Incremental Decision Procedures
for Modal Logic with
Eventualities and Nominals
Gert Smolka
Saarland University
DL 2011
Barcelona, July 15, 2011
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Introduction
My First Encounter with DL
1984 Brachman and Levesque (AAAI, Austin)
C , D ::= A | C D | ∀R.C | ∃R.C
Concepts describe sets of individuals
Roles are binary relations on individuals
Notation for fragment of FOL
Subsumption coNP-hard (but O(n2 ) if ∃R.)
1988 Schmidt-Schauß and Smolka (AI Journal, 1991)
Closure under complement, ALC
C , D ::= A | ¬C | C D | C D | ∀R.C | ∃R.C
Subsumption ∼
= satisfiability, PSPACE-complete
New decision method, evolved in tableau-based method
1991 Klaus Schild (IJCAI)
ALC ∼
= modal logic K
PSPACE-completeness first shown by Ladner 1977
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Introduction
Problem Considered in This Talk
ALC plus nominals plus R ∗ (reflexive transitive closure)
s, t ::= p | s ∧ t | ∀R.s | ∀R ∗ .s
| ¬p | s ∨ t | ∃R.s | ∃R ∗ .s
Eventualities are formulas of the form ∃R ∗ .s
Nominals are p’s that hold for exactly one individual
Incremental decision procedures for satisfiability
Challenge comes from combination of eventualities and nominals
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Introduction
Acknowledgement: Joint work with Mark Kaminski
Terminating Tableaux for Hybrid Logic with Eventualities
IJCAR 2010, Edinburgh
Clausal tableaux for hybrid PDL
Tech. Report 2010
Clausal Graph Tableaux for Hybrid Logic with Eventualities and Difference
LPAR 2010, Yogyakarta
Correctness and Worst-case Optimality of Pratt-style Decision Procedures
for Modal and Hybrid Logics
Tableaux 2011, Bern (with Thomas Schneider)
Correctness of an Incremental and Worst-case Optimal Decision Procedure
for Modal Logic with Eventualities
Tech. Report 2011
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Introduction
Box and Diamond Notation
We restrict to a single atomic role R
(extension to multiple atomic roles is straightforward)
We use box and diamond notation
s := ∀R.s
∗ s := ∀R ∗ .s
+ s := ∀R.∀R ∗.s
♦s := ∃R.s
♦∗ s := ∃R ∗ .s
♦+ s := ∃R.∃R ∗.s
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Introduction
We Think of Models as Transition Systems
x, p
2
q
1
3
4
p, q
Individuals appear as states
Atomic concepts and nominals label states
Nominals label exactly one state
s holds at a state w if all successors of w satisfy s
∗ s holds at a state w if all states reachable from w satisfy s
♦s holds at a state w if some successor of w satisfy s
♦∗ s holds at a state w if some state reachable from w satisfy s
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Introduction
Equivalences
∗ s ≡ s ∧ + s
♦∗ s ≡ s ∨ ♦+ s
¬∗ s ≡ ♦∗ ¬s
¬♦∗ s ≡ ∗ ¬s
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Introduction
Satisfiability and Demos
A formula is satisfiable if it holds at some state of some model
Our decision procedures search for syntactic models called demos
A formula is satisfiable iff it is satisfied
by a demo obtained from its subformulas
Finite search space: Given a formula,
only finitely many demos need to be considered
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Introduction
Satisfiability and Demos
A formula is satisfiable if it holds at some state of some model
Our decision procedures search for syntactic models called demos
A formula is satisfiable iff it is satisfied
by a demo obtained from its subformulas
Finite search space: Given a formula,
only finitely many demos need to be considered
The states of a demo are finite sets of formulas called clauses
Gert Smolka (Saarland University)
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Introduction
Satisfiability and Demos
A formula is satisfiable if it holds at some state of some model
Our decision procedures search for syntactic models called demos
A formula is satisfiable iff it is satisfied
by a demo obtained from its subformulas
Finite search space: Given a formula,
only finitely many demos need to be considered
The states of a demo are finite sets of formulas called clauses
A state of a demo satisfies every formula contained in it
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Introduction
Demo Search for ♦+¬p ∧ p ∧ p
♦+ ¬p, p, p
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Introduction
Demo Search for ♦+¬p ∧ p ∧ p
♦+ ¬p, p, p
♦+ ¬p, p, p
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Introduction
Demo Search for ♦+¬p ∧ p ∧ p
♦+ ¬p, p, p
♦+ ¬p, p, p
♦+ ¬p, p
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Introduction
Demo Search for ♦+¬p ∧ p ∧ p
♦+ ¬p, p, p
♦+ ¬p, p, p
♦+ ¬p, p
¬p
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Introduction
Demo Search with Nominals
♦+ ¬p, (x ∧ p), ♦p
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Introduction
Demo Search with Nominals
♦+ ¬p, (x ∧ p), ♦p
♦+ ¬p, x, p
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Introduction
Demo Search with Nominals
♦+ ¬p, (x ∧ p), ♦p
♦+ ¬p, x, p, p
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Introduction
Demo Search with Nominals
♦+ ¬p, (x ∧ p), ♦p
♦+ ¬p, x, p, p
♦+ ¬p, p
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Introduction
Demo Search with Nominals
♦+ ¬p, (x ∧ p), ♦p
♦+ ¬p, x, p, p
♦+ ¬p, p
¬p
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Introduction
Cyclic Demo
G := ∗ (♦∗ p ∧ ♦∗ ¬p)
Every reachable state can reach both p and ¬p
Every finite model must cycle
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Introduction
Cyclic Demo
G := ∗ (♦∗ p ∧ ♦∗ ¬p)
Every reachable state can reach both p and ¬p
Every finite model must cycle
♦+ p, G , ¬p
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Introduction
Cyclic Demo
G := ∗ (♦∗ p ∧ ♦∗ ¬p)
Every reachable state can reach both p and ¬p
Every finite model must cycle
♦+ p, G , ¬p
p, G , ♦+ ¬p
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Introduction
Cyclic Demo
G := ∗ (♦∗ p ∧ ♦∗ ¬p)
Every reachable state can reach both p and ¬p
Every finite model must cycle
♦+ p, G , ¬p
p, G , ♦+ ¬p
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Introduction
Plan of the Talk
Foundations for efficient decision procedures
for modal logics with eventualities and nominals
1
Hintikka demos and pruning
2
Expansion and graph search
3
Backtracking search
In each part, nominals will first be ignored
and then be added in a second step
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Hintikka Demos and Pruning
I Hintikka Demos and Pruning
Basic theory we need
Pruning yields complexity-optimal decision procedure
Fischer and Ladner 1977 (PDL)
Pratt 1979 (PDL)
Emerson and Halpern 1985 (CTL)
Kaminski, Schneider, and Smolka, Tableaux 2011
(PDL with nominals, difference, and converse)
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Hintikka Demos and Pruning
Formula Decomposition and Finite Syntactic Closure
A formula is either conjunctive (α), disjunctive (β), or literal
s, t ::= s ∧ t | ∗ s | s ∨ t | ♦∗ s | p | ¬p | s | ♦s
conjunctive
Gert Smolka (Saarland University)
disjunctive
Incremental Decision Procedures
literal
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Hintikka Demos and Pruning
Formula Decomposition and Finite Syntactic Closure
A formula is either conjunctive (α), disjunctive (β), or literal
s, t ::= s ∧ t | ∗ s | s ∨ t | ♦∗ s | p | ¬p | s | ♦s
conjunctive
disjunctive
literal
Compatible formula decomposition
s ∧ t α s, t
s ∨ t β s, t
∗ s
♦∗ s
α s, + s
β s, ♦+ s
s
♦s
μ s
μ s
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Hintikka Demos and Pruning
Formula Decomposition and Finite Syntactic Closure
A formula is either conjunctive (α), disjunctive (β), or literal
s, t ::= s ∧ t | ∗ s | s ∨ t | ♦∗ s | p | ¬p | s | ♦s
conjunctive
disjunctive
literal
Compatible formula decomposition
s ∧ t α s, t
s ∨ t β s, t
∗ s
♦∗ s
α s, + s
β s, ♦+ s
s
♦s
μ s
μ s
Gert Smolka (Saarland University)
A denotes finite set of formulas
A is decomposition closure of A
Size of A is linear in the size of A
α-β-decomposition terminates
μ is modal literal
+ s := ∗ s
♦+ s := ♦♦∗ s
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Hintikka Demos and Pruning
Hintikka Sets
A is a Hintikka set if
1
2
3
If ¬p ∈ A, then p ∈
/A
If s ∈ A is of type α, then all constituents of s are in A
If s ∈ A is of type β, then at least one constituent of s is in A
H ≡ LH if H Hintikka set (LH is set of literals in H)
Every state w of every model M yields a finite Hintikka set in A:
Hw := { s ∈ A | M, w |= s }
The Hintikka sets for M and A contain exactly the formulas s ∈ A
that are satisfiable in M
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Hintikka Demos and Pruning
Hintikka Systems
A Hintikka system is a set of Hintikka sets
A Hintikka system H describes a model:
The states are the sets H ∈ H
A state H is labeled with p iff p ∈ H
There is an edge H → H iff { s | s ∈ H } ⊆ H We call RH := { s | s ∈ H } the request of H
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Hintikka Demos and Pruning
Demos
A demo is a Hintikka system H such that:
If ♦s ∈ H ∈ H, then ∃ H ∈ H. H →H H and s ∈ H If ♦+ s ∈ H ∈ H, then ∃ H ∈ H. H →+
H H and s ∈ H
Lemma
H, H |= s if H is a demo and s ∈ H ∈ H
Small Model Theorem
A
s ∈ A satisfiable ⇐⇒ ∃ demo H ∈ 22 ∃H ∈ H. s ∈ H
Naive decision procedure is double exponential
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Hintikka Demos and Pruning
Pruning (Pratt 1979)
Start from maximal Hintikka system over A
Stepwise delete Hintikka sets that violate a demo condition
Terminates with largest demo over A
demos are closed under union
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Hintikka Demos and Pruning
Pruning (Pratt 1979)
Start from maximal Hintikka system over A
Stepwise delete Hintikka sets that violate a demo condition
Terminates with largest demo over A
demos are closed under union
Yields exponential decision procedure
complexity-optimal
decides satisfiability of every formula s ∈ A in one go
not practical since it starts with all Hintikka sets
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Hintikka Demos and Pruning
Pruning Generalizes
Pruning can be applied to every Hintikka system
Always terminates with largest demo
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Hintikka Demos and Pruning
Pruning Generalizes
Pruning can be applied to every Hintikka system
Always terminates with largest demo
Correctness: H0 → H1 → H2 → · · · → Hn
1
2
3
4
H0 ⊃ H1 ⊃ H2 ⊃ · · · ⊃ Hn
Hn is a demo
Hk+1 contains all demos contained in Hk
Hn is greatest demo contained in H0
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Hintikka Demos and Pruning
Everything Extends to Nominals
Require Hintikka systems to be nominally coherent
(every nominal appears in at most one Hintikka set)
Exponentially many maximal Hintikka systems
Pruning still yields exponential decision procedure
(every demo is contained in a maximal Hintikka system)
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Expansion and Graph Search
II Expansion and Graph Search
Represent Hintikka sets by their literals
Start from input clauses
Stepwise add clauses by expansion
Determine satisfiable and unsatisfiable clauses
Stop if input clauses are determined
Kaminski and Smolka 2010, 2011
Related to Goré and Widmann’s [2009, 2010]
decision procedure for PDL with converse
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Expansion and Graph Search
Clauses = Literal Hintikka Sets
A literal is a formula that is neither α nor β
(p, ¬p, s, ♦s)
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Expansion and Graph Search
Clauses = Literal Hintikka Sets
A literal is a formula that is neither α nor β
(p, ¬p, s, ♦s)
The non-literal formulas of a Hintikka set are logically redundant
(H ≡ LH)
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Expansion and Graph Search
Clauses = Literal Hintikka Sets
A literal is a formula that is neither α nor β
(p, ¬p, s, ♦s)
The non-literal formulas of a Hintikka set are logically redundant
(H ≡ LH)
A clause is a Hintikka set just containing literals
(i.e., a conflict-free set of literals)
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Expansion and Graph Search
Clauses = Literal Hintikka Sets
A literal is a formula that is neither α nor β
(p, ¬p, s, ♦s)
The non-literal formulas of a Hintikka set are logically redundant
(H ≡ LH)
A clause is a Hintikka set just containing literals
(i.e., a conflict-free set of literals)
Support: C A : ⇐⇒ ∃ Hintikka set H. A ⊆ H ∧ LH ⊆ C
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Expansion and Graph Search
Clausal Demos
A clause system is a set of clauses
Every clause system S comes with a transition relation →S :
C →S C : ⇐⇒ C ∈ S ∧ C ∈ S ∧ C RC
Clausal demo: set S of clauses satisfying the demo conditions:
If ♦s ∈ C ∈ S, then ∃ C ∈ S. C →S C and C s
If ♦+ s ∈ C ∈ S, then ∃ C ∈ S. C →+
S C and C s
Hintikka demo yields clausal demo and vice versa
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Expansion and Graph Search
Demonstrated Clauses
Let S be a clause system
A clause C is demonstrated in S if ∃ demo D ⊆ S. C ∈ D
Demonstrated clauses are satisfiable
The clauses demonstrated in S can be computed by pruning
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Expansion and Graph Search
Disjunctive Normal Forms (DNF)
Sets of formulas are interpreted conjunctively
i.e., {s1 , . . . , sn } is interpreted as s1 ∧ · · · ∧ sn
A set {C1 , . . . , Cn } of clauses is a DNF of A if
A ≡ C1 ∨ · · · ∨ Cn
Ci A and Ci ⊆ A for all i ∈ {1, . . . , n}
Every finite A has a DNF, can be obtained by α-β-decomposition
DNFs are not unique
If A has an empty DNF, then A is unsatisfiable
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Expansion and Graph Search
Example of DNF Computation
♦∗ (p ∧ q), ∗ (¬p ∨ ¬q)
Expansion and Graph Search
Example of DNF Computation
♦∗ (p ∧ q), ∗ (¬p ∨ ¬q)
¬p ∨ ¬q
+ (¬p
p∧q
p
q
∨ ¬q)
♦+ (p ∧ q)
¬p ¬q
¬p ¬q
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Expansion and Graph Search
Example of DNF Computation
♦∗ (p ∧ q), ∗ (¬p ∨ ¬q)
¬p ∨ ¬q
+ (¬p
p∧q
p
q
∨ ¬q)
♦+ (p ∧ q)
¬p ¬q
¬p ¬q
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Expansion and Graph Search
Example of DNF Computation
♦∗ (p ∧ q), ∗ (¬p ∨ ¬q)
¬p ∨ ¬q
+ (¬p
p∧q
p
q
¬p ¬q
∨ ¬q)
♦+ (p ∧ q)
¬p ¬q
C1 C2
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Expansion and Graph Search
Expansions
♦s ∈ C needs a successor clause in a demo
An expansion of ♦s ∈ C is a set of possible successor clauses
If a set S contains an expansion for every ♦s ∈ C ,
then it contains a demo for every satisfiable clause
expansion of ♦s ∈ C if s is not an eventuality
DNF of {s} ∪ RC
expansion of ♦+ s ∈ C
(DNF of {s} ∪ RC ) ∪ (DNF of {♦+ s} ∪ RC )
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Expansion and Graph Search
Expansions
Expansion of {♦p, (q1 ∨ q2 )}:
{p, q1 }
{p, q2 }
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Expansion and Graph Search
Expansions
Expansion of {♦p, (q1 ∨ q2 )}:
{p, q1 }
{p, q2 }
Expansion of {♦+ p, + ♦+ p}:
{p, ♦+ p, + ♦+ p}
{♦+ p, + ♦+ p}
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Expansion and Graph Search
Expansions
Expansion of {♦p, (q1 ∨ q2 )}:
{p, q1 }
{p, q2 }
Expansion of {♦+ p, + ♦+ p}:
{p, ♦+ p, + ♦+ p}
{♦+ p, + ♦+ p}
If a state w satisfies C and w is a successor of w satisfying s,
then w satisfies some clause in every expansion of ♦+ s ∈ C
♦+ s ∈ C
s
w →M w Gert Smolka (Saarland University)
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Expansion and Graph Search
Expansions
Expansion of {♦p, (q1 ∨ q2 )}:
{p, q1 }
{p, q2 }
Expansion of {♦+ p, + ♦+ p}:
{p, ♦+ p, + ♦+ p}
{♦+ p, + ♦+ p}
If a state w satisfies C and w is a successor of w satisfying s,
then w satisfies some clause in every expansion of ♦+ s ∈ C
♦ + s ∈ C →S
C
s
w →M w Gert Smolka (Saarland University)
Incremental Decision Procedures
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Expansion and Graph Search
Saturation under Expansion
A set of clauses S is saturated if
it contains an expansion for every ♦s ∈ C ∈ S
Theorem Let S be saturated. Then:
C ∈ S is satisfiable iff C is demonstrated in S
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Expansion and Graph Search
Saturation under Expansion
A set of clauses S is saturated if
it contains an expansion for every ♦s ∈ C ∈ S
Theorem Let S be saturated. Then:
C ∈ S is satisfiable iff C is demonstrated in S
Decision procedure
1
Start with S := {C0 }
2
Expand S until either C0 demonstrated in S or S saturated
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Expansion and Graph Search
Saturation under Expansion
A set of clauses S is saturated if
it contains an expansion for every ♦s ∈ C ∈ S
Theorem Let S be saturated. Then:
C ∈ S is satisfiable iff C is demonstrated in S
Decision procedure
1
Start with S := {C0 }
2
Expand S until either C0 demonstrated in S or S saturated
Can be efficient if C is satisfiable
Inefficient if C is unsatisfiable
How can we determine unsatisfiable clauses?
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Expansion and Graph Search
Sinks
A sink is a sufficiently expanded set of clauses so that it is clear that
some eventuality cannot be fulfilled
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Expansion and Graph Search
Sinks
A sink is a sufficiently expanded set of clauses so that it is clear that
some eventuality cannot be fulfilled
A sink is a pair (S, ♦+ s) such that every clause C ∈ S satisfies
1
2
♦+ s ∈ C and C s
S contains every satisfiable clause of an expansion of ♦+ s ∈ C
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Expansion and Graph Search
Sinks
A sink is a sufficiently expanded set of clauses so that it is clear that
some eventuality cannot be fulfilled
A sink is a pair (S, ♦+ s) such that every clause C ∈ S satisfies
1
2
♦+ s ∈ C and C s
S contains every satisfiable clause of an expansion of ♦+ s ∈ C
Theorem
Every clause of a sink is unsatisfiable
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Expansion and Graph Search
Sinks
A sink is a sufficiently expanded set of clauses so that it is clear that
some eventuality cannot be fulfilled
A sink is a pair (S, ♦+ s) such that every clause C ∈ S satisfies
1
2
♦+ s ∈ C and C s
S contains every satisfiable clause of an expansion of ♦+ s ∈ C
Theorem
Every clause of a sink is unsatisfiable
Example of a 2-clause sink
G := (¬p ∨ ¬q)
♦+ (p ∧ q), + G , ¬p
♦+ (p ∧ q), + G , ¬q
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Expansion and Graph Search
Sinks
A sink is a sufficiently expanded set of clauses so that it is clear that
some eventuality cannot be fulfilled
A sink is a pair (S, ♦+ s) such that every clause C ∈ S satisfies
1
2
♦+ s ∈ C and C s
S contains every satisfiable clause of an expansion of ♦+ s ∈ C
Theorem
Every clause of a sink is unsatisfiable
Example of a 2-clause sink
G := (¬p ∨ ¬q) ∧ ♦D
♦+ (p ∧ q), + G , ¬p, ♦D
♦+ (p ∧ q), + G , ¬q, ♦D
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Expansion and Graph Search
Refutation
1
If every clause of an expansion of ♦s ∈ C is unsatisfiable,
then C is unsatisfiable
2
Every superset of an unsatisfiable clause is unsatisfiable
3
Every clause of a sink is unsatisfiable
Theorem
If S is saturated, then the refutation rules
can refute every unsatisfiable clause in S.
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Expansion and Graph Search
Incremental Decision Procedure
1
Maintains a set S of a clauses and two disjoint subsets
D ⊆ S and R ⊆ S of demonstrated and refuted clauses
D
S
R
2
A clause is determined if it is in D or R
3
Starts with D := R := ∅
4
Expands undetermined clauses in S
5
Grows D and R by pruning and refutation
6
Stops once initial clauses are determined
Complexity-optimal
Can be efficient for clauses with small refutations or small demos
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Expansion and Graph Search
Extension to Nominals
Additional expansion rule that adds unions of nominal clauses
Soundness of demonstration and refutation is preserved
Completeness of demonstration is preserved
(prune every maximal nominally coherent subsystem)
Completeness of refutation is lost
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Backtracking Search
III Backtracking Search
Simple algorithm
Works well in presence of nominals
DL reasoners for expressive logics are backtracking
Kaminski and Smolka 2010 (IJCAR)
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Backtracking Search
Functional Demos
A functional demo is a clause system that
♦+ ¬p, p, p
♦+ ¬p, p, p
♦+ ¬p, p
¬p
fixes a unique successor for every diamond by a link
contains no delegation loop (red links are delegating)
(ensures satisfaction of eventualities)
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Backtracking Search
Backtracking Demo Search
Construct a functional demo by single successor expansion of
diamonds (don’t know choice)
Every clause added is part of the final demo
Fail if a diamond has no successor
Fail if a delegation loop is introduced
Succeeds with functional demo iff input clause is satisfiable
Minimal bureaucracy (no pruning, no refutation)
Extends smoothly to nominals
Worst-case runtime is double exponential
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Backtracking Search
Example
♦+ ¬p, p, p
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Backtracking Search
Example
♦+ ¬p, p, p
♦+ ¬p, p, p
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Incremental Decision Procedures
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Backtracking Search
Example
♦+ ¬p, p, p
♦+ ¬p, p, p
♦+ ¬p, p
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Incremental Decision Procedures
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Backtracking Search
Example
♦+ ¬p, p, p
♦+ ¬p, p, p
♦+ ¬p, p
¬p
Success
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Incremental Decision Procedures
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Backtracking Search
Example
♦+ ¬p, p, p
♦+ ¬p, p, p
♦+ ¬p, p
♦+ ¬p
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Incremental Decision Procedures
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Backtracking Search
Example
♦+ ¬p, p, p
♦+ ¬p, p, p
♦+ ¬p, p
♦+ ¬p
¬p
Success
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Incremental Decision Procedures
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Backtracking Search
Example
♦+ ¬p, p, p
♦+ ¬p, p, p
♦+ ¬p, p
♦+ ¬p
Failure because of delegation loop
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Incremental Decision Procedures
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Backtracking Search
Completeness Argument
Why is a demo found if the initial clause is satisfiable?
Clauses added by expansion are satisfied by initial model
Every proper delegation link C (♦+ s)D reduces δ-distance:
1
δM Ds > 0
2
δM Cs = 0 or δM Cs > δM Ds
δM As := minimal distance
from a state satisfying A to a state satisfying s
Minimal distance idea appears in [Baader 1990]
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Incremental Decision Procedures
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Backtracking Search
Extension to Nominals
Start with nominally coherent set of clauses
If an expansion step introduces a nominal clause,
regain nominal coherence as follows:
Union new clause with all clauses containing a common nominal
Fail if union is not a clause; otherwise:
Redirect links pointing to subsumed clauses to union clause
Delete subsumed clauses
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Incremental Decision Procedures
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Backtracking Search
Example
♦+ p, ¬p, (x ∧ ¬p), ♦¬p
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Incremental Decision Procedures
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Backtracking Search
Example
♦+ p, ¬p, (x ∧ ¬p), ♦¬p
x, ♦+ p, ¬p
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Incremental Decision Procedures
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Backtracking Search
Example
♦+ p, ¬p, (x ∧ ¬p), ♦¬p
x, ♦+ p, ¬p
p
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Incremental Decision Procedures
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Backtracking Search
Example
♦+ p, ¬p, (x ∧ ¬p), ♦¬p
x, ♦+ p, ¬p
x, ¬p, ¬p
p
Nominal union needed
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Incremental Decision Procedures
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Backtracking Search
Example
♦+ p, ¬p, (x ∧ ¬p), ♦¬p
x, ♦+ p, ¬p
♦+ p, x, ¬p, ¬p
p
Delete subsumed nominal clause
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Incremental Decision Procedures
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Backtracking Search
Example
♦+ p, ¬p, (x ∧ ¬p), ♦¬p
♦+ p, x, ¬p, ¬p
p
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Incremental Decision Procedures
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Backtracking Search
Example
♦+ p, ¬p, (x ∧ ¬p), ♦¬p
♦+ p, x, ¬p, ¬p
p
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♦+ p, ¬p
Incremental Decision Procedures
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Backtracking Search
Example
♦+ p, ¬p, (x ∧ ¬p), ♦¬p
♦+ p, x, ¬p, ¬p
p
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♦+ p, ¬p
Incremental Decision Procedures
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Conclusions
Conclusions
Decision procedures for ALC with eventualities and nominals
Approach: Goal-directed demo search
Graph search and backtracking search
Backtracking search easier to implement
Graph search has not been implemented with nominals
Impressive graph prover for PDL by Goré and Widmann
Backtracking search performs best on most K benchmarks
Everything extends to PDL (union and composition of roles)
Fischer-Ladner decomposition does not satisfy H ≡ LH
(e.g., {a∗∗ ⊥, a∗ a∗∗ ⊥} is an unsatisfiable Hintikka set)
Expansion requires terminating α-β-decomposition (can be arranged)
Open: Extension to converse roles
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Incremental Decision Procedures
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