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T alk outline 1. Higher-Order Modal Logic (HOML) 2. Automating - - PowerPoint PPT Presentation

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Theorem Provers for Every Normal Modal Logic 1 T obias Gleiner Alexander Steen Christoph Benzmller Freie Universitt Berlin 21st Int. Conf. on Logic for Programming, Artificial Intelligence and Reasoning 1This work has been supported by

slide-1
SLIDE 1

Theorem Provers for Every Normal Modal Logic1

T

  • bias Gleißner

Alexander Steen Christoph Benzmüller Freie Universität Berlin 21st Int. Conf. on Logic for Programming, Artificial Intelligence and Reasoning

1This work has been supported by the DFG under grant BE 2501/11-1 (Leo-III).

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SLIDE 2

T alk outline

  • 1. Higher-Order Modal Logic (HOML)
  • 2. Automating HOML
  • 3. Evaluation
  • 4. Example / Demo

, Theorem Provers for Every Normal Modal Logic, LPAR-21 2

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SLIDE 3

Introduction

Reasoning in Non-Classical Logics

◮ Increasing interest various fields

◮ Artificial Intelligence

(e.g. Agents, Knowledge)

◮ Computer Linguistics

(e.g. Semantics)

◮ Mathematics

(e.g. Geometry, Category theory)

◮ Theoretical Philsophy

(e.g. Metaphysics)

◮ Most powerful ATP/ITP: Classical logic only

Our focus: Modal logics

◮ Prover for (propositional) modal logics exist

◮ ModLeanTAP, Molle, Bliksem, FaCT++, ◮ MOLTAP, KtSeqC, ST

eP, TRP

◮ ...

◮ Only few for quantified variants

◮ MleanTAP, MleanCoP, MleanSeP (J. Otten) ◮ f2p+MSPASS , Theorem Provers for Every Normal Modal Logic, LPAR-21 3

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SLIDE 4

Introduction

Reasoning in Non-Classical Logics

◮ Increasing interest various fields

◮ Artificial Intelligence

(e.g. Agents, Knowledge)

◮ Computer Linguistics

(e.g. Semantics)

◮ Mathematics

(e.g. Geometry, Category theory)

◮ Theoretical Philsophy

(e.g. Metaphysics)

◮ Most powerful ATP/ITP: Classical logic only

Our focus: Modal logics

◮ Prover for (propositional) modal logics exist

◮ ModLeanTAP, Molle, Bliksem, FaCT++, ◮ MOLTAP, KtSeqC, ST

eP, TRP

◮ ...

◮ Only few for quantified variants

◮ MleanTAP, MleanCoP, MleanSeP (J. Otten) ◮ f2p+MSPASS , Theorem Provers for Every Normal Modal Logic, LPAR-21 3

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SLIDE 5

Introduction

Reasoning in Non-Classical Logics

◮ Increasing interest various fields

◮ Artificial Intelligence

(e.g. Agents, Knowledge)

◮ Computer Linguistics

(e.g. Semantics)

◮ Mathematics

(e.g. Geometry, Category theory)

◮ Theoretical Philsophy

(e.g. Metaphysics)

◮ Most powerful ATP/ITP: Classical logic only

Our focus: Modal logics

◮ Prover for (propositional) modal logics exist

◮ ModLeanTAP, Molle, Bliksem, FaCT++, ◮ MOLTAP, KtSeqC, ST

eP, TRP

◮ ...

◮ Only few for quantified variants

◮ MleanTAP, MleanCoP, MleanSeP (J. Otten) ◮ f2p+MSPASS , Theorem Provers for Every Normal Modal Logic, LPAR-21 3

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SLIDE 6

Automation of Quantified Modal Logic

Motivation

  • 1. First-order quantification is (sometimes) not enough
  • 2. Semantic diversity/flexibility needed

, Theorem Provers for Every Normal Modal Logic, LPAR-21 4

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SLIDE 7

Automation of Quantified Modal Logic

Motivation

  • 1. First-order quantification is (sometimes) not enough
  • 2. Semantic diversity/flexibility needed

Studies in Metaphysics (e.g. Ontological Argument), Studies in Computer Ethics

, Theorem Provers for Every Normal Modal Logic, LPAR-21 4

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SLIDE 8

Automation of Quantified Modal Logic

Motivation

  • 1. First-order quantification is (sometimes) not enough
  • 2. Semantic diversity/flexibility needed

, Theorem Provers for Every Normal Modal Logic, LPAR-21 4

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SLIDE 9

Automation of Quantified Modal Logic

Motivation

  • 1. First-order quantification is (sometimes) not enough
  • 2. Semantic diversity/flexibility needed

Automation approach

◮ Indirect: Via encoding into (classical) HOL ◮ Use existing general purpose HOL reasoners

, Theorem Provers for Every Normal Modal Logic, LPAR-21 4

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SLIDE 10

Automation of Quantified Modal Logic

Motivation

  • 1. First-order quantification is (sometimes) not enough
  • 2. Semantic diversity/flexibility needed

Automation approach

◮ Indirect: Via encoding into (classical) HOL ◮ Use existing general purpose HOL reasoners

Advantages

◮ Sophisticated existing systems

◮ ATPs: TPS, agsyHOL, Satallax, LEO-II, Leo-III ◮ Further: Isabelle, Nitpick

◮ Not fixed to a proving system ◮ Semantic variations with minor adjustments

◮ Axiomatization ◮ Quantification semantics ◮ ... , Theorem Provers for Every Normal Modal Logic, LPAR-21 4

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SLIDE 11

Automation of Quantified Modal Logic

Motivation

  • 1. First-order quantification is (sometimes) not enough
  • 2. Semantic diversity/flexibility needed

Automation approach

◮ Indirect: Via encoding into (classical) HOL ◮ Use existing general purpose HOL reasoners

Advantages

◮ Sophisticated existing systems

◮ ATPs: TPS, agsyHOL, Satallax, LEO-II, Leo-III ◮ Further: Isabelle, Nitpick

◮ Not fixed to a proving system ◮ Semantic variations with minor adjustments

◮ Axiomatization ◮ Quantification semantics ◮ ... , Theorem Provers for Every Normal Modal Logic, LPAR-21 4

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SLIDE 12

Higher Order Modal Logic – Syntax

Based on Simple type theory

[Church, J.Symb.L., 1940]

augmented with modalities

◮ Simple types T generated by base types and → ◮ T

ypically, base types are o and ι

, Theorem Provers for Every Normal Modal Logic, LPAR-21 5

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SLIDE 13

Higher Order Modal Logic – Syntax

Based on Simple type theory

[Church, J.Symb.L., 1940]

augmented with modalities

◮ Simple types T generated by base types and → ◮ T

ypically, base types are o and ι T ype of truth-values

, Theorem Provers for Every Normal Modal Logic, LPAR-21 5

slide-14
SLIDE 14

Higher Order Modal Logic – Syntax

Based on Simple type theory

[Church, J.Symb.L., 1940]

augmented with modalities

◮ Simple types T generated by base types and → ◮ T

ypically, base types are o and ι T ype of individuals

, Theorem Provers for Every Normal Modal Logic, LPAR-21 5

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SLIDE 15

Higher Order Modal Logic – Syntax

Based on Simple type theory

[Church, J.Symb.L., 1940]

augmented with modalities

◮ Simple types T generated by base types and → ◮ T

ypically, base types are o and ι

◮ T

erms defined by (α,β ∈ T , cα ∈ Σ, Xα ∈ V, i ∈ I) s, t ::= cα | Xα

◮ Allow infix notation for binary logical connectives ◮ Remaining logical connectives can be defined as usual ◮ Formulae of HOML are those terms with type o

, Theorem Provers for Every Normal Modal Logic, LPAR-21 5

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SLIDE 16

Higher Order Modal Logic – Syntax

Based on Simple type theory

[Church, J.Symb.L., 1940]

augmented with modalities

◮ Simple types T generated by base types and → ◮ T

ypically, base types are o and ι

◮ T

erms defined by (α,β ∈ T , cα ∈ Σ, Xα ∈ V, i ∈ I) s, t ::= cα | Xα | (λXα.sβ)α→β | (sα→β tα)β

◮ Allow infix notation for binary logical connectives ◮ Remaining logical connectives can be defined as usual ◮ Formulae of HOML are those terms with type o

, Theorem Provers for Every Normal Modal Logic, LPAR-21 5

slide-17
SLIDE 17

Higher Order Modal Logic – Syntax

Based on Simple type theory

[Church, J.Symb.L., 1940]

augmented with modalities

◮ Simple types T generated by base types and → ◮ T

ypically, base types are o and ι

◮ T

erms defined by (α,β ∈ T , cα ∈ Σ, Xα ∈ V, i ∈ I) s, t ::= cα | Xα | (λXα.sβ)α→β | (sα→β tα)β | (i

  • →oso)o

◮ Allow infix notation for binary logical connectives ◮ Remaining logical connectives can be defined as usual ◮ Formulae of HOML are those terms with type o

, Theorem Provers for Every Normal Modal Logic, LPAR-21 5

slide-18
SLIDE 18

Higher Order Modal Logic – Syntax

Based on Simple type theory

[Church, J.Symb.L., 1940]

augmented with modalities

◮ Simple types T generated by base types and → ◮ T

ypically, base types are o and ι

◮ T

erms defined by (α,β ∈ T , cα ∈ Σ, Xα ∈ V, i ∈ I) s, t ::= cα | Xα | (λXα.sβ)α→β | (sα→β tα)β | (i

  • →oso)o

◮ Allow infix notation for binary logical connectives ◮ Remaining logical connectives can be defined as usual ◮ Formulae of HOML are those terms with type o

, Theorem Provers for Every Normal Modal Logic, LPAR-21 5

slide-19
SLIDE 19

Higher Order Modal Logic – Syntax

Based on Simple type theory

[Church, J.Symb.L., 1940]

augmented with modalities

◮ Simple types T generated by base types and → ◮ T

ypically, base types are o and ι

◮ T

erms defined by (α,β ∈ T , cα ∈ Σ, Xα ∈ V, i ∈ I) s, t ::= cα | Xα | (λXα.sβ)α→β | (sα→β tα)β | (i

  • →oso)o

◮ Allow infix notation for binary logical connectives ◮ Remaining logical connectives can be defined as usual ◮ Formulae of HOML are those terms with type o

, Theorem Provers for Every Normal Modal Logic, LPAR-21 5

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SLIDE 20

Higher Order Modal Logic – Semantics

Extend HOL models with Kripke structures M =

  • W , {Ri}i∈I , {Dw}w∈W , {Iw}w∈W
  • ,

Theorem Provers for Every Normal Modal Logic, LPAR-21 6

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SLIDE 21

Higher Order Modal Logic – Semantics

Extend HOL models with Kripke structures M =

  • W , {Ri}i∈I , {Dw}w∈W , {Iw}w∈W
  • Set of possible worlds

, Theorem Provers for Every Normal Modal Logic, LPAR-21 6

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SLIDE 22

Higher Order Modal Logic – Semantics

Extend HOL models with Kripke structures M =

  • W , {Ri}i∈I , {Dw}w∈W , {Iw}w∈W
  • Family of accessibility relations Ri ⊆ W × W

, Theorem Provers for Every Normal Modal Logic, LPAR-21 6

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SLIDE 23

Higher Order Modal Logic – Semantics

Extend HOL models with Kripke structures M =

  • W , {Ri}i∈I , {Dw}w∈W , {Iw}w∈W
  • Family of frames, one for every world

Notion of frames D = (Dτ)τ∈T as in HOL: Dι = ∅ Do = {T, F} Dτ→ν = DDτ

ν

, Theorem Provers for Every Normal Modal Logic, LPAR-21 6

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SLIDE 24

Higher Order Modal Logic – Semantics

Extend HOL models with Kripke structures M =

  • W , {Ri}i∈I , {Dw}w∈W , {Iw}w∈W
  • Family of interpretation functions Iw

cτ

Iw

→ d ∈ Dτ ∈ Dw Assume Iw(¬), Iw(∨) . . . is standard.

, Theorem Provers for Every Normal Modal Logic, LPAR-21 6

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SLIDE 25

Higher Order Modal Logic – Semantics

Extend HOL models with Kripke structures M =

  • W , {Ri}i∈I , {Dw}w∈W , {Iw}w∈W
  • Value of a term given by

XτM,g,w = gw(X) cτM,g,w = Iw(X) (λXτ. sν)τ→ν M,g,w = y ∈ Dτ → sνM,g[Xτ/y]w,w (sτ→ν tτ)ν M,g,w = sτ→νM,g,w tτM,g,w i

  • →o soM,g,w =
  • T

if soM,g,v = T for all v ∈ W s.t. (w, v) ∈ Ri F

  • therwise

, Theorem Provers for Every Normal Modal Logic, LPAR-21 6

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SLIDE 26

Semantic variants of HOML

  • 1. Axiomatization of i
  • 2. Quantification
  • 3. Rigidity
  • 4. Consequence

, Theorem Provers for Every Normal Modal Logic, LPAR-21 7

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SLIDE 27

Semantic variants of HOML

  • 1. Axiomatization of i

◮ What properties does the box operators have? ◮ Depending on the application domain

Some popular axiom schemes:

Name Axiom scheme Condition on ri

  • Corr. formula

K i(s ⊃ t) ⊃ (is ⊃ it) — — B s ⊃ i◊is symmetric wRiv ⊃ vRiw D is ⊃ ◊is serial ∃v.wRiv T/M is ⊃ s reflexive wRiw 4 is ⊃ iis transitive

  • wRiv ∧ vRiu
  • ⊃ wRiu

5 ◊is ⊃ i◊is euclidean

  • wRiv ∧ wRiu
  • ⊃ vRiu

... ... ... ...

  • 2. Quantification
  • 3. Rigidity
  • 4. Consequence

, Theorem Provers for Every Normal Modal Logic, LPAR-21 7

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SLIDE 28

Semantic variants of HOML

  • 1. Axiomatization of i

◮ What properties does the box operators have?

  • 2. Quantification

◮ What is the meaning of ∀? ◮ Several popular choices exist

(1) Varying domains: As introduced (unrestricted frames) (2) Constant domains: Dw = Dv for all worlds w, v ∈ W (3) Cumulative domains: Dw ⊆ Dv whenever (w, v) ∈ Ri (4) Decreasing domains: Dw ⊇ Dv whenever (w, v) ∈ Ri

  • 3. Rigidity
  • 4. Consequence

, Theorem Provers for Every Normal Modal Logic, LPAR-21 7

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SLIDE 29

Semantic variants of HOML

  • 1. Axiomatization of i

◮ What properties does the box operators have?

  • 2. Quantification

◮ What is the meaning of ∀?

  • 3. Rigidity

◮ Do all constants c ∈ Σ denote the same object at every world? ◮ Several popular choices exist

(1) Flexible constants: As introduced (unrestricted Iw) (2) Rigid constants: Iw(c) = Iv(c) for all worlds w, v ∈ W and all c ∈ Σ

  • 4. Consequence

, Theorem Provers for Every Normal Modal Logic, LPAR-21 7

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SLIDE 30

Semantic variants of HOML

  • 1. Axiomatization of i

◮ What properties does the box operators have?

  • 2. Quantification

◮ What is the meaning of ∀?

  • 3. Rigidity

◮ Do all constants c ∈ Σ denote the same object at every world?

  • 4. Consequence

◮ What is an appropriate notion of logical consequence |=HOML? ◮ Many choices exist, two of them are

(1) Local consequence: ... not displayed here ... (2) Global consequence: ... not displayed here ...

, Theorem Provers for Every Normal Modal Logic, LPAR-21 7

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SLIDE 31

Semantic variants of HOML

  • 1. Axiomatization of i

◮ What properties does the box operators have?

  • 2. Quantification

◮ What is the meaning of ∀?

  • 3. Rigidity

◮ Do all constants c ∈ Σ denote the same object at every world?

  • 4. Consequence

◮ What is an appropriate notion of logical consequence |=HOML?

−→ at least 10 × 4 × 2 × 2 = 160 distinct logics

, Theorem Provers for Every Normal Modal Logic, LPAR-21 7

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SLIDE 32

Embedding of HOML within HOL

Automation approach: Encode HOML semantics within (classical) HOL HOL (meta-logic): s, t ::= HOML (target logic): s, t ::= Embedding of in (1) Introduce new type μ for worlds HOML formulas so are mapped to HOL predicates sμ→o (2) Introduce new constants ri

μ→μ→o for each i ∈ I

(3) Connectives: = = = = (4) Meta-logical notions: =

, Theorem Provers for Every Normal Modal Logic, LPAR-21 8

slide-33
SLIDE 33

Embedding of HOML within HOL

Automation approach: Encode HOML semantics within (classical) HOL HOL (meta-logic): s, t ::= HOML (target logic): s, t ::= Embedding of in (1) Introduce new type μ for worlds HOML formulas so are mapped to HOL predicates sμ→o (2) Introduce new constants ri

μ→μ→o for each i ∈ I

(3) Connectives: = = = = (4) Meta-logical notions: =

, Theorem Provers for Every Normal Modal Logic, LPAR-21 8

slide-34
SLIDE 34

Embedding of HOML within HOL

Automation approach: Encode HOML semantics within (classical) HOL HOL (meta-logic): s, t ::= HOML (target logic): s, t ::= Embedding of in (1) Introduce new type μ for worlds HOML formulas so are mapped to HOL predicates sμ→o (2) Introduce new constants ri

μ→μ→o for each i ∈ I

(3) Connectives: = = = = (4) Meta-logical notions: =

, Theorem Provers for Every Normal Modal Logic, LPAR-21 8

slide-35
SLIDE 35

Embedding of HOML within HOL

Automation approach: Encode HOML semantics within (classical) HOL HOL (meta-logic): s, t ::= HOML (target logic): s, t ::= Embedding of in (1) Introduce new type μ for worlds HOML formulas so are mapped to HOL predicates sμ→o (2) Introduce new constants ri

μ→μ→o for each i ∈ I

(3) Connectives: ¬o→o = λSσ.λWμ. ¬(S W) ∨o→o→o = λSσ.λTσ.λWμ. (S W) ∨ (T W) Πτ

(τ→o)→o = λPτ→σ.λWμ.∀Xτ. P X W

  • →o = λSσ.λWμ.∀Vμ. ¬(ri W V) ∨ S V

(4) Meta-logical notions: =

, Theorem Provers for Every Normal Modal Logic, LPAR-21 8

slide-36
SLIDE 36

Embedding of HOML within HOL

Automation approach: Encode HOML semantics within (classical) HOL HOL (meta-logic): s, t ::= HOML (target logic): s, t ::= Embedding of in (1) Introduce new type μ for worlds HOML formulas so are mapped to HOL predicates sμ→o (2) Introduce new constants ri

μ→μ→o for each i ∈ I

(3) Connectives: ¬o→o = λSσ.λWμ. ¬(S W) ∨o→o→o = λSσ.λTσ.λWμ. (S W) ∨ (T W) Πτ

(τ→o)→o = λPτ→σ.λWμ.∀Xτ. P X W

  • →o = λSσ.λWμ.∀Vμ. ¬(ri W V) ∨ S V

(4) Meta-logical notions: valid = λsσ.∀Wμ. s W

, Theorem Provers for Every Normal Modal Logic, LPAR-21 8

slide-37
SLIDE 37

Embedding of HOML within HOL #2

Embedding semantic variants

  • 1. Axiomatization of i
  • 2. Quantification
  • 3. Rigidity
  • 4. Consequence

, Theorem Provers for Every Normal Modal Logic, LPAR-21 9

slide-38
SLIDE 38

Embedding of HOML within HOL #2

Embedding semantic variants

  • 1. Axiomatization of i

Recall correspondences:

Name Axiom scheme Condition on ri

  • Corr. formula

... ... ... ... B s ⊃ i◊is symmetric wRiv ⊃ vRiw ... ... ... ...

For each desired axiom scheme for i: Postulate frame condition on ri as HOL axiom

  • 2. Quantification
  • 3. Rigidity
  • 4. Consequence

, Theorem Provers for Every Normal Modal Logic, LPAR-21 9

slide-39
SLIDE 39

Embedding of HOML within HOL #2

Embedding semantic variants

  • 1. Axiomatization of i

Postulate frame condition on ri as HOL axiom

  • 2. Quantification

Choose appropriate definition/axiomatization of quantifier: Constant domains quantifier: Πτ

(τ→o)→o = λPτ→σ.λWμ.∀Xτ. P X W

Varying domains quantifier: Πτ(τ→o)→o,va = λPτ→σ.λWμ.∀Xτ. ¬(eiw X W) ∨ (P X W) Cumulative/decreasing domains quantifier: Add axioms on eiw

  • 3. Rigidity
  • 4. Consequence

, Theorem Provers for Every Normal Modal Logic, LPAR-21 9

slide-40
SLIDE 40

Embedding of HOML within HOL #2

Embedding semantic variants

  • 1. Axiomatization of i

Postulate frame condition on ri as HOL axiom

  • 2. Quantification

Choose appropriate definition/axiomatization of quantifier

  • 3. Rigidity

Rigid constants: Only translate Boolean types to predicates: o = μ → o Rigid constants: Also translate individuals types to predicates: ι = μ → ι

  • 4. Consequence

, Theorem Provers for Every Normal Modal Logic, LPAR-21 9

slide-41
SLIDE 41

Embedding of HOML within HOL #2

Embedding semantic variants

  • 1. Axiomatization of i

Postulate frame condition on ri as HOL axiom

  • 2. Quantification

Choose appropriate definition/axiomatization of quantifier

  • 3. Rigidity

Appropriate type lifting

  • 4. Consequence

Global consequence: Apply valid(μ→o)→o to all translated sμ→o so = valid(μ→o)→o sμ→o Local consequence: Apply actuality operator A to all translated sμ→o so = A(μ→o)→o sμ→o where A = λSσ. s wactual and wactual is an uninterpreted symbol

, Theorem Provers for Every Normal Modal Logic, LPAR-21 9

slide-42
SLIDE 42

Problem representation

Ongoing work: Extension of TPTP THF syntax for modal logic (1) Formula syntax

thf( classical, axiom, ! [X:$i]: (p @ X)).

↓ Extend syntax with modalities

thf( modal, axiom, ! [X:$i]: ($box @ (p @ X))). thf( multi_modal, axiom, ! [X:$i]: ($box_int @ 1 @ (p @ X))).

(2) Semantics configuration Add ”logic”-annotated statements to the problem:

thf(simple_s5, logic, ($modal := [ $constants := $rigid, $quantification := $constant, $consequence := $global, $modalities := $modal_system_S5 ])).

◮ Intended semantics is attached to the problem

, Theorem Provers for Every Normal Modal Logic, LPAR-21 10

slide-43
SLIDE 43

Problem representation

Ongoing work: Extension of TPTP THF syntax for modal logic (1) Formula syntax

thf( classical, axiom, ! [X:$i]: (p @ X)).

↓ Extend syntax with modalities

thf( modal, axiom, ! [X:$i]: ($box @ (p @ X))). thf( multi_modal, axiom, ! [X:$i]: ($box_int @ 1 @ (p @ X))).

(2) Semantics configuration Add ”logic”-annotated statements to the problem:

thf(simple_s5, logic, ($modal := [ $constants := $rigid, $quantification := $constant, $consequence := $global, $modalities := $modal_system_S5 ])).

◮ Intended semantics is attached to the problem

, Theorem Provers for Every Normal Modal Logic, LPAR-21 10

slide-44
SLIDE 44

Problem representation

Ongoing work: Extension of TPTP THF syntax for modal logic (1) Formula syntax

thf( classical, axiom, ! [X:$i]: (p @ X)).

↓ Extend syntax with modalities

thf( modal, axiom, ! [X:$i]: ($box @ (p @ X))). thf( multi_modal, axiom, ! [X:$i]: ($box_int @ 1 @ (p @ X))).

(2) Semantics configuration Add ”logic”-annotated statements to the problem:

thf(simple_s5, logic, ($modal := [ $constants := $rigid, $quantification := $constant, $consequence := $global, $modalities := $modal_system_S5 ])).

◮ Intended semantics is attached to the problem

, Theorem Provers for Every Normal Modal Logic, LPAR-21 10

slide-45
SLIDE 45

Problem representation

Ongoing work: Extension of TPTP THF syntax for modal logic (1) Formula syntax

thf( classical, axiom, ! [X:$i]: (p @ X)).

↓ Extend syntax with modalities

thf( modal, axiom, ! [X:$i]: ($box @ (p @ X))). thf( multi_modal, axiom, ! [X:$i]: ($box_int @ 1 @ (p @ X))).

(2) Semantics configuration Add ”logic”-annotated statements to the problem:

thf( mydomain_type , type , ( human : $tType ) ). thf( myconstant_declaration , type , ( myconstant : $i ) ). thf( complicated_s5 , logic , ( $modal := [ $constants := [ $rigid , myconstant := $flexible ] , $quantification := [ $constant , human := $varying ] , $consequence := [ $global , myaxiom := $local ] , $modalities := [ $modal_system_S5, $box_int @ 1 := $modal_system_T ] ] ) ).

◮ Intended semantics is attached to the problem

, Theorem Provers for Every Normal Modal Logic, LPAR-21 11

slide-46
SLIDE 46

Stand-alone tool

Embedding procedure implemented as stand-alone tool

◮ Semantic specification is analyzed first ◮ Adequate definitions of logical and meta-logical notions are included

as axioms and definitions

◮ The problem is translated as presented ◮ Output format: Modal THF ◮ Integrated as pre-processor into Leo-III

, Theorem Provers for Every Normal Modal Logic, LPAR-21 12

slide-47
SLIDE 47

Evaluation

Evaluation setting:

◮ Translated all 580 mono-modal QMLTP problems to modal THF ◮ Semantic setting:

  • 1. Modal operator axiom system ∈ {K, D, T, S4, S5}
  • 2. Quantification semantics ∈ {constant, varying, cumul., decreasing}
  • 3. Rigid constants
  • 4. Consequence ∈ {local, global}

◮ Native modal logic prover: MleanCoP (J. Otten) ◮ HOL reasoners: Satallax, LEO-II, Nitpick ◮ Timeout 60s (2x AMD Opteron 2376 Quad Core/16 GB RAM)

Comments on evaluation result:

◮ MleanCoP not applicable to modal logic K ◮ MleanCoP not applicable to decreasing domains semantics ◮ MleanCoP not applicable to problems with equality symbol ◮ MleanCoP not applicable for global consequence ◮ Only first-order modal logic problems ◮ Embedding approach not restricted to benchmark settings

, Theorem Provers for Every Normal Modal Logic, LPAR-21 13

slide-48
SLIDE 48

Evaluation

Evaluation setting:

◮ Translated all 580 mono-modal QMLTP problems to modal THF ◮ Semantic setting:

  • 1. Modal operator axiom system ∈ {K, D, T, S4, S5}
  • 2. Quantification semantics ∈ {constant, varying, cumul., decreasing}
  • 3. Rigid constants
  • 4. Consequence ∈ {local, global}

◮ Native modal logic prover: MleanCoP (J. Otten) ◮ HOL reasoners: Satallax, LEO-II, Nitpick ◮ Timeout 60s (2x AMD Opteron 2376 Quad Core/16 GB RAM)

Comments on evaluation result:

◮ MleanCoP not applicable to modal logic K ◮ MleanCoP not applicable to decreasing domains semantics ◮ MleanCoP not applicable to problems with equality symbol ◮ MleanCoP not applicable for global consequence ◮ Only first-order modal logic problems ◮ Embedding approach not restricted to benchmark settings

, Theorem Provers for Every Normal Modal Logic, LPAR-21 13

slide-49
SLIDE 49

Evaluation #2

Result excerpt: Theorems

D vary D const T vary T const S4 vary S4 const S5 vary S5 const

100 200 300 400 Theorems found LEO-II Satallax MleanCoP

, Theorem Provers for Every Normal Modal Logic, LPAR-21 14

slide-50
SLIDE 50

Evaluation #3

Result excerpt: Counter satisfiable (CSA)

D vary D const T vary T const S4 vary S4 const S5 vary S5 const

50 100 150 200 250 CSA found Nitpick MleanCoP

, Theorem Provers for Every Normal Modal Logic, LPAR-21 15

slide-51
SLIDE 51

The penultimate slide

Related work

◮ Generic theorem proving systems:

The Logics Workbench,MetT eL2, LoTREC

◮ Embedding of further logics:

Conditional logics, hybrid logics, many-valued logics, ... Conclusion

◮ Provided a quite general semantics for HOML ◮ Presented a procedure that automatically converts HOML into HOL ◮ Implemented a stand-alone tool (e.g. as preprocessor)

◮ standard HOL provers can be used to reason about problems encoded in

the modal THF syntax

◮ Approach feasible (no evaluation for higher-order problems yet) ◮ Many new problems contributed in the modal THF format

, Theorem Provers for Every Normal Modal Logic, LPAR-21 16

slide-52
SLIDE 52

The penultimate slide

Related work

◮ Generic theorem proving systems:

The Logics Workbench,MetT eL2, LoTREC

◮ Embedding of further logics:

Conditional logics, hybrid logics, many-valued logics, ... Conclusion

◮ Provided a quite general semantics for HOML ◮ Presented a procedure that automatically converts HOML into HOL ◮ Implemented a stand-alone tool (e.g. as preprocessor)

◮ standard HOL provers can be used to reason about problems encoded in

the modal THF syntax

◮ Approach feasible (no evaluation for higher-order problems yet) ◮ Many new problems contributed in the modal THF format

, Theorem Provers for Every Normal Modal Logic, LPAR-21 16

slide-53
SLIDE 53

The ultimate slide

Thank you for your attention!

, Theorem Provers for Every Normal Modal Logic, LPAR-21 17