14] [15, 8]
TRS, Natal, 2015 – 41 / 43
Example - Continued Introduction Resolution Destructive Non-Clausal Resolution Clausal Resolution
⊲ for Modal Logics Introduction Inference rules Example Another Example S4 Example Example -
⊲ Continued Some notes References
C. Nalon
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
( a ∨ s1 ) (s1 ∨ s2 ) ( ♦¬s2 ∨ s3 ) ( ♦ a ∨ s1 ) (s4 ∨ ¬s3 ) (s4 ∨ ¬s1 ) (¬s4 ∨ ¬s5 ) ( s5 ∨ ¬s6 ) ♦ s6 s4 ∨ s2 s2 ∨ ¬s5 s4 ∨ ♦¬s2 ♦s2 ∨ ¬s5 s2 ∨ ¬ ♦ s5 ♦ s5 ¬s6 false
[6, 2] [10, 7] [5, 3] [12, 7] [11, T, 4] [13, 14] [15, 8] [16, 9]
TRS, Natal, 2015 – 41 / 43
Some notes Introduction
Resolution Destructive Non-Clausal Resolution Clausal Resolution
⊲ for Modal Logics
Introduction Inference rules Example Another Example S4 Example Example - Continued Some notes References
The resolution method proposed by Mints can be extended to deal with other modalities (e.g. S5). The calculi are all complete with respect to the intended semantics. Global reasoning can be achieved by adding the universal modality to the basic modal language.
⊲
C. Nalon
TRS, Natal, 2015 – 42 / 43
References [Fit90]
Melvin Fitting. Destructive modal resolution. Journal of Logic and Computation, 1(1):83–97, July 1990.
[Min90] G. Mints. Gentzen-type systems and resolution rules, part I: Propositional logic. Lecture Notes in Computer Science, 417:198–231, 1990.
C. Nalon
TRS, Natal, 2015 – 43 / 43
