The World's Most Widely Applicable Modal Logic Theorem...
Transcript of The World's Most Widely Applicable Modal Logic Theorem...
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The World’s Most Widely ApplicableModal Logic Theorem Proverand its Associated Infrastructure
Alexander SteenFreie Universität Berlin
RuleML Webinar September 29th
0jww C. Benzmüller, T. Gleißner
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Talk outline
1. Motivation
2. Flavours of modal logics
3. How it works (roughly)
4. Evaluation
,
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Introduction
Reasoning in Non-Classical Logics
É Increasing interest in various fieldsÉ Artificial Intelligence (e.g. Agents, Knowledge)É Computer Linguistics (e.g. Semantics)É Mathematics (e.g. Geometry, Category theory)É Theoretical Philosophy (e.g. Metaphysics)É Legal Informatics (e.g. Computable/Smart contracts)
É Most powerful ATP/ITP: Classical logic only
Focus here: Modal logics
É Prover for (propositional) modal logics existÉ ModLeanTAP, Molle, Bliksem, FaCT++,É MOLTAP, KtSeqC, STeP, TRPÉ ...
É Only few for quantified variantsÉ MleanTAP, MleanCoP, MleanSeP (J. Otten)É f2p+MSPASS
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 3
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Introduction
Reasoning in Non-Classical Logics
É Increasing interest in various fieldsÉ Artificial Intelligence (e.g. Agents, Knowledge)É Computer Linguistics (e.g. Semantics)É Mathematics (e.g. Geometry, Category theory)É Theoretical Philosophy (e.g. Metaphysics)É Legal Informatics (e.g. Computable/Smart contracts)
É Most powerful ATP/ITP: Classical logic only
Focus here: Modal logics
É Prover for (propositional) modal logics existÉ ModLeanTAP, Molle, Bliksem, FaCT++,É MOLTAP, KtSeqC, STeP, TRPÉ ...
É Only few for quantified variantsÉ MleanTAP, MleanCoP, MleanSeP (J. Otten)É f2p+MSPASS
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 3
![Page 5: The World's Most Widely Applicable Modal Logic Theorem ...ruleml.org/...LeoIIIModalProver-RuleMLWebinar...29.pdf · Automation of Quantified Modal Logic Motivation 1.First-order](https://reader033.fdocuments.net/reader033/viewer/2022052717/5f0446af7e708231d40d2d68/html5/thumbnails/5.jpg)
Introduction
Reasoning in Non-Classical Logics
É Increasing interest in various fieldsÉ Artificial Intelligence (e.g. Agents, Knowledge)É Computer Linguistics (e.g. Semantics)É Mathematics (e.g. Geometry, Category theory)É Theoretical Philosophy (e.g. Metaphysics)É Legal Informatics (e.g. Computable/Smart contracts)
É Most powerful ATP/ITP: Classical logic only
Focus here: Modal logics
É Prover for (propositional) modal logics existÉ ModLeanTAP, Molle, Bliksem, FaCT++,É MOLTAP, KtSeqC, STeP, TRPÉ ...
É Only few for quantified variantsÉ MleanTAP, MleanCoP, MleanSeP (J. Otten)É f2p+MSPASS
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 3
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Automation of Quantified Modal Logic
Motivation1. First-order quantification is (sometimes) not enough2. Semantic diversity/flexibility needed
See studies in Metaphysics, e.g.
◦ Gödel’s Ontological Argument [BenzmüllerW.-Paleo,2017]
and several variants of it◦ Anderson-Hájek Controversy [Benzm.WeberW.-Paleo,2017]
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 4
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Automation of Quantified Modal Logic
Motivation1. First-order quantification is (sometimes) not enough2. Semantic diversity/flexibility needed:
Properties of modal operators necessary (�) and possibly (◊)
... but that’s not all of it!,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 4
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Automation of Quantified Modal Logic
Motivation1. First-order quantification is (sometimes) not enough
2. Semantic diversity/flexibility needed
Automation approachÉ Indirect: Via encoding into (classical) HOLÉ Use existing general purpose HOL reasoners
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 4
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Automation of Quantified Modal Logic
Motivation1. First-order quantification is (sometimes) not enough
2. Semantic diversity/flexibility needed
Automation approachÉ Indirect: Via encoding into (classical) HOLÉ Use existing general purpose HOL reasoners
AdvantagesÉ Sophisticated existing systems
É ATPs: TPS, agsyHOL, Satallax, LEO-II, Leo-IIIÉ Further: Isabelle, Nitpick
É Not fixed to any one proving systemÉ Semantic variations with minor adjustments
É AxiomatizationÉ Quantification semanticsÉ ...
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The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 4
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Higher Order Modal Logic – Syntax
Based on Simple type theory [Church, J.Symb.L., 1940]
augmented with modalities
É Simple types T generated by base types and mappings (→)É Usually, base types are o and ι
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 5
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Higher Order Modal Logic – Syntax
Based on Simple type theory [Church, J.Symb.L., 1940]
augmented with modalities
É Simple types T generated by base types and mappings (→)É Usually, base types are o and ι
Type of truth-values
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 5
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Higher Order Modal Logic – Syntax
Based on Simple type theory [Church, J.Symb.L., 1940]
augmented with modalities
É Simple types T generated by base types and mappings (→)É Usually, base types are o and ι
Type of individuals
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 5
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Higher Order Modal Logic – Syntax
Based on Simple type theory [Church, J.Symb.L., 1940]
augmented with modalities
É Simple types T generated by base types and mappings (→)É Usually, base types are o and ι
É Terms defined by (α,β ∈ T , cα ∈ Σ, Xα ∈ V, i ∈ I)
s, t ::= cα | Xα
| (λXα.sβ)α→β | (sα→β tα)β
| (�io→o
so)o
É Allow infix notation for binary logical connectivesÉ Remaining logical connectives can be defined as usualÉ Formulae of HOML are those terms with type o
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 5
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Higher Order Modal Logic – Syntax
Based on Simple type theory [Church, J.Symb.L., 1940]
augmented with modalities
É Simple types T generated by base types and mappings (→)É Usually, base types are o and ι
É Terms defined by (α,β ∈ T , cα ∈ Σ, Xα ∈ V, i ∈ I)
s, t ::= cα | Xα| (λXα.sβ)α→β | (sα→β tα)β
| (�io→o
so)o
É Allow infix notation for binary logical connectivesÉ Remaining logical connectives can be defined as usualÉ Formulae of HOML are those terms with type o
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 5
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Higher Order Modal Logic – Syntax
Based on Simple type theory [Church, J.Symb.L., 1940]
augmented with modalities
É Simple types T generated by base types and mappings (→)É Usually, base types are o and ι
É Terms defined by (α,β ∈ T , cα ∈ Σ, Xα ∈ V, i ∈ I)
s, t ::= cα | Xα| (λXα.sβ)α→β | (sα→β tα)β
| (�io→o
so)o
É Allow infix notation for binary logical connectivesÉ Remaining logical connectives can be defined as usualÉ Formulae of HOML are those terms with type o
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 5
![Page 16: The World's Most Widely Applicable Modal Logic Theorem ...ruleml.org/...LeoIIIModalProver-RuleMLWebinar...29.pdf · Automation of Quantified Modal Logic Motivation 1.First-order](https://reader033.fdocuments.net/reader033/viewer/2022052717/5f0446af7e708231d40d2d68/html5/thumbnails/16.jpg)
Higher Order Modal Logic – Syntax
Based on Simple type theory [Church, J.Symb.L., 1940]
augmented with modalities
É Simple types T generated by base types and mappings (→)É Usually, base types are o and ι
É Terms defined by (α,β ∈ T , cα ∈ Σ, Xα ∈ V, i ∈ I)
s, t ::= cα | Xα| (λXα.sβ)α→β | (sα→β tα)β
| (�io→o
so)o
É Allow infix notation for binary logical connectivesÉ Remaining logical connectives can be defined as usualÉ Formulae of HOML are those terms with type o
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 5
![Page 17: The World's Most Widely Applicable Modal Logic Theorem ...ruleml.org/...LeoIIIModalProver-RuleMLWebinar...29.pdf · Automation of Quantified Modal Logic Motivation 1.First-order](https://reader033.fdocuments.net/reader033/viewer/2022052717/5f0446af7e708231d40d2d68/html5/thumbnails/17.jpg)
Higher Order Modal Logic – Syntax
Based on Simple type theory [Church, J.Symb.L., 1940]
augmented with modalities
É Simple types T generated by base types and mappings (→)É Usually, base types are o and ι
É Terms defined by (α,β ∈ T , cα ∈ Σ, Xα ∈ V, i ∈ I)
s, t ::= cα | Xα| (λXα.sβ)α→β | (sα→β tα)β
| (�io→o
so)o
É Allow infix notation for binary logical connectivesÉ Remaining logical connectives can be defined as usualÉ Formulae of HOML are those terms with type o
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 5
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Higher Order Modal Logic – Semantics
Extend HOL models with Kripke structures
M =�
W , {Ri}i∈I , {Dw}w∈W , {Iw}w∈W�
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 6
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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
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 6
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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
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 6
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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 worldNotion of frames D = (Dτ)τ∈T as in HOL:
Dι 6= ∅
Do = {T,F}
Dτ→ν = DDτν
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 6
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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 Iwcτ
Iw7→ d ∈ Dτ ∈ Dw
Assume Iw(¬),Iw(∨) . . . is standard.
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 6
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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 (wrt. var. assignment g):
‖Xτ‖M,g,w = gw(X)
... (not all shown here ...)
‖�io→o
so‖M,g,w =
¨
T if ‖so‖M,g,v = T for all v ∈W s.t. (w,v) ∈ Ri
F otherwise
Assume Henkin semantics
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 6
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Semantic variants of HOML
1. Axiomatization of �i
2. Quantification
3. Rigidity
4. Consequence
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 7
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Semantic variants of HOML
1. Axiomatization of �i
É What properties does the box operators have?É 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 ⊃ vRiwD �
is ⊃ ◊is serial ∃v.wRivT/M �
is ⊃ s reflexive wRiw4 �
is ⊃ �i�is transitive�
wRiv∧ vRiu�
⊃ wRiu5 ◊
is ⊃ �i◊is euclidean�
wRiv∧wRiu�
⊃ vRiu... ... ... ...
2. Quantification
3. Rigidity
4. Consequence
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 7
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Semantic variants of HOML
1. Axiomatization of �i
É What properties does the box operators have?
2. QuantificationÉ What is the meaning of ∀?É 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
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 7
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Semantic variants of HOML
1. Axiomatization of �i
É What properties does the box operators have?
2. QuantificationÉ What is the meaning of ∀?
3. RigidityÉ Do all constants c ∈ Σ denote the same object at every world?É 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
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 7
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Semantic variants of HOML
1. Axiomatization of �i
É What properties does the box operators have?
2. QuantificationÉ What is the meaning of ∀?
3. RigidityÉ Do all constants c ∈ Σ denote the same object at every world?
4. ConsequenceÉ What is an appropriate notion of logical consequence |=HOML?É Many choices exist, two of them are
(1) Local consequence: ... not displayed here ...
(2) Global consequence: ... not displayed here ...
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 7
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Semantic variants of HOML
1. Axiomatization of �i
É What properties does the box operators have?
2. QuantificationÉ What is the meaning of ∀?
3. RigidityÉ Do all constants c ∈ Σ denote the same object at every world?
4. ConsequenceÉ What is an appropriate notion of logical consequence |=HOML?
−→ at least 10× 4× 2× 2 = 160 distinct logics
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 7
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Semantic variants of HOML
1. Axiomatization of �i
É What properties does the box operators have?
2. QuantificationÉ What is the meaning of ∀?
3. RigidityÉ Do all constants c ∈ Σ denote the same object at every world?
4. ConsequenceÉ What is an appropriate notion of logical consequence |=HOML?
Example: Modal logic S5, constant domains, rigid symbolsGenerated by 5: ◊is ⊃ �i◊isTheorems: all classical tautologies,
s ⊃ �i◊is, (symmetric ri)�is ⊃ s, (reflexive ri)
∀X.�if X ⊃ �i∀X.f X (Barcan formula)...
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 7
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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 riμ→μ→o
for each i ∈ I
(3) Connectives:
¬o→o =
∨o→o→o =
Πτ(τ→o)→o
=
�o→o =
(4) Meta-logical notions:
valid = ,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 8
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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 riμ→μ→o
for each i ∈ I
(3) Connectives:
¬o→o =
∨o→o→o =
Πτ(τ→o)→o
=
�o→o =
(4) Meta-logical notions:
valid = ,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 8
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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 riμ→μ→o
for each i ∈ I
(3) Connectives:
¬o→o =
∨o→o→o =
Πτ(τ→o)→o
=
�o→o =
(4) Meta-logical notions:
valid = ,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 8
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Stand-alone tool
Embedding procedure implemented as stand-alone tool
É Semantic specification is analyzed firstÉ (Meta-)logical notions are included as axioms/definitionsÉ Output format: ”Plain THF” (TH0)É Integrated as pre-processor into Leo-III
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 9
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Evaluation
Evaluation setting:É Translated all 580 mono-modal QMLTP problems to modal THFÉ Semantic setting:
1. Modal operator axiom system ∈ {K,D,T,S4,S5}2. Quantification semantics ∈ {constant,varying,cumul.,decreasing}3. Rigid constants4. Consequence ∈ {local,global}
É Native modal logic prover: MleanCoP (J. Otten)É HOL reasoners: Satallax, LEO-II, NitpickÉ Timeout 60s (2x AMD Opteron 2376 Quad Core/16 GB RAM)
Comments on evaluation result:É MleanCoP not applicable to modal logic KÉ MleanCoP not applicable to decreasing domains semanticsÉ MleanCoP not applicable to problems with equality symbolÉ MleanCoP not applicable for global consequenceÉ Only first-order modal logic problemsÉ Embedding approach not restricted to benchmark settings
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 10
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Evaluation
Evaluation setting:É Translated all 580 mono-modal QMLTP problems to modal THFÉ Semantic setting:
1. Modal operator axiom system ∈ {K,D,T,S4,S5}2. Quantification semantics ∈ {constant,varying,cumul.,decreasing}3. Rigid constants4. Consequence ∈ {local,global}
É Native modal logic prover: MleanCoP (J. Otten)É HOL reasoners: Satallax, LEO-II, NitpickÉ Timeout 60s (2x AMD Opteron 2376 Quad Core/16 GB RAM)
Comments on evaluation result:É MleanCoP not applicable to modal logic KÉ MleanCoP not applicable to decreasing domains semanticsÉ MleanCoP not applicable to problems with equality symbolÉ MleanCoP not applicable for global consequenceÉ Only first-order modal logic problemsÉ Embedding approach not restricted to benchmark settings
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 10
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Evaluation #2
Result excerpt: Theorems
D vary D const T vary T const S4 vary S4 const S5 vary S5 const0
100
200
300
400
Theo
rem
sfo
und
LEO-IISatallaxMleanCoP
,
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Evaluation #3
Result excerpt: Counter satisfiable (CSA)
D vary D const T vary T const S4 vary S4 const S5 vary S5 const0
50
100
150
200
250
CS
Afo
und
NitpickMleanCoP
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 12
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The penultimate slide
Related workÉ Generic theorem proving systems:
The Logics Workbench,MetTeL2, LoTRECÉ Embedding of further logics:
Conditional logics, hybrid logics, many-valued logics, free logic, ...
ConclusionÉ Provided a quite general semantics for HOMLÉ Presented a procedure that automatically converts HOML into HOLÉ Implemented a stand-alone tool (e.g. as preprocessor)
É standard HOL provers can be used to reason about problems encoded inthe modal THF syntax
É Approach feasible (no evaluation for higher-order problems yet)É Many new problems contributed in the modal THF format
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 13
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The penultimate slide
Related workÉ Generic theorem proving systems:
The Logics Workbench,MetTeL2, LoTRECÉ Embedding of further logics:
Conditional logics, hybrid logics, many-valued logics, free logic, ...
ConclusionÉ Provided a quite general semantics for HOMLÉ Presented a procedure that automatically converts HOML into HOLÉ Implemented a stand-alone tool (e.g. as preprocessor)
É standard HOL provers can be used to reason about problems encoded inthe modal THF syntax
É Approach feasible (no evaluation for higher-order problems yet)É Many new problems contributed in the modal THF format
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 13
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The ultimate slide
Thank you for your attention!
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 14
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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 riμ→μ→o
for each i ∈ I
(3) Connectives:
=
=
=
=
(4) Meta-logical notions:
=,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 15
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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 riμ→μ→o
for each i ∈ I
(3) Connectives:
=
=
=
=
(4) Meta-logical notions:
=,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 15
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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 riμ→μ→o
for each i ∈ I
(3) Connectives:
=
=
=
=
(4) Meta-logical notions:
=,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 15
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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 riμ→μ→o
for each i ∈ I
(3) Connectives:
¬o→o = λSμ→o.λWμ. ¬(SW)
∨o→o→o = λSμ→o.λTμ→o.λWμ. (SW)∨ (T W)
Πτ(τ→o)→o
= λPτ→μ→o.λWμ.∀Xτ . P X W
�o→o = λSμ→o.λWμ.∀Vμ. ¬(ri W V)∨ S V
(4) Meta-logical notions:
=,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 15
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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 riμ→μ→o
for each i ∈ I
(3) Connectives:
¬o→o = λSμ→o.λWμ. ¬(SW)
∨o→o→o = λSμ→o.λTμ→o.λWμ. (SW)∨ (T W)
Πτ(τ→o)→o
= λPτ→μ→o.λWμ.∀Xτ . P X W
�o→o = λSμ→o.λWμ.∀Vμ. ¬(ri W V)∨ S V
(4) Meta-logical notions:
valid = λsμ→o.∀Wμ. s W,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 15
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Embedding of HOML within HOL #2
Embedding semantic variants
1. Axiomatization of �i
2. Quantification3. Rigidity4. Consequence
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 16
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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. Quantification3. Rigidity4. Consequence
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 16
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Embedding of HOML within HOL #2
Embedding semantic variants
1. Axiomatization of �i
Postulate frame condition on ri as HOL axiom
2. QuantificationChoose appropriate definition/axiomatization of quantifier:Constant domains quantifier:
Πτ(τ→o)→o
= λPτ→μ→o.λWμ.∀Xτ . P X W
Varying domains quantifier:
Πτ(τ→o)→o,va = λPτ→μ→o.λWμ.∀Xτ . ¬(eiw X W)∨ (P X W)
Cumulative/decreasing domains quantifier:Add axioms on eiw
3. Rigidity4. Consequence
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 16
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Embedding of HOML within HOL #2
Embedding semantic variants
1. Axiomatization of �i
Postulate frame condition on ri as HOL axiom
2. QuantificationChoose appropriate definition/axiomatization of quantifier
3. RigidityRigid constants:Only translate Boolean types to predicates: o = μ→ o
Rigid constants:Also translate individuals types to predicates: ι = μ→ ι
4. Consequence
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 16
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Embedding of HOML within HOL #2
Embedding semantic variants
1. Axiomatization of �i
Postulate frame condition on ri as HOL axiom
2. QuantificationChoose appropriate definition/axiomatization of quantifier
3. RigidityAppropriate type lifting
4. ConsequenceGlobal 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 = λSμ→o. s wactual and wactual is an uninterpreted symbol
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 16
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Problem representation
Ongoing work: Extension of TPTP THF syntax for modal logic
(1) Formula syntaxthf( classical, axiom, ! [X:$i]: (p @ X)).
↓ Extend syntax with modalitiesthf( modal, axiom, ! [X:$i]: ($box @ (p @ X))).thf( multi_modal, axiom, ! [X:$i]: ($box_int @ 1 @ (p @ X))).
(2) Semantics configurationAdd ”logic”-annotated statements to the problem:
thf(simple_s5, logic, ($modal := [$constants := $rigid,$quantification := $constant,$consequence := $global,$modalities := $modal_system_S5 ])).
É Intended semantics is attached to the problem
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 17
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Problem representation
Ongoing work: Extension of TPTP THF syntax for modal logic
(1) Formula syntaxthf( classical, axiom, ! [X:$i]: (p @ X)).
↓ Extend syntax with modalitiesthf( modal, axiom, ! [X:$i]: ($box @ (p @ X))).thf( multi_modal, axiom, ! [X:$i]: ($box_int @ 1 @ (p @ X))).
(2) Semantics configurationAdd ”logic”-annotated statements to the problem:
thf(simple_s5, logic, ($modal := [$constants := $rigid,$quantification := $constant,$consequence := $global,$modalities := $modal_system_S5 ])).
É Intended semantics is attached to the problem
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 17
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Problem representation
Ongoing work: Extension of TPTP THF syntax for modal logic
(1) Formula syntaxthf( classical, axiom, ! [X:$i]: (p @ X)).
↓ Extend syntax with modalitiesthf( modal, axiom, ! [X:$i]: ($box @ (p @ X))).thf( multi_modal, axiom, ! [X:$i]: ($box_int @ 1 @ (p @ X))).
(2) Semantics configurationAdd ”logic”-annotated statements to the problem:
thf(simple_s5, logic, ($modal := [$constants := $rigid,$quantification := $constant,$consequence := $global,$modalities := $modal_system_S5 ])).
É Intended semantics is attached to the problem
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 17
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Problem representation
Ongoing work: Extension of TPTP THF syntax for modal logic
(1) Formula syntaxthf( classical, axiom, ! [X:$i]: (p @ X)).
↓ Extend syntax with modalitiesthf( modal, axiom, ! [X:$i]: ($box @ (p @ X))).thf( multi_modal, axiom, ! [X:$i]: ($box_int @ 1 @ (p @ X))).
(2) Semantics configurationAdd ”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 ] ] ) ).
É Intended semantics is attached to the problem
,
The World’s Most Widely Applicable Modal Logic Theorem Prover, RuleML Webinar 18