IS
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UN
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Integrated Proof Transformation
Services
Jürgen Zimmer
IV
joint work with Andreas Meier, Geoff Sutcliffe, Yuan Zhang
S
IS
S
SA
R
University of Edinburgh, Scotland
E R SIT
A
UN
Universität des Saarlandes, Germany
A VIE N
*The author is supported by the European Union
C ALCULEMUS IHP Training Network HPRN-CT-2000-00102
c J.Zimmer
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.1
Overview
M ATH S ERV: Semantic Reasoning Web Services
Some Systems Integrated in M ATH S ERV
Theorem Proving and Proof Transformation Services
Brokering of Proof Transformation Services
Conclusion
c J.Zimmer
Future Work
Demo
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.2
The MathServ Framework
MathWeb Software
Bus
Web Services
Experience
Infrastructure
MathServ
Service Combination
AI
Planning
c J.Zimmer
Semantic Markup/
Ontologies
Semantic Web
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.3
The MathServ Framework
A new framework for semantic reasoning services:
Based on Web Service technology.
Semantic markup for web services in the
Mathematical Service Description Language (MSDL):
Developed by MONET and MathBroker project.
Based on commonly agreed ontology.
Brokering mechanism retrieves and combines reasoning
services using modified POP planner.
c J.Zimmer
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.4
Benefits of MathServ
The MathServ framework can be used by humans or
machines to...
retrieve reasoning services (by human machine)
given a semantic description of a problem.
automatically combine services to tackle a problem.
tackle subproblems in automatic or interactive theorem
proving.
No need to know the underlying reasoning system!
c J.Zimmer
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.5
Systems Integrated as Web Services
1) Automated theorem proving systems:
EP, Otter, SPASS.
For classical first-order predicate logic with equality.
2) Otterfier Tool for proof transformation [Sutcliffe’04]:
CNF refutation
CNF refutation (BrFP) calculus
BrFP = Binary resolution, Factoring, Paramodulation
Calls Otter to replace “alien” inference steps
c J.Zimmer
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.6
More Systems Integrated
3) The Tramp system [Meier’00]:
ND proof at assertion level.
FOF problem
+
CNF refutation (BrFP)
DEF
Formal proof
system [Fiedler’01]:
!
4) The
( DEF) :
assertion level step [Huang’94]:
Natural Language (NL) proof.
Proof quality depends on linguistic knowledge.
c J.Zimmer
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.7
An Ontology for Service Descriptions
Thing
proofOf
Proving−Problem
Proof
1..1
logic
calc
1..1
Calculus
Result
1..1
Logic
InFormal−Proof
is−a
FO−Proving−Problem
TSTP
CNF−Problem
NL−Proof
TSTP
FOF−Problem
is−a
Formal−Proof
proof
ND−Proof
CNF−Refutation
Twega
ND−Proof
TSTP−
CNF−Refutation
0..1
FO−ATP−
Result
status
1..1
FO−ATP−
Status
...
Theorem
A
B
p
Property
a..b
i
c J.Zimmer
A subconcept of B
C
TSTP−CNF−
LC−Refutation
...
Satisfiable
Unsatisfiable
p
i individual of C
TSTP−CNF−
BrFP−Refutation
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.8
An ATP Service in MSDL
'
&%
Service:
# "
$
The central part of an MSDL description [MICAI’04]:
problem::TSTP-CNF-Problem (Concept)
output parameters:
result::FO-ATP-Result
post-conditions:
proof(?result, ?proof)
)
pre-conditions:
(
input parameters:
type(?proof, TSTP-CNF-Refutation)
We completely omit XML details.
Conditions in Semantic Web Rule Language (SWRL)
(RDF-triples, conjunction, implications).
c J.Zimmer
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.9
oldResult::FO-ATP-Result
output parameters:
newResult::FO-ATP-Result
pre-conditions:
proof(?oldResult, ?oldProof)
post-conditions:
proof(?newResult, ?newProof)
type(?newProof, TSTP-CNF-BrFP-Refutation)
altProof(?newProof, ?oldProof)
altProof = alternative proof
c J.Zimmer
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.10
5
input parameters:
5
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1
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0
The Otterfier Service in MSDL
<
;
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:
9 8
Service:
67
The Services of Tramp and P.rex
fofProblem::TSTP-FOF-Problem
input parameters:
atpResult::FO-ATP-Result
ndProof::Twega-ND-Proof
pre-conditions:
proof(atpResult, ?proof)
=
output parameters:
type(?proof, TSTP-CNF-BrFP-Refutation)
A 67
@?
>
:
Service:
6B
proofOf(ndProof, fofProblem)
post-conditions:
ndProof::Twega-ND-Proof
output parameters:
nlProof::NL-Proof
post-conditions:
proofOf(?ndProof, ?p)
=
pre-conditions:
C
input parameters:
informalProofOf(?nlProof, ?p)
c J.Zimmer
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.11
Example Conjecture
Scenario: Given a first-order conjecture:
is a group,
,
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and some theory axioms, e.g.:
Queries:
Peter: Give me a first-order ATP system result!
Susan: Give me a ND calculus proof!
Mary: Give me a NL proof!
c J.Zimmer
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.12
c J.Zimmer
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Example: TSTP Encoding
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.13
Peter’s Query: Execution Plan
proofOf
TSTP
FOF
Problem
Clause
Normalizer
sat−equiv
c J.Zimmer
TSTP
CNF
Problem
EP
ATP
FO−ATP−Result
proofOf
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.14
c J.Zimmer
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Peter’s Query: A Resolution Proof
EP delivers a proof with 19 clauses in 31ms.
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.15
Execution plan for Susan’s Query
proofOf
TSTP
FOF
Problem
Clause
Normalizer
sat−equiv
c J.Zimmer
Ep
ATP
proofOf
Otterfier
Service
NDforFOF
Twega−ND−Proof
altProof
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.16
c J.Zimmer
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ND Proof answers Susan’s Query
Tramp’s ND proof with 10 steps (6 assertion level) after 20 seconds:
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Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
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– p.17
Mary’s Query: Execution Plan with P.rex
proofOf
TSTP
FOF
Problem
Clause
Normalizer
sat−equiv
Ep
ATP
proofOf
Otterfier
Service
NDforFOF
PrexND2NL
NL−Proof
altProof
informalProofOf
c J.Zimmer
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.18
c J.Zimmer
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Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
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Mary’s Query: NL Proof
’ NL proof after 80 seconds with BASIC linguistic knowledge:
– p.19
Conclusion
The MathServ framework offers
Semantic retrieval of reasoning services.
Our broker can provide customized execution plans for a
given query.
In Computer-supported theory development:
Interactive theorem proving useful for
closing subgoals.
retrieve useful lemmas (HELM web services).
Proofs in different calculi.
c J.Zimmer
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.20
Ongoing and Future Work
Service Execution
MONET plan executor?
Description of other reasoning systems
(e.g., model generators, decision procedures?).
More fine-grained services (like MONET).
, prove that is prime).
(e.g., given
Advanced brokering with
reasoning on ontology (subsumption test, etc.).
disjunctive plans (or re-planning).
c J.Zimmer
Workshop on Computer-Supported Mathematical Theory Development, Cork, 5th July, 2004
– p.21