Can Proofs be Animated by Games

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Can Proofs be Animated by Games?

• Susumu Hayashi
• Humanistic Informatics
• Graduate School of Letters
• Kyoto University
• April 22, 2005, TLCA05, Nara, Japan

2
What is the talk about?

• The subject is
• 1-backtracking game
• A join work with S. Berardi and Th. Coquand.

3
1-backtracking game semantics

• A restriction of the full backtracking game
  semantics, introduced by Th. Coqunad in 1991-2
  , 1995.
• Coquand introduce a form of 1-backtracking game
  already in 1991-2

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Game semantics for PCF?

• No! It is a semantics for logic.
• However, it seems related to game semantics of
  PCF and related calculi.
• It is conjectured that Coquands semantics is
  isomorphic to J. Lairds game semantics for
  PCFControl, which is an extension of the game
  semantics by Hyland-Ong. (S. Berardi)

5
A different motivation

• Although our semantics is likely to be related to
  the game semantics by Hyland-Ong and Laird, our
  motivation is not full-abstraction.
• Our motivation is Proof Animation.

6
Proof Animation
Proof Score Method for CafeOBJ by Futatsugi et
al. is a typical example of Proof Engineering.
(This afternoon at WRS 05.)

• A technique of Proof Engineering.
• Proof Engineering is my terminology for the
  engineering to build formal proofs, e.g., the
  researches and activities in the projects of
  CafeOBJ, Coq, HOL, Mizar, PVS,

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An example of Proof Animation
-- ASSUMPTION -- There is a bag. And some white
or black marbles are in it.
-- CONCLUSION -- All marbles in the bag are of
the same color.
This is wrong.
However, we prove it by mathematical induction!
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Proof of the theorem

• Base case n1 is easy
• The induction step

• The theorem holds for groups A and B, since they
  have only n marbles. All the marbles are of the
  same color, since they share an.

What is wrong?
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The proof is constructive and executable.
A wrong lemma was used!groups A and B share a
marble. You can introduce the wrong lemma as a
subgoal and prove the theorem formally with a
proof checker. Then
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Proof animation helps to debug formal
constructive proofs

• The proof was constructive and the wrong lemma
  was detected quickly by executing the proof by
  Curry-Howard isomorphism.
• I often used such a technique in my PX project
  in 1980s. I could very quickly find bugs in
  definitions, goals and subgoals by the technique.
  
• PX was a constructive proof animator.

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Proof animation project

• Build a proof animator which helps formal proof
  developments not only for constructive
  mathematics but also for proof developments in
  general.
• We must find a means to execute non-constructive
  proofs.

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Proof animator for non-constructive proofs?

• Classical proofs are not directly executable.
• However, there are many works to execute
  classical proofs CPS translations,
  C-combinator, lm-calculus,

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Constructive interpretations of classical proofs
are inadequate

• These works are theoretically good, but are not
  adequate for proof animation.
• Locally legible each computation step in these
  semantics is legible enough.
• Globally illegible interpretations of proofs
  with several steps combinatorially explode.
  Algorithms resulting from even small proofs
  cannot be understood.

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An important REMARK

• The global illegibility is not bad for logicians.
• If the aim is to unwind classical proofs, such as
  works by logicians Kreisel, Kohlenbach, and
  Schwichtenberg, then the illegibility implies
  non-triviality of their mathematical works.
• However, our aim is a technology of proof
  engineering. If one can write an academic paper
  when he or she could execute a proof by a method
  executing classical proofs, then the method is
  bad for proof animation.

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What we need for proof animation

• We need a lightweight method executing proofs in
  everyday proof developments.
• A tool for proof animation must be easy to use as
  a test tool for programming languages.
• Its underlying theory must be easy to understand.
  It is a tool, not an objective.

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A solution Inductive inference from Learning
Theory

• Algorithmic Learning Theory a discipline to
  investigate machine learning from the viewpoint
  of theory of computation. (a.k.a. computational
  learning theory)
• Inductive inference the oldest mathematical
  definition of learning in algorithmic learning
  theory.

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An example of learning process by inductive
inference (1)

• MNP (Minimal Number Principle)Let f be a
  function from Nat to Nat. Then, there is n Nat
  such that f(n) is the smallest value among
  f(0), f(1), f(2), Nat the set of natural
  numbers

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An example of learning process by inductive
inference (2)

• Such an n is not Turing-computable from f.
• However, the number n is inferred in finite time
  from f by a non-stopping algorithm of inductive
  inference.

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The inductive inference algorithm for MNP

• Consider a box containing a natural number.
  Denote the content of the box by x.
• Initialize the box by setting x0.
• Regard f , as a stream f(0), f(1), f(2),
• Compare f(x) with the next element of the stream,
  say f(n). If the new one is smaller than f(x),
  then put n in the box. Otherwise, keep the old
  value in the box.
• Repeat it forever.

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It gives the right answer in finite time

• We have a sequence of natural numbers
  f(n0)gtf(n1)gtf(n2)gt
• Thus, the content of the box will eventually
  become a correct answer and after then the
  content x will never change.
• In this sense, the non-terminating process
  infers (or learns) the right answer in finite
  time.
• You will eventually get a right answer, although
  you will never know when you got it.

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Limit-computable functions

• The process inferring x is expressed by the
  limit
• lim n 8 h(n) x
• The functions defined by g(x)lim n 8 f(n,x),
  for a recursive function f, are called
  limit-computable functions.
• The limit-computable functions coincide with the
  D02-functions.

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Logic based on limit-computable functions

• Semantics of constructive mathematics is given by
  the realizability interpretation based on
  recursive functions.
• The D02-functions constitute a domain of abstract
  recursion theory.
• Thus, we may replace recursive functions with
  D02-functions to define a mathematics.
• The defined mathematics is called
  Limit-Computable Mathematics (LCM)

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Execution of LCM proofs

• All proofs of LCM are executable by
  non-stopping inductive inference algorithms.
• We can observe that LCM-proofs perpetually
  approximate right answers, and eventually reach
  right answers.

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What kind of mathematics holds in LCM?

• Not all classical theorems hold. For example,
  Law of Excluded Middle holds for S01-formulas but
  not for S02-formulas.
• However, an unexpectedly large fragment of
  classical theorems hold.
• Dixons lemma, Hilberts invariant theory,
  Gödel's completeness theorem, Hahn-Banach
  theorem,
• There are reverse mathematics-like researches on
  the extent of LCM. (Akama et al. LICS 04,
  Toftdal ICALP 04. in the references of the
  proceedings paper.)

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It looks fine, however...A technical problem

• If proofs are interpreted by limits over time
  parameter t0,1,2, as the original theory of
  inductive inference, then plural inductive
  inference processes are merged into one process
  to interpret logical inference rules with plural
  premises.
• The merged inference process behaves like a CPU
  executing plural programs in the time-sharing
  way.
• Thus its behavior is not legible.

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Possible solutions

• Design a calculus of communicating inductive
  inference processes.
• Use generalized limits. S. Berardi has introduced
  limit-interpretations based on such generalized
  limits.
• However, there is a much better way.
• Game theoretical semantics

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A semantics based on 1-backtracking game

• There is a game theoretical semantics equivalent
  to LCM.
• Good points of games
• Avoid the problem of global clock.
• More interactive.
• Much easier to understand than realizability
  interpretation.

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Game theoretical semantics of logic (1)

• Due to P. Lorenzen and J. Hinttika.
• In the semantics, validating a logical formula is
  counted as a game between two players Abelard
  (opponent) and Eloise (proponent).

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Game theoretical semantics of logic (2)

• For simplicity, we illustrate the semantics by
  prenex normal forms x1."y1.,,xn."yn.A(x1,y1,,
  xn,yn) ,where A is a decidable formula.
• A play is a sequence of moves by Eloise and
  Abelard ".
• Eloise wins by making A(x1,y1,,xn,yn) true.
  Otherwise Eloise loses and Abelard wins.

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A play for x1."y1.x2."y2.A(x1,y1,x2,y2)

• Eloise moves x15.
• Abelard moves y111.
• Eloise moves x27.
• Abelard moves y22.
• If A(5,11,7,2) is true, then Eloise wins.
• If A(5,11,7,2) is false, then Abelard wins.

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The definition of truth

• A formula is defined to be true, if and only if,
  there is a winning strategy for Eloise.
• A strategy str of Eloise is a set-theoretical
  function, which returns her next move from the
  preceding moves, e.g., str(x1,y1 ) x2 for
  x1."y1.x2."y2.A(x1,y1,x2,y2)

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Constructive truth and game theoretical semantics

• Giving a strategy for Eloise means giving Skolem
  functions.
• Thus, the game theoretical truth definition is
  equivalent to Tarski semantics.
• And, a formula is constructively true
  (recursively realizable) iff Eloise has a
  constructive (recursive) strategy.

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1-backtracking game

• We introduce a new rule
• Eloise is allowed to backtrack to any preceding
  position of the current situation of play and
  restart from the position.
• Eloises strategy may have a memory to record
  information on past moves by Abelard and Eloise.
• Everything is the same besides these two.

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A recursive winning strategy for
x."a.((xgt0ÙA(x-1))Ú(x0ÙØA(a)))

• Eloise moves x0.
• Abelard moves a24.
• If ØA(24) holds, Eloise stops and she wins. If
  A(24) holds, she backtracks to the stage 1, and
  moves with x25, i.e. x241.
• Then, Abelard moves. However, Eloise always wins,
  since A(x-1) holds with x241.

A(x) is assumed to be decidable. Thus the formula
(xgt0ÙA(x-1))Ú(x0ÙØA(a)) is the decidable part of
prenex form.
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Stack presentation of the strategyx."a.((xgt0ÙA(
x-1))Ú(x0ÙØA(a)))

• We consider the case of backtracking, i.e. the
  case A(24) holds.

• The stack behaviour
• x0
• x0, a24
• backtrackand
• x25 new move
• x25,a743
• Eloise wins, since 25gt0ÙA(24) holds.

• Eloise moves x0.
• Abelard moves a24.
• Since A(24) holds, Eloise backtracksand moves
  with x241.
• Abelard moves, say a743
• Eloise wins.

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A play for x1."y1.x2."y2.A(x1,y1,x2,y2)

• Eloise moves x15.
• Abelard moves y111.
• Eloise moves x27.
• Abelard moves y22.
• If A(5,11,7,2) is true, then Eloise wins.
• If A(5,11,7,2) is false, then Abelard wins.

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The equivalence theorem

• For any prenex normal formula A, the following
  conditions are equivalent
• Eloise has a recursive winning strategy for A.
• A is LCM-correct, i.e., it has a limit-recursive
  realizer.

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Other logical signs

• Conjunctions and disjunctions can be treated as
  special kind of quantifiers.
• Semantics of implication can be given by
  Hinttikas notion of subgame.

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S01-EM is true in the sense of 1-backtracking
game

• x."a.((xgt0ÙA(x-1))Ú(x0ÙØA(a))) is
  constructively equivalent to S01-EM
• x.A(x) Ú "a.ØA(a)
• Eloise has a recursive winning strategy for
  S01-EM.

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A play with disjunction x.A(x) Ú "a.ØA(a)

• right
• right, a24
• backtrack and go with two new moves
  left, x25.
• left, x25, a743
• Eloise wins.

• Eloise moves with right
• Abelard moves a24.
• Since A(24) holds, Eloise backtracksand moves
  withleft and x241.
• Abelard moves, say a743
• Eloise wins.

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The convergence property of 1-backtracking
winning strategy
In the proceedings paper, I called it
stability, but convergence property is
better. I changed the name.

• The Convergence Property
• As Abelard attacks Eloise with more and more
  moves, Eloises move after a winning strategy
  eventually converges in the manner of inductive
  inference to the right values given by Tarski
  semantics.
• The convergences take place from the outside of
  the formula to the inside of the formula.

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The Convergence Propertycaution over simplified
for explanation

• x1."y1.x2."y2.A(x1,y1,x2,y2)

X1a1
When Abelard tries all possible moves for him,
a1, a2, a3, given by Eloises winning strategy
converges to the right value in the sense of
Tarski semantics. In this figure, it is a5.
X1a2
X2b1
X1a3
X1a4
X1a5
X2b2
X2b3
X2b4
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Remarks

• The figure is a little bit simplified. In
  reality, the sequence a1, a2, a3, converges when
  all possible plays are considered. But, the
  figure represents only one play. Eloise may win
  accidentally with wrong values.
• The parameter space for the convergence or
  limit of a1, a2, a3, is the directed set of
  the finite sets of Abelards moves with the usual
  set ordering.

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The Convergence Property and Proof Animation (1)

• When one animates a proof by an animation tool,
  he tests the proof by providing test inputs, sets
  of Abelards moves.
• The user of animator expects particular values
  are returned for existential quantifiers for the
  test inputs by the winning strategy associated to
  the proof.

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The Convergence Property and Proof Animation (2)

• The expected value is the limit of the sequence
  of trial values a1, a2, a3,
• It is just as the inductive inference of MNP
  example.
• The behavior of 1-backtracking winning strategy
  is always in this pattern ! You do not need to
  worry about other patterns.

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Full backtracking game and Proof Animation

• In Coquands full backtracking game, Eloise is
  allowed to backtrack to any point of the past.
• Even if a stack configuration was flushed away
  (popped away) by her own backtracks, she is
  allowed to return to positions of configurations
  once flushed away.
• A strategy for S02-EM already cannot have
  convergence property. Values returned by the
  strategy are locally correct, but never globally
  correct. Thus, it is difficult to understand the
  behavior of the strategy (proof).

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Towards Proof Animator with 1-backtracking game

• A proof animator via 1-backtracking game is now
  planned.
• The ultimate goal is to animate proofs of David
  Hilberts theory of algebraic invariants in his
  1890 Mathematische Annalen paper.

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Hilberts invariant theory

• This is the theory that Paul Gordan called not
  mathematics, but theology.
• In 19th century algebra, solutions had to be
  given by algorithms. Gordan, who was the king of
  invariant theory then, realized Hilberts proof
  of the finite basis theorem embodies no
  algorithm.
• Hilbert used S01-EM repeatedly in the proof. All
  other parts were constructive.

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The theology is executable

• Theology was S01-EM.
• When the 1-backtracking animator is built,
  Hilberts theology will run on a computer!
• Remark LCM was found through my investigation of
  history of mathematic on Hilberts invariant
  theory thanks to help of a learning theorist
  Akihiro Yamamoto.

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Generalized equivalence theorem

• Berardi has defined a 1-backtracking game Back(G)
  for every game G in the sense of set theory, and
  proved the following theorem
• For any recursion theoretic degree a, the
  following are equivalent
• The degree a contains a winning strategy for
  Back(G).
• The jump of the degree a contains a winning
  strategy for G.

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Iteration

• Berardis Back(-) can be iterated.
• Thus, we can climb up the arithmetical
  hierarchy by iterating 1-backtracking extension.
• It might be possible to animate beyond LCM using
  Berardis iteration.

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Conclusion

• 1-backtracking game will serve as the right
  foundations for a proof animation tool.
• Hilberts invariant theory will be animated by
  the proof animation tool.
• It might be possible to animate beyond LCM using
  Berardis iteration.
• It seems to be related to game semantics for the
  full abstraction problems.

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Proof Animation/ LCM home page

• For more information, visit our home page
• http//www.shayashi.jp/PALCM/