Developments and Applicationsautexier/talks/CISA2006.pdf& CISA, University of Edinburgh, Scotland...

The New Proof Assistant ΩMEGA:Developments and Applications

Serge Autexier

(joint work with the ΩMEGA Group∗

∗ Jorg Siekmann, Christoph Benzmuller, Chad E. Brown, Mark Buckley, Dominik Dietrich, Armin Fiedler (part-time), Andreas Franke,

Henri Lesourd, Marvin Schiller, Frank Theiß, Marc Wagner, Jurgen Zimmer

)

[email protected]

DFKI GmbH & CS Department, Saarland University, Saarbrucken, Germany

& CISA, University of Edinburgh, Scotland (May-July)

CISA Seminar, 20th July 2006

Edinburgh, Scotland

CISA Seminar, 20th July 2006 – p.1

Source: Autexier

Old ΩMEGA

Exp

ansio

n

Ab

stra

ctio

n

Abstract Proof Plan

Higher Order Natural Deduction

Proof Object

Struct

ure

Data

VerbalizationProofComputation

Transformation

MULTI

OANTS

TRAMP

P.REX

LOUI

OANTS−R

OMDOC

MBASEMAYA

MATHWEB

ATPs & C

ASs

LEO

COSIE

SAPPER

LEARNOMATIC

Proof

Knowledge−basedProof Planning

Agent−oriented Theorem Proving

ExternalSystems Transformation

Proof

Learning GUI

MathematicalServices

MathsRepositories

MathsDocuments

AssertionRetrieval

OMEGA

Proof-Planning(MULTI, E. Melis, A. Meier)

Agent-based TheoremProving (ΩANTS, V. Sorge)

NL Verbalisation of Proofs(P.rex , A. Fiedler)

External systems (ATP, CAS)(MATHWEB-SB J. Zimmer)

Proof transformation(TRAMP, A. Meier)

Math. Knowledge base(MBASE A. Franke)

OMDOC (M. Kohlhase)


Source: Autexier

Old ΩMEGA

Exp

ansio

n

Ab

stra

ctio

n

Abstract Proof Plan

Higher Order Natural Deduction

Proof Object

Struct

ure

Data

VerbalizationProofComputation

Transformation

MULTI

OANTS

TRAMP

P.REX

LOUI

OANTS−R

OMDOC

MBASEMAYA

MATHWEB

ATPs & C

ASs

LEO

COSIE

SAPPER

LEARNOMATIC

Proof


Agent−oriented Theorem Proving

ExternalSystems Transformation

Proof

Learning GUI


MathsRepositories

MathsDocuments

AssertionRetrieval

OMEGA

Base Calculus: simplytyped HOL, NaturalDeduction

Hierarchical proofdatastructure (PDS) tailoredto

Base ND calculus

Proof planning methods

Tactics

Heavily overcrowded

Graphical user interfaceLΩUI


Source: Autexier

Why and What did we Change?

Natural Deduction Calculus too cumbersome (not natural enough)


Source: Autexier



Want direct reasoning with the assertions (interactive & automatic)

⇒ CORE-calculus that directly supports that [Autexier, CADE’05]


Source: Autexier





The proof datastructure was completely overcrowded


Source: Autexier






New generic proof datastructure (PDS)


Source: Autexier







Definition of the task-layer as uniform reasoning platform


Source: Autexier








GUI LΩUI nice, but did not meet requirements of targeted users,say mathematicians


Source: Autexier









Plug ΩMEGA as reasoning service provider into tools anyway used

by users, e.g. WYSIWYG text-editors like TEXMACS (analogy:

grammar checkers)


Source: Autexier









Plug ΩMEGA as reasoning service provider into tools anyway used

by users, e.g. WYSIWYG text-editors like TEXMACS (analogy:

grammar checkers)

Add development graph manager MAYA to efficiently maintainformal theories and proofs


Source: Autexier

Towards the New ΩMEGA


Source: Autexier

New Architecture


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Rat

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Int

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DevelopmentGraph

Agent−orientedTheoremProving

ComputationTransformation

RetrievalAssertion

Learning

DeveloperGUI

Task Layer

ConstraintSolver

MathematicalRepositoriesProof

Verbalization

LOUI

OANTS

MULTI

LEARNOMATIC

SAPPER

MATHSERV

COSIE

OANTS−R

MBASE

PREX

MAYA


CORE

PLATO

MEDIATOR


Source: Autexier

This Talk


DIALOG

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DevelopmentGraph



RetrievalAssertion

Learning

DeveloperGUI

Task Layer

ConstraintSolver


Verbalization

LOUI

OANTS

MULTI

LEARNOMATIC

SAPPER

MATHSERV

COSIE

OANTS−R

MBASE

PREX

MAYA


CORE

PLATO

MEDIATOR


Source: Autexier

Development graphs:Maintaining structured mathematical

theories


Source: Autexier

Context

Evolutionary Formal Development

Structured Specification

Theorem Prover Structured Internal Representation

Translation intologicalrepresentation

Proof Obligations + Structured Database

Changingspecificationsdue to prooffailures


Source: Autexier

Development Graphs

LIST[ELEM]Local Signature:

· Local sorts List[Elem]

· Local constants [] : List[Elem], . . .

Local Axioms: · [] ++ K = K, . . .

Local Lemmata:

· reverse(K ++ L) = reverse(L) ++ reverse(K)

· . . .

MONOID

Local Signature:

· Local sorts: Elem

· Local constants: ∗ : Elem × Elem → Elem

Local Axioms: x ∗ e = x, . . .

Local Signature:

· Local sorts: Elem

σ

id

[Hutter’00], [AutexierHutter’02&’05], [MossakowskiAutexierHutter’01&’06]CISA Seminar, 20th July 2006 – p.9

Source: Autexier

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SigOrder

basic−42

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BooleanAlgebra

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ExtBooleanAlgebra

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RichBooleanAlgebra

ExtPartialOrder

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TotalOrder

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basic−2

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Source: Autexier

Role of Development Graphs

Development Graph

=

Structured Logical Content of Specifications Structured Specification

+

Status of proof obligations Verification +

(pending, proven, used axioms, . . . ) Management

Deals with evolution of formal specifications and proofs

⇒ Save as much proof work as possible!

Exploits graph structure of formal specifications to reduce effects ofchanges on existing proofs

Difference analysis of formal specifications to determine fine grainedchanges


Source: Autexier

This Talk


DIALOG

daVinci

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union−59

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union−82

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basic−89

SigOrder

basic−51

union−50

basic−49

basic−110

inst−111

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union−107

BooleanAlgebra

union−104

inst−105

union−106

inst−130

union−129

ExtBooleanAlgebra

basic−112

basic−109

inst−120

union−119

inst−121

inst−116

union−115

inst−118

union−117

RichBooleanAlgebra

ExtPartialOrder

basic−87

inst−100

union−99

inst−101

inst−96

union−95

TotalOrder

inst−63

union−62

union−92

inst−93

union−94

inst−127

union−126

ExtTotalOrder

basic−97

RichTotalOrder

union−75

inst−76

inst−78

union−77

Rat

basic−21

inst−20

union−19

Int

basic−16

inst−15

union−14

Nat

basic−11

union−67

inst−68

inst−66

union−65

union−72

inst−73

inst−71

union−70

inst−124

union−123

RichPartialOrder

union−46

inst−47

EquivalenceRelation

inst−45

union−44


inst−42

union−41

SymmetricRelation

inst−29

union−28

union−34

inst−35

SimilarityRelation

inst−37

union−36

inst−40

union−39

DevelopmentGraph



RetrievalAssertion

Learning

DeveloperGUI

Task Layer

ConstraintSolver


Verbalization

LOUI

OANTS

MULTI

LEARNOMATIC

SAPPER

MATHSERV

COSIE

OANTS−R

MBASE

PREX

MAYA


CORE

PLATO

MEDIATOR


Source: Autexier

The Task Layer:Developing and maintaining proofs, proof

plans, proof sketches, and alternatives


Source: Autexier

The TASK LAYER at a Glance

The uniform proof construction interface used by both the human user andthe automated proof search procedures

Instance of the new proof datastructure (PDS)

Tasks = Gentzen-style multi-conclusion sequents + focuses of attentionon subformulas

Proof construction steps (task justifications):

1. introduction of a proof sketch [Wiedijk, TYPES’03]

2. deep structural rules for weakening and decomposition of subformulas

3. the application of a lemma that can be postulated on the fly

4. the application of a CORE calculus rule

5. the application of an inference (⇐ Proof Planning, ΩANTS)


Source: Autexier

The new Proof Datastructure (PDS)


Source: Autexier

Proof Data Structure (PDS)

Proof simultaneously at different levels of granularity

Representation of abstract proof ideas and their refinement

(Proof Planning)

Representation of external systems proofs/computations and theirrefinement

Definable level of granularity (slices through the hierarchy) Views

• Interactive proof development• Adaptive natural language proof explanations

Allows to postpone verification (expansion) of higher-level proof steps

Alternative proof attempts (on the same level of granularity)

Support for lemmatization (forest of PDS trees with links)

[AutexierBenzmüllerDietrichMeierWirth,MKM 2005]


Source: Autexier

Example Abstract PDS

⊢ ¬rat(√

12)

Sketch

rat(√

12) ⊢ ⊥

Sketch

rat(√

12), int(n), int(m),¬commondiv(n, m),√

12 = nm

⊢ ⊥

Sketch

Σ ⊢ div(n, 3) ∧ div(m, 3)

Sketch

Σ ⊢ div(n, 3) Σ ⊢ div(m, 3)

Sketch

Σ ⊢ div(n, 2) ∧ div(m, 2)

Sketch

Σ ⊢ div(n, 2) Σ ⊢ div(m, 2)


Source: Autexier

Example Complete PDSComplete PDS

⊢ ¬rat(√

12)

Sketch

rat(√

12) ⊢ ⊥

Sketch

rat(√


12 = nm

⊢ ⊥

Sketch

Σ ⊢ div(n, 3) ∧ div(m, 3)

Sketch

Σ ⊢ div(n, 3) Σ ⊢ div(m, 3)

Sketch

Σ ⊢ div(n, 2) ∧ div(m, 2)

Sketch

Σ ⊢ div(n, 2) Σ ⊢ div(m, 2)

ApplyLemma(Rat − Criterion)

rat(√

12),∃y:int, z:int√

12 = yz∧ ¬commondiv(y, z) ⊢ ⊥

Decomposition

h

¬Ih


Source: Autexier

Example Complete PDSComplete PDS A PDS View

⊢ ¬rat(√

12)

Sketch

rat(√

12) ⊢ ⊥

Sketch

rat(√


12 = nm

⊢ ⊥

Sketch

Σ ⊢ div(n, 3) ∧ div(m, 3)

Sketch

Σ ⊢ div(n, 3) Σ ⊢ div(m, 3)

Sketch

Σ ⊢ div(n, 2) ∧ div(m, 2)

Sketch

Σ ⊢ div(n, 2) Σ ⊢ div(m, 2)


rat(√



Decomposition

h

¬Ih

⊢ ¬rat(√

12)

Sketch

rat(√

12) ⊢ ⊥


rat(√



Decomposition

rat(√

12), int(n), int(m),¬commondiv(n,m),√

12 = nm

⊢ ⊥

Sketch

Σ ⊢ div(n, 3) ∧ div(m, 3)

Sketch

Σ ⊢ div(n, 3) Σ ⊢ div(m, 3)

Sketch

Σ ⊢ div(n, 2) ∧ div(m, 2)

Sketch

Σ ⊢ div(n, 2) Σ ⊢ div(m, 2)


Source: Autexier

PDSs

Alternative subproofs require to allow for alternative substitutions ofMetavariables =⇒ Mechanism to support that efficiently

[Dietrich, Diploma Thesis, 2006]

PDS provided basis to use λµµ-calculus proof terms as semantics forOMDOC proofs extended by alternatives (= PDS)

[Autexier&Sacerdoti-Coen, MKM’06]

(λµeµ-calculus proposed by Curien and Herbelin, Curry-Howard Isomorphism toclassical SK, allows call-by-value/call-by-name)


Source: Autexier

Proof Construction by Inference Application


Source: Autexier

From Tactics and Methods. . .. . . to Inferences

In the old ΩMEGA system there were

(ΩMEGA-)tactics (interactive theorem proving)

methods (proof planning)

For each of them, again each application direction (AD) had to be specified manually:

P1 : F P2 : U = V

C : G=subst-m-fw(π)

Application Condition: −

Outline functions:

〈C compute-=subst-prem(P1, P2, π)〉

P1 : F P2 : U = V

C : G=subst-m-bw1(π)


Outline functions:

〈P1 compute-=subst-prem(C, P2, π)〉

Having P1 and P2, we can get C (fw)

Having C and P2, we can get P1 (bw1)

Having C and P1, we can get P2 (bw2)

Having P1, P2 and C, we check the

proof step


Source: Autexier

The same as a single Inference. . .

P1 : F P2 : U = V

C : G=subst-m(π)

Application Condition: −Outline functions:

〈C compute-=subst-prem(P1,P2, π)〉〈P1 compute-=subst-prem(C,P2, π)〉

There were 4 tactics and 2 methods in the old ΩMEGA system!

Inferences unify (ΩMEGA-)tactics and methods, overcome manualspecification of ADs


Source: Autexier

Examples of Inferences

P1 : A ⊆ B P2 : B ⊆ C

C : A ⊆ C



Source: Autexier


[ǫ > 0,D > 0, 0 <| x − A |, | x − A |< D]...

P :| F(x) − L |< ǫ

C : limA

F = LLimit(ǫ, x)

Application Condition: EV(ǫ, F,A,L) ∧ EV(x, F,A,L,D)


Source: Autexier


P : O(U,V)

C : O(U′,V′)Maple-Simplify

Application Condition: O ∈ =,≤, <,≥, >Outline functions: 〈U maple-simplify(U′)〉

〈U′ maple-simplify(U)〉〈V maple-simplify(V′)〉〈V′ maple-simplify(V)〉


Source: Autexier

General Form of Inferences[ H1

1 , . . . , H1

n1]

.

.

.

P1 : F1 . . .

[ Hk

1 , . . . , Hk

nk]

.

.

.

Pk : Fk

C1 : G1 . . . Cm : Gm

Name(ω1, . . . , ωn)

Outline Functions: 〈i1, f1(i11, . . . , i1j1)〉 Application Conditions: P(i01, . . . , i0m)

...〈il, f l(il1, . . . , iljl)〉


Source: Autexier

General Form of Inferences[ H1

1 , . . . , H1

n1]

.

.

.

P1 : F1 . . .

[ Hk

1 , . . . , Hk

nk]

.

.

.

Pk : Fk

C1 : G1 . . . Cm : Gm

Name(ω1, . . . , ωn)

Outline Functions: 〈i1, f1(i11, . . . , i1j1)〉 Application Conditions: P(i01, . . . , i0m)

...〈il, f l(il1, . . . , iljl)〉

Procedure to automatically synthesise inferences from arbitrary formulascontained in the mathematical theories.

∀A, B, C : Set.(A ⊆ B ∧ B ⊆ C) ⇒ (A ⊆ C)

∀f, a, l.∀ǫ.ǫ > 0 ⇒ ∃δ.δ > 0 ⇒ ∀x.(0 <| x − a | ∧ | x − a |< δ) ⇒| f(x) − l |< ǫ

⇒ lima f = l

(Had all to be encoded manually in the old ΩMEGA system. . . )

Only the other inferences must be manually encoded (e.g. =subst-m)CISA Seminar, 20th July 2006 – p.26

Source: Autexier

Application of Inferences

Inference are applied on subformulas of a task

Not all premises and conclusions need to be mapped.

Premises are mapped to negative positions in a task

Conclusions are mapped to positive positions in a task

Example:

Φ

ΦSame

A ∧ B, (A ∧ B) ⇒ C ) ⊢ P ⇒ C


Source: Autexier

Application of Inferences

Inference are applied on subformulas of a task

Not all premises and conclusions need to be mapped.

Premises are mapped to negative positions in a task

Conclusions are mapped to positive positions in a task

Example:

Φ

ΦSame

A ∧ B, ( A ∧ B︸︷︷︸conditions

) ⇒ C ) ⊢ P ⇒ C

A ∧ B,A ∧ B ⇒ C ⊢ P ⇒ A ∧ B


Source: Autexier

A more complex example

P ⇒ (A ⊂ B) ⊢ Q ⇒ (A ⊂ C)


Source: Autexier


P ⇒ (A ⊂ B) ⊢ Q ⇒ (A ⊂ C)P1 : (A ⊂ B) P2 : (B ⊂ C)

C : (A ⊂ C)Subset


Source: Autexier


P ⇒ (A ⊂ B) ⊢ Q ⇒ (A ⊂ C)P1 : (A ⊂ B) P2 : (B ⊂ C)

C : (A ⊂ C)Subset

Suppose we map P1 7→ 〈1, 2〉 and C 7→ 〈2, 2〉 (Hence P2 → B ⊂ C).


Source: Autexier


P ⇒ (A ⊂ B) ⊢ Q ⇒ (A ⊂ C)P1 : (A ⊂ B) P2 : (B ⊂ C)

C : (A ⊂ C)Subset

Suppose we map P1 7→ 〈1, 2〉 and C 7→ 〈2, 2〉 (Hence P2 → B ⊂ C).

P ⇒ (A ⊂ B) ⊢ Q ⇒ (P ∧ (B ⊂ C))


Source: Autexier

Interesting Questions

When is an inference applicable?

P1 : F P2 : U = V

C : G=subst-m(π)

Application Condition: −Outline functions:

〈C compute-=subst-prem(P1,P2, π)〉〈P1 compute-=subst-prem(C,P2, π)〉

As soon as we have matched enough premises and conclusions such

that we can compute any missing premise and conclusion using the

outline functions

Application directions = These sets of premises and conclusions=⇒ Sufficient information for the proof planner

=⇒ No need for extra methods as in the old ΩMEGA system

More enhanced static analysis of inferences allows for automaticgeneration of inference-specific agents required by ΩANTS

=⇒ Overcomes manual definition of the agents as in the oldΩMEGA system [Autexier&Dietrich, MKM’06]


Source: Autexier

Comparing wrt. Proofs

old ΩMEGA new ΩMEGA

long proofs

low branching

short proofs

high branching

Decomposition methods become superfluous

LIM+

53 inference applications in total

28 applications of decomposition rules (> 50%)


Source: Autexier

Next Steps

Automation of proof search

Reactivate ΩANTS on that basis

Reimplement proof planner MULTI

Evaluation:

Coding effort is already drastically reduced

Less inferences required by deep application of inferences

Benefit for automated proof search?

Specific services required to support proof development inTEXMACS via PLATΩ


Source: Autexier

Comparison

Old ΩMEGA New ΩMEGA

tactics for math. knowledge manually automatically

tactics for procedures manually manually

application directions manually automatically

agent specification manually automatically

control knowledge manually manually

assertion level simulated directly

OR parallelism no yes

abstract proof operators get independent of logical details


Source: Autexier

This Talk


DIALOG

daVinci

V2.

1

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TransitiveRelation

inst−32

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union−25

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union−55

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PreOrder

union−59

inst−60

PartialOrder

union−82

inst−83

union−84

inst−86

union−85

inst−90

basic−89

SigOrder

basic−51

union−50

basic−49

basic−110

inst−111

inst−108

union−107

BooleanAlgebra

union−104

inst−105

union−106

inst−130

union−129

ExtBooleanAlgebra

basic−112

basic−109

inst−120

union−119

inst−121

inst−116

union−115

inst−118

union−117

RichBooleanAlgebra

ExtPartialOrder

basic−87

inst−100

union−99

inst−101

inst−96

union−95

TotalOrder

inst−63

union−62

union−92

inst−93

union−94

inst−127

union−126

ExtTotalOrder

basic−97

RichTotalOrder

union−75

inst−76

inst−78

union−77

Rat

basic−21

inst−20

union−19

Int

basic−16

inst−15

union−14

Nat

basic−11

union−67

inst−68

inst−66

union−65

union−72

inst−73

inst−71

union−70

inst−124

union−123

RichPartialOrder

union−46

inst−47

EquivalenceRelation

inst−45

union−44


inst−42

union−41

SymmetricRelation

inst−29

union−28

union−34

inst−35

SimilarityRelation

inst−37

union−36

inst−40

union−39

DevelopmentGraph



RetrievalAssertion

Learning

DeveloperGUI

Task Layer

ConstraintSolver


Verbalization

LOUI

OANTS

MULTI

LEARNOMATIC

SAPPER

MATHSERV

COSIE

OANTS−R

MBASE

PREX

MAYA


CORE

PLATO

MEDIATOR


Source: Autexier

Mediating between Texteditors and ProofAssistance Systems

–The PLATΩ System


Source: Autexier

Plato at a Glance

The user authors his mathematical documents with a scientificWYSIWYG text-editor

in the informal language he is used to, that is a mixture of naturallanguage and formulas

using semantic annotations (PL) that follow the textual structure


Source: Autexier

Plato at a Glance


Generate from this informal semantic representation the correspondingformal representation for a proof assistant


Source: Autexier

Plato at a Glance


The primary task of PLATΩ(Marc Wagner, Diploma Thesis, ?July 2006?)

Maintain consistent formal (PA) and informal representations(Text-editor)

Relay service requests from the Text-editor to the PA

Propagate changes from the PA to the Text-editorCISA Seminar, 20th July 2006 – p.35

Source: Autexier

The initial Document in the Te xt-Editor


Source: Autexier

Uploaded Theory in the M AYA system


Source: Autexier

Uploaded Proof in the T ASK LAYER


Source: Autexier

User-Modified Document in the Text-Editor


Source: Autexier

Transformed User-Modified Proof inthe TASK LAYER


Source: Autexier

Menu Interaction in the Text-Editor


Source: Autexier

Patched Menu in the Text-Editor


Source: Autexier

System-Modified Proof in the T ASK

LAYER


Source: Autexier

Patched Document in the Text-Editor


Source: Autexier

My work during the visit to the DREAM Group:

“Automated Reasoning in Large StructuredTheories”


Source: Autexier

Motivation

daVinci

V2.

1

inst−45

inst−17

ReflexiveRelation

inst−47

union−48

PreOrder

union−50

inst−51

PartialOrder

union−73

inst−74

union−75

inst−77

union−76

inst−81

basic−80

SigOrder

basic−42

basic−101

inst−102

inst−99

inst−96

union−97

inst−121

ExtBooleanAlgebra

basic−103

basic−100

inst−111

inst−112

inst−107

inst−109

union−108

RichBooleanAlgebra

ExtPartialOrder

basic−78

inst−91

union−90

inst−92

inst−87

union−86

TotalOrder

inst−54

union−53

union−83

inst−84

union−85

inst−118

union−117

ExtTotalOrder

basic−88

RichTotalOrder

inst−115

union−114

RichPartialOrder

union−37

inst−38

EquivalenceRelation

inst−36

union−35


inst−33

union−32

SymmetricRelation

inst−20

union−25

inst−26

SimilarityRelation

inst−28

union−27

inst−31

ATP(Proof Planner)

Proofor

Fail

Failures of automatic proof search procedures (e.g. classical ATP, ProofPlanner, . . . ) and reasons:

Too much knowledge/Search space too large (Restrict Knowledge)

Not a theorem

Only part of knowledge has been provided:

(Adjust provided knowledge)

All knowledge has been provided: (Definitely not a theorem)CISA Seminar, 20th July 2006 – p.46

Source: Autexier

Question

How to automate restriction and adjustment of knowledge provided to theATPs? (Based on Feedback provided in case of failure)

daVinci

V2.

1

inst−45

inst−17

ReflexiveRelation

inst−47

union−48

PreOrder

union−50

inst−51

PartialOrder

union−73

inst−74

union−75

inst−77

union−76

inst−81

basic−80

SigOrder

basic−42

basic−101

inst−102

inst−99

inst−96

union−97

inst−121

ExtBooleanAlgebra

basic−103

basic−100

inst−111

inst−112

inst−107

inst−109

union−108

RichBooleanAlgebra

ExtPartialOrder

basic−78

inst−91

union−90

inst−92

inst−87

union−86

TotalOrder

inst−54

union−53

union−83

inst−84

union−85

inst−118

union−117

ExtTotalOrder

basic−88

RichTotalOrder

inst−115

union−114

RichPartialOrder

union−37

inst−38

EquivalenceRelation

inst−36

union−35


inst−33

union−32

SymmetricRelation

inst−20

union−25

inst−26

SimilarityRelation

inst−28

union−27

inst−31

ATP(Proof Planner)

Proofor

Fail

TheoryFormation

(MathsAid)

ATLAS *

∗The titan in the Greek mythology punished by Zeus by giving him the task to hold up the skyCISA Seminar, 20th July 2006 – p.47

Source: Autexier

Main Objectives

System environment that supports the definition and evaluation ofknowledge filters and in-the-large reasoning procedures.

Criteria characterising the knowledge that can be provided byATP, ATF and systems maintaining structured theories

Basic set of filters to formulate queries for each subsystems

Criteria to rate the response given by a subsystem to a query.

Sample in-the-large reasoning heuristics:

Given the goal to prove a conjecture

Automate cooperation among the subsystems by formulatingqueries, evaluating responses and formulating new queries.


Source: Autexier

Status

Interfacing systems:

MAYA (DG), MathServ (ATP) , MATHsAid (ATF)

Sounds easy, is simple in principle, but tedious and long inpractice

Different logics, but especially syntax restrictions

Basic set of 4 filters (Goal dependent and adjustable)Theory distance, Definitorial hierarchy, Reformulation distance,

Proving importance

Formalised Class, Set, Relations, Functions, and Groupsfollowing closely Hungerford’s “Algebra” (in CASL)

(69 axioms, 42 conjectures in 6 theories)


Source: Autexier

Status

ATLAS state-machineCHANGE−FILTERCHANGE−GRAVITY

CHANGE−FILTERCHANGE−GRAVITY

SWITCH−MODE

SWITCH−MODE

Prove Explore

OuterLoop

InitializationResult

NextGoal?

No Next Goal or all proven

parameterised overspecific heuristics foreach state to decide

Its running, someerrors in MATHSERVon some inputs

Sofar 2 out of 42 can beproven by ATP alone

TODO:

Find a heuristic combining ATP and ATF that is better thanATF and ATP alone


Source: Autexier

Summary Development Graphs to maintain formal specifications and proofs

Hierarchical proof datastructure that allows for alternative proofs ondifferent levels of granularity, manages alternative substitutions

Unifying tactics and methods into inferences

deep application of inferences (inspired by CORE’s assertionapplication)

automatic synthesis of inferences from mathematical knowledge automatic computation of additional information required by proof

planner and ΩANTS

PLATΩ tool to plug ΩMEGA into texteditors:

mediates between text structure and logical structure, relays servicerequests and propagates changes (ask for screenshots of a workedexample)

ATLAS Project: “Automated Reasoning in Large Structured Theories”CISA Seminar, 20th July 2006 – p.51

Developments and Applicationsautexier/talks/CISA2006.pdf& CISA, University of Edinburgh, Scotland...

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Transcript of Developments and Applicationsautexier/talks/CISA2006.pdf& CISA, University of Edinburgh, Scotland...