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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

2

x, p

1 3

q

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,

• nly 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,

• nly finitely many demos need to be considered

The states of a demo are finite sets of formulas called clauses

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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,

• nly 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

• conjunctive

| s ∨ t | ♦∗s

• disjunctive

| p | ¬p | s | ♦s

• 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

• conjunctive

| s ∨ t | ♦∗s

• disjunctive

| p | ¬p | s | ♦s

• literal

Compatible formula decomposition s ∧ t α s, t s ∨ t β s, t ∗s α s, +s ♦∗s β s, ♦+s s µ s ♦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

• conjunctive

| s ∨ t | ♦∗s

• disjunctive

| p | ¬p | s | ♦s

• literal

Compatible formula decomposition s ∧ t α s, t s ∨ t β s, t ∗s α s, +s ♦∗s β s, ♦+s s µ s ♦s µ s 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

If ¬p ∈ A, then p / ∈ A

2

If s ∈ A is of type α, then all constituents of s are in A

3

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 s ∈ A satisfiable ⇐ ⇒ ∃ demo H ∈ 22A ∃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

H0 ⊃ H1 ⊃ H2 ⊃ · · · ⊃ Hn

2

Hn is a demo

3

Hk+1 contains all demos contained in Hk

4

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 Gor´ e 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 ∨ ¬q) p ∧ q p 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 ∨ ¬q) p ∧ q p 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 ∨ ¬q) p ∧ q p q ¬p ¬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′

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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′

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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

♦+s ∈ C and C ⋫ s

2

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

♦+s ∈ C and C ⋫ s

2

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

♦+s ∈ C and C ⋫ s

2

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

♦+s ∈ C and C ⋫ s

2

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 R

• S

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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Backtracking Search

Example

♦+¬p, p, p ♦+¬p, p, p ♦+¬p, p

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Backtracking Search

Example

♦+¬p, p, p ♦+¬p, p, p ♦+¬p, p ¬p Success

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Backtracking Search

Example

♦+¬p, p, p ♦+¬p, p, p ♦+¬p, p ♦+¬p

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Backtracking Search

Example

♦+¬p, p, p ♦+¬p, p, p ♦+¬p, p ♦+¬p ¬p Success

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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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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

δMDs > 0

2

δMCs = 0 or δMCs > δMDs

δMAs := minimal distance from a state satisfying A to a state satisfying s Minimal distance idea appears in [Baader 1990]

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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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Backtracking Search

Example

♦+p, ¬p, (x ∧ ¬p), ♦¬p

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Backtracking Search

Example

♦+p, ¬p, (x ∧ ¬p), ♦¬p x, ♦+p, ¬p

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Backtracking Search

Example

♦+p, ¬p, (x ∧ ¬p), ♦¬p x, ♦+p, ¬p p

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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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Backtracking Search

Example

♦+p, ¬p, (x ∧ ¬p), ♦¬p x, ♦+p, ¬p ♦+p, x, ¬p, ¬p p Delete subsumed nominal clause

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Backtracking Search

Example

♦+p, ¬p, (x ∧ ¬p), ♦¬p ♦+p, x, ¬p, ¬p p

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Backtracking Search

Example

♦+p, ¬p, (x ∧ ¬p), ♦¬p ♦+p, x, ¬p, ¬p p ♦+p, ¬p

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Backtracking Search

Example

♦+p, ¬p, (x ∧ ¬p), ♦¬p ♦+p, x, ¬p, ¬p p ♦+p, ¬p

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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 Gor´ e 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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