Towards Natural Style for Resolution Proofs in Theorema

1
Towards Natural Style for Resolution Proofs in
Theorema

• Diana Dubu
• West University of Timisoara
• eAustria Research Institute
• Supervisor
• Prof. Dr. Tudor Jebelean

2
Presentation Outline

• Work context
• Motivation
• Tool support
• Approach
• Notions
• My work
• Example
• Future work...

3
Presentation Outline

• Work context
• Motivation
• Tool support
• Approach
• Notions
• My work
• Example
• Future work...

4
Research Environment

• Scholarship at the Research Institute for
  Symbolic Computation (RISC), Johannes Kepler
  Universität, Linz, Austria
• Program Coordinator Prof. Dr. Tudor Jebelean
• Attendance at the Automated Theorem Proving II
  lecture and Theorema Seminars

• Further collaborations with Theorema Group members

5
Presentation Outline

• Work context
• Motivation
• Tool support
• Approach
• Notions
• My work
• Example
• Future work...

6
Why Natural Style?

• Problems with proofs generated by clausa
  reasoners
• too long (intermediary steps)
• machine-oriented formalism
• hard to follow by the user (even if experienced)
• different system representations
• Solution
• construct a uniform machine-independent
  representation
• translate machine-found proofs into a
  human-comprehensible format

7
Why Natural Style?

• Problems with proofs generated by clausa
  reasoners
• too long (intermediary steps)
• machine-oriented formalism
• hard to follow by the user (even if experienced)
• different system representations
• Solution
• construct a uniform machine-independent
  representation
• translate machine-found proofs into a
  human-comprehensible format

8
Presentation Outline

• Work context
• Motivation
• Tool support
• Approach
• Notions
• My work
• Example
• Future work...

9
Toolkit

• Theorema
• built on top of Mathematica at RISC by the
  Theorema Research Group initiated by Prof. Dr.
  Bruno Buchberger
• integrates the computing capabilities of a CAS
  with the deduction capabilities of ATPs
• interacts with the user in the language of
  predicate logic (the natural language for
  expressing mathematical properties and algorithms)

10
Proving in Theorema

• methods for several mathematical domains
• propositional logic
• general predicate logic
• induction over integers and over lists
• set theory
• boolean combinations of polynomial
  inequalities (using Groebner Bases)
• combinatorial summation (using PauleSchornZeilbe
  rger)
• PCS (provingcomputingsolving) for proving in
  higher-order logic with equality Buchberger

11
Tma Proof Object

• generated as a result of the proof
• contains the proof tree - information about
• assumptions list
• subgoals at each proof step
• formulae used at each step
• formulae generated at each step
• status (proved, failed, pending)

12
Tma Proof Object

• generated as a result of the proof
• contains the proof tree - information about
• assumptions list
• subgoals at each proof step
• formulae used at each step
• formulae generated at each step
• status (proved, failed, pending)

13
Tma Proof Object

• generated as a result of the proof
• contains the proof tree - information about
• assumptions list
• subgoals at each proof step
• formulae used at each step
• formulae generated at each step
• status (proved, failed, pending)

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Example - TmaProofObject

• TheoremaProversCommonProofObjectPrivateAndNod
  e TheoremaProversCommonProofObjectPrivat
  eProofInfo"DoneMatching", usedFormulae,
• generatedFormulae, TheoremaProversCo
  mmonProofObjectPrivateSubgoals
  TheoremaProversCommonProofObjectPrivateAndNod
  eTheoremaProversCommonProofObjectPrivateProo
  fInfo "ConclusionIsAssumption", usedFormulae
  "Proposition (3)", "Proposition (1)/1",
  generatedFormulae, TheoremaProversCommonPro
  ofObjectPrivateSubgoals, TheoremaProversComm
  onProofObjectPrivateMainProofSituation
  lf"Proposition (3)", MORTALConfucius,
  finfo, asmllf "Proposition (1)/1",
  MORTALConfucius, finfo"", lf"Proposition
  (1)", ForAllrangesimpleRangevarx,
  True, ImpliesMANvarx,
• MORTALvarx, finfo"",
  lf"Proposition (2)", MANConfucius,
  finfo"", lkTab"ProversHistory", PND,
  "LastProver", PND, "PND", lkTab"ModusPonen
  sFacts",
• lkTab, "NewFormulae",
  , TheoremaProversPredicateLogicAuxiliaryPri
  vateoldForms, "GoalHistory",
• MORTALConfucius,
  "MatchingFacts", lkTab"Proposition (1)",
  "Proposition (2)", True,
  TheoremaProversCommonProofObjectPrivateConstr
  aints, "proved", TheoremaProversCommonProof
  ObjectPrivateMainProofSituationlf"Proposition
  (3)",
• MORTALConfucius, finfo,
  asmllf"Proposition (1)", ForAll
  rangesimpleRangevarx, True,
• ImpliesMANvarx, MORTAL
  varx, finfo"", lf"Proposition (2)",
  MANConfucius,
• finfo"", lkTab"ProversHistory",
  , "LastProver", PND, "PND",
  lkTab"ModusPonensFacts", lkTab,
  "MatchingFacts", lkTab, "NewFormulae",
  , TheoremaProversPredicateLogicAuxiliaryPri
  vateoldForms, "GoalHistory",
  MORTALConfucius, TheoremaProversComm
  onProofObjectPrivateConstr\
• aints, "proved"

15
Example - TmaProofObject

• TheoremaProversCommonProofObjectPrivateAndNod
  e TheoremaProversCommonProofObjectPrivat
  eProofInfo"DoneMatching", usedFormulae,
• generatedFormulae, TheoremaProversCo
  mmonProofObjectPrivateSubgoals
  TheoremaProversCommonProofObjectPrivateAndNod
  eTheoremaProversCommonProofObjectPrivateProo
  fInfo "ConclusionIsAssumption", usedFormulae
  "Proposition (3)", "Proposition (1)/1",
  generatedFormulae, TheoremaProversCommonPro
  ofObjectPrivateSubgoals, TheoremaProversComm
  onProofObjectPrivateMainProofSituation
  lf"Proposition (3)", MORTALConfucius,
  finfo, asmllf "Proposition (1)/1",
  MORTALConfucius, finfo"", lf"Proposition
  (1)", ForAllrangesimpleRangevarx,
  True, ImpliesMANvarx,
• MORTALvarx, finfo"",
  lf"Proposition (2)", MANConfucius,
  finfo"", lkTab"ProversHistory", PND,
  "LastProver", PND, "PND", lkTab"ModusPonen
  sFacts",
• lkTab, "NewFormulae",
  , TheoremaProversPredicateLogicAuxiliaryPri
  vateoldForms, "GoalHistory",
• MORTALConfucius,
  "MatchingFacts", lkTab"Proposition (1)",
  "Proposition (2)", True,
  TheoremaProversCommonProofObjectPrivateConstr
  aints, "proved", TheoremaProversCommonProof
  ObjectPrivateMainProofSituationlf"Proposition
  (3)",
• MORTALConfucius, finfo,
  asmllf"Proposition (1)", ForAll
  rangesimpleRangevarx, True,
• ImpliesMANvarx, MORTAL
  varx, finfo"", lf"Proposition (2)",
  MANConfucius,
• finfo"", lkTab"ProversHistory",
  , "LastProver", PND, "PND",
  lkTab"ModusPonensFacts", lkTab,
  "MatchingFacts", lkTab, "NewFormulae",
  , TheoremaProversPredicateLogicAuxiliaryPri
  vateoldForms, "GoalHistory",
  MORTALConfucius, TheoremaProversComm
  onProofObjectPrivateConstr\
• aints, "proved"

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Example - TmaProofObject

• TheoremaProversCommonProofObjectPrivateAndNod
  e TheoremaProversCommonProofObjectPrivat
  eProofInfo"DoneMatching", usedFormulae,
• generatedFormulae, TheoremaProversCo
  mmonProofObjectPrivateSubgoals
  TheoremaProversCommonProofObjectPrivateAndNod
  eTheoremaProversCommonProofObjectPrivateProo
  fInfo "ConclusionIsAssumption", usedFormulae
  "Proposition (3)", "Proposition (1)/1",
  generatedFormulae, TheoremaProversCommonPro
  ofObjectPrivateSubgoals, TheoremaProversComm
  onProofObjectPrivateMainProofSituation
  lf"Proposition (3)", MORTALConfucius,
  finfo, asmllf "Proposition (1)/1",
  MORTALConfucius, finfo"", lf"Proposition
  (1)", ForAllrangesimpleRangevarx,
  True, ImpliesMANvarx,
• MORTALvarx, finfo"",
  lf"Proposition (2)", MANConfucius,
  finfo"", lkTab"ProversHistory", PND,
  "LastProver", PND, "PND", lkTab"ModusPonen
  sFacts",
• lkTab, "NewFormulae",
  , TheoremaProversPredicateLogicAuxiliaryPri
  vateoldForms, "GoalHistory",
• MORTALConfucius,
  "MatchingFacts", lkTab"Proposition (1)",
  "Proposition (2)", True,
  TheoremaProversCommonProofObjectPrivateConstr
  aints, "proved", TheoremaProversCommonProof
  ObjectPrivateMainProofSituationlf"Proposition
  (3)",
• MORTALConfucius, finfo,
  asmllf"Proposition (1)", ForAll
  rangesimpleRangevarx, True,
• ImpliesMANvarx, MORTAL
  varx, finfo"", lf"Proposition (2)",
  MANConfucius,
• finfo"", lkTab"ProversHistory",
  , "LastProver", PND, "PND",
  lkTab"ModusPonensFacts", lkTab,
  "MatchingFacts", lkTab, "NewFormulae",
  , TheoremaProversPredicateLogicAuxiliaryPri
  vateoldForms, "GoalHistory",
  MORTALConfucius, TheoremaProversComm
  onProofObjectPrivateConstr\
• aints, "proved"

17
Theorema Proof Notebook

• A\Link1\-home-info-www-people-knakagaw-ex-indexln
  k2.htm
• Predicate Logic Prover
• Prove
• (Proposition (6))((Q)\Or(R)),
• under the assumptions
• (Proposition (1))P\OrQ,
• (Proposition (2))Q\OrR,
• (Proposition (3))R\OrW,
• (Proposition (4))(R)\Or(P),
• (Proposition (5))(W)\Or(Q).
• We prove (Proposition (6)) by contradiction.
• We assume
• (1)(Q)\Or(R),
• and show a contradiction.
• We prove (a contradiction) by case distinction
  using (1).
• Case (1.1) Q
• We delete (Proposition (5)) because it is
  subsumed by (1.1).
• From (1.1) and (Proposition (1)) we obtain by
  resolution
• (2)P.

• From (5) and (Proposition (5)) we obtain by
  resolution
• (6)W.
• From (1.2) and (Proposition (3)) we obtain by
  resolution
• (7)W.
• Formula (a contradiction) is proved because (7)
  and (6) are contradictory.
• Additional Proof Generation Information
• The Proof Call
• ProveProposition"6",
• using\RuleProposition"1",Proposition"2",P
  roposition"3",
• Proposition"4",Proposition"5",SearchDep
  th-gt35
• Formulae Occuring during the Proof
• (1)(Q)\Or(R)
• (1.1)Q
• (1.2)R
• (2)P
• (3)R

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and...

• Otter
• a resolution-style theorem proving program for
  first order with equality
• includes the inference rules binary resolution,
  hyperresolution, UR-resolution and binary
  paramodulation
• transforms formulae into normal form
• there is a direct link with Theorema

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Theorema and Otterblack box link
Translating component
T h e o r e m a Proof in a notebook a
TranslatorStep 2
ExternalSystemStep 4
Theorema callProve
Step 3
Step 1
Step 5
Linking component to external system
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Theorema and Otterwhite box link
Translating component
T h e o r e m a Proof in a notebook a
TranslatorStep 2
ExternalSystemStep 4
Theorema callProve
Step 3
Step 1
BackTranslatorStep 6
Step 5
Step 7
Linking component to external system
21
Presentation Outline

• Work context
• Motivation
• Tool support
• Approach
• Notions
• My work
• Example
• Future work...

22
How?

• Understand underlying mechanisms of proving in
  Theorema
• Understand the interaction with external provers
  (i.e. Otter) of Theorema
• Study the current status of research w.r.t.
  Natural Style (i.e. Transformation of
  Machine-Found Proofs into Assertion Level Proofs,
  Andreas Meier)

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Other Approaches1.

• Search for an optimal proof by applying the
  transformation rules on-the-fly
• Rewrite-rules (Buchberger, Jebelean)
• S-decomposition in Jebelean

24
Other Approaches2.

• Transform Resolution proofs into Natural
  Deduction proofs
• Andrews, Miller, Schmitt Kreitz,
  Lingenfelder
• Problems
• many levels of indirect parts
• Cause use of ND-rules for eliminating
  quantifiers and connectors
• representation
• Cause a large number of low-level syntactical
  manipulations of logical quantifiers and
  connectives

25
This approach

• Andreas Meier -gt perform transformations at the
  assertion level
• Assertions
• theorems and definitions

26
Schemata of the Approach
Theorema
TmaProofObject
Notebook
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Which proofs?

• Proofs obtained by resolution through refutation
• A set of clauses is unsatisfiable
  (inconsistent) iff there is a resolution
  deduction of the empty clause ? from S.

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Refutation

• Definition
• A refutation of ? is a derivation in which some
  finite subset of ground formulas is unsatisable.
• (? a finite set of closed formulae in normal
  form)
• How?
• Add negated goal to the set of axioms and prove
  the inconsistency of the new set by producing the
  empty clause through a set of inferences.

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Resolution Principle

• (Robinson, 1965) For any two clauses C and D,
  if there is a literal L1 in C that is
  complementary to a literal L2 in D, then delete
  L1 and L2 from C and D, respectively, and
  construct the disjunction of the remaining
  clauses. The constructed clause is a resolvent
  of C and D (Chang and Lee, Symbolic Logic and
  Mechanical Theorem Proving)

30

• Problem
• Proofs format vary with the systeme
  (automated theorem prover)
• Solution
• Find a common representation of machine-found
  proofs.
• Refutation Graphs
• (Transformation of Machine-Found Proofs into
  Assertion Level Proofs, Andreas Meier)

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Presentation Outline

• Work context
• Motivation
• Tool support
• Approach
• Notions
• My work
• Example
• Future work...

32
Definitions

• Clause graph - a quadruple G (L,C,MLit,?),
  where
• L is a finite set elements are literal nodes
• C?2L is a partition of the set of literal nodes
  elements are clause nodes of G
• MLit is a mapping from L to the set of literals,
  labelling literal nodes with literals
• ?, the set of links, is a partition of a subset
  of L, s.t. for all ??? the following hold
• ?1 All the literal nodes in one link are labeled
  with literals whos atoms are unifible
• ?2 There must be at least one positive shore and
  one negative literal literal in a link

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• Literal nodes not belonging to any link are
  called pure
• Each link ? has two opposite shores a positive
  shore S(?) and a negative shore S-(?) (i.e.
  literal nodes with positive and negative
  literals, respectively)
• trail - a walk in which all links are distrinct
  joins start and end clause nodes a trail to a
  link ? - a trail whose last clause has a literal
  in ?
• cycle - a trail joining a clause node to itself
  a graph with such a cicle is called cyclic

34

• Deduction graph - a non-empty, ground (i.e. all
  literals are ground) and acyclic clause graph
• Refutation graph a deduction graph without pure
  literal nodes
• Minimal deduction (refutation) graph - one
  containing no proper subgraph which is itself a
  deduction (refutation) graph)

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Presentation Outline

• Work context
• Motivation
• Tool support
• Approach
• Notions
• My work
• Example
• Future work...

36
How to obtain refutation graphs?

• Call Otter for proving by resolution (automatic
  transformation in normal form)
• Retrieve the result in TmaProofObject
• Parse TmaProofObject and extract relevant
  information (used formulae, generated formulae)
• Build the data structure

37
Algorithm

• C1C2 Cn - initial set of clauses -
  represent the nodes in the refutation graph
• Identify the resolvents R1R2 Rm generated
  by the inference rules
• Determine the links in the refutation graph
• 1. Identify in each resolvent Ri from the set
  R1R2 Rm which original clause has been
  used. For the inferences using resolvents to
  generate new ones, identify from which original
  clauses have the former been generated
• 2. Extract from the initial clauses the literals
  remaining after the resolution step and connect
  them s. t. each link has a positive shore and a
  negative one.

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Remarks

• It is possible that a literal in Ri originates
  from more than one initial clause ?all labels are
  stored s.t. all possible links between the nodes
  of the refutation graph are established
• Parsing has been performed on strings (black box
  link)

39
Presentation Outline

• Work context
• Motivation
• Tool support
• Approach
• Notions
• My work
• Example
• Future work...

40
Simple example
prove

• Considering

Q ? R, R? (P? Q), P? (Q? R)
P? Q
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Refutation graph
1
6
4
1.1
1.2
6.2
6.1
2
4.1
4.2
2.1
2.2
3
5
r
3.2
3.1
5.3
5.1
5.2
Theorem A deduction graph is minimal iff it
has one more clauses as links. Eisinger
42
Data Structure
43
Presentation Outline

• Work context
• Motivation
• Tool support
• Approach
• Notions
• My work
• Example
• Future work...

44
Future directions

• Implementation of transformation procedure in
  Theorema
• Extend the analysis to predicate logic
• Analize proofs obtained from other provers or
  with other proving methods

45
Whats next?

• Analyse and transform the refutation graphs s.t.
  proofs are simplified
• How?
• The solution offered by Andreas Meier
• with possible(?) improvements

46
Definitions

• Unit Clause Step (UCS)
• G - refutation graph. AC, UC1,, UCn, Rlit is
  a UCS in G if
• AC, UC1,, UCn - clauses in G, Rlit - literal of
  AC
• UC1,, UCn - unit clauses, AC - not an unit
  clause
• each literal of AC (except Rlit) - linked with
  the some unit clausess literal of UC1,, Ucn
• UC1,, UCn - unit clauses , AC - assertion
  clause, Rlit - result literal of the UCS

47
UCS - Replacement

• G - refutation graph. AC, UC1,, UCn, Rlit a
  UCS in G, UCnew a new unit clause consisting of
  Rlit. UCS-replacement
• Remove AC from G
• Add UCnew to G
• ? - link connecting Rlit of AC. If ? was removed
  at first step, add ? connecting Rlit of UCnew
  and literals connected by ?. Otherwise, add Rlit
  of UCnew to ?.
• Each of UC1,, UCn, whose literal became pure at
  1st step is removed

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UCS-Decomposition Algorithm

• G - refutation graph
• Initialization-step D(G) si GcurrG
• UCS - replacement step While Gcurr is not an end
  step
• Seek and UCS ? in Gcurr
• Replace ? in Gcurr and assign Gcurr to the
  resulting refutation graph
• Assign D(G)D(G) ? ?
• If there is no UCS in Gcurr stop with error
  message
• Final step If Gcurr has the form of an end step
  S, assign D(G)D(G) ? S and finish

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Obtain UCS-decomposable graphs
Liquidation-Strategy with the Direct-Decomposition
Method
Transformation rules
?
Not minimal gt ? is liquidated
?
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Obtain UCS-decomposable graphs

• Separation Strategy with the Direct Decomposition
  Method

Transformation rules
51
Obtain UCS-decomposable graphs

• Liquidation Strategy with the Symetrical
  Simplification Method

Transformation rules
52
Obtain UCS-decomposable graphs

• Separation Strategy with the Symetrical
  Simplification Method

Transformation rules
53
Thank you