Continuations, Backtracking and Limits: a notion of

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Continuations, Backtracking and Limitsa notion
of Construction for Classical Logic(Ongoing
work)Kyoto, November 2004
Stefano Berardi, Semantic of Computation
group C.S. Dept., Turin University,
http//www.di.unito.it/stefano
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Abstract of the talk

• We describe Classical Principles in term of
  constructions, of a general kind, learning a
  value by trial-and-error, rather than computing
  it exactly.
• This notion of construction may be described
  in term of continuations and backtracking, or by
  Coquands game interpretation, or by Hayashis
  Limit Interpretation.

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1. Heytings notion of construction

• Heyting defined a notion of construction for
  a formula F.
• For Heyting, a construction of F is something
  computing all informations about F we expect from
  a proof of F.
• We will now introduce Heytings notion of
  construction in some details.
• Later, we will consider a generalization of it
• Limit Realizability.

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An inductive definition of a construction for a
formula F

• If F P(t1, , tn), with P decidable predicate,
  then a construction of F is anything, provided F
  is true.
• If F F1?F2, then a construction r of F is any
  pair lti,sgt, with i?1,2, and s construction of
  Fi.
• If F F1?F2, then a construction r of F is any
  pair ltr1,r2gt, with r1 construction of F1, and r2
  construction of F2.

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An inductive definition of a construction for a
formula F

• If F ?x?I.G(x), then a construction r of F is
  any computable map
• f I?constructions of G(x), for some x?I
• such that, for all x?I, f(x) is a construction of
  G(x).
• If F ?x.G(x), a construction r of F is a pair
  lti,sgt, with i?I, and s construction of G(i).

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An inductive definition of a construction for a
formula F

• If F F1?F2, then a construction r of F is any
  computable map
• f constructions of F1?constructions of F2
• If F ?G, a construction of F is anything,
  provided there is no construction of G.

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?0n- and ?02-formulas

• ?0n-formulas are all formulas
• A(x1, x2, ) ?x1?N. ?x2?N. . P(x1, x2, )
• with n alternating quantifiers, starting with ?,
  and with P decidable.
• ?0n-formulas are all formulas
• B(x1, x2, ) ?x1?N. ?x2?N. . P(x1, x2, )
• with n alternating quantifiers, starting with ?,
  and with P decidable.

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Constructions as programs

• By unfolding definitions, a construction r for a
  closed ?02-formula B ?x?N. ?y?N. P(x,y)
  provides some map f, satisfying P(x,f(x)) for all
  x?N.
• If we think of B as a specification, a
  construction for B provides some program f
  satisfying the specification B.

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Intuitionistic and Classical Proofs

• EM (Excluded Middle) is the axiom schema
• ?x?N. A(x) ? ?A(x)
• n-EM is the axiom schema EM restricted to ?0n-
  formulas.
• Intuitionistic proofs are all proofs not using
  Excluded Middle (in any equivalent form).
• Classical proofs are proofs possibly using
  Excluded Middle (in some form).

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Intuitionistic Proofs are constructions

• We may interpret any intuitionistic proof of F as
  defining some construction of F, and therefore a
  program if F is ?02-formula.
• Programs extracted from proofs in this way are
  usually very naive, but they are a starting point
  in order to design real programs.
• Constructive interpretation of Heyting is a
  bridge between Mathematical proofs and
  applications.

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Classical Proofs are not Heytings constructions

• There is no construction for Excluded Middle. Let
  A(x) ?y?N.P(x,y) be any ?01-formula. By
  unfolding definition, there is some construction
  for the formula
• ?x?N. A(x) ? ?A(x)
• if and only if there is some computable
  characteristic map in N for A(x) (i.e. some
  fN?N such that f(x)0 if and only if A(x) is
  true). But for some P, there is no computable
  characteristic map in N of A(x).

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Preliminary Conclusion

• Most Mathematical proofs are Classical. By what
  we said, apparently, such proofs define no
  program. Therefore they have no applicative
  interest.
• This preliminary conclusion, however, is false.

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Plan of the Talk

• We will define a relaxed notion of construction,
  obtained from Heytings by replacing everywhere
  computable with limit computable.
• With this new notion of construction there is a
  construction for Excluded Middle. Constructions
  of closed ?02-formulas still correspond to
  programs.
• Before introducing limit computability, we give a
  quick account of what is known about extraction
  of programs from Classical Proofs.

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2. A constructive content for Classical Logic

• Call PA (Peano Arithmetic) Arithmetic with
  quantification over integers, Induction axiom for
  all formulas, and Excluded Middle.
• Call HA (Heyting Arithmetic) PA without Excluded
  Middle.
• Godel proved ?02-conservativity of PA w.r.t. HA
  If B is a ?02-formula and PA proves B, then HA
  proves B. Besides, there is a computable map
  turning every proof of B in PA into a proof of
  the same B in HA

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Classical proofs are programs

• If we interpret proofs in HA by constructions,
  ?02-conservativity implies that every proof in PA
  of a ?02-formula B ?x?N. ?y?N. P(x,y) may be
  turned into a program f such that P(x,f(x)).
• ?02-conservativity is a bridge between
  Mathematics and applications.
• However, this result has also some limitations,
  as we will see.

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Some limitations of ?02-conservativity result

• No intuitive explanation is given. How can
  classical proofs of ?02-statements be programs,
  if they use Excluded Middle, and there is no
  construction for Excluded Middle?
• As a consequence, we miss a global understanding
  of the program we extract from a proof. This
  means that, if the program is naive, as usually
  is, we have no way of making it better. Therefore
  we have no real way of using it.

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A refinement of ?02-conservativity result

• The first refinement of ?02-conservativity is due
  to Griffin (around 1980).
• Griffin defined a programming language in which
  classical proofs could both be written and
  executed lambda calculus with Continuations.
• Griffins programming language was simplified by
  Parigot (??calculus), and by Berardi (symmetric
  ? calculus). Eventually Curien and Herbelin
  merged the last two calculus, defining symmetric
  ??calculus (possibly the best so far).

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Continuations and Classical Logic

• There is a common idea underlying Griffins
  calculus and those who came later to interpret
  Excluded Middle by means of Continuations.
• Let ? be any computation. The continuations of ?
  are some set of computation states M1, , Mn,
  stored in the memory of ?.
• At any moment, ? can store the current state E
  (or part of it) in the set of continuations.

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Backtracking

• At any moment, the computation ? can either
• move from the current state E to the next state
  E (an ordinary step)
• move to some state M1, , Mn in the set of
  continuation (an exceptional step).
• If Mi is some previous state of the computation,
  we call this second choice backtracking.

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Continuations and simply typed lambda ?calculus

• We may add continuations to simply typed lambda
  ?calculus by adding a constant C of type (??F ?
  F), together with the rewrite rule
• (C) EC(M) False ? M(?x.Ex)
  False
• The axiom schema (??F ? F) is intuitionistically
  equivalent to Excluded Middle. This is the
  connection between C and Classical Logic.
• The rule (C) say that M is a continuation the
  computation of E can move to M. A copy of E, in
  the form ?x.Ex, is saved as argument of M. E
  can added by M to the set of continuations of the
  computation.

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Advantagesof continuations

• Using continuations, we may describe step-by-step
  the evaluation of the classical program extracted
  from a proof.
• Continuation are a major advance in extracting
  programs out of classical proofs.

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Disadvantagesof continuations

• Using continuations, we miss a global and
  intuitive understanding of the program.
• This means we cannot rewrite naive programs in
  order to obtain efficient programs. In fact, we
  still cannot use them.
• In the rest of the talk, we will outline a
  semantical constructive interpretations of
  classical proofs.

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3. Semantic of Classical Proofs

• By a Tarski structure M for a language L we mean
  a set X, equipped with an equivalence relation ,
  and, for each relation R and map f of L, some
  relation RM and some map fM, both compatible with
  .
• A formula F of L is true in M if it is is true
  the formula obtained by replacing, in F, the
  symbol with , and all symbols R, f with RM,
  fM.
• A Tarski model of a theory T is a Tarski
  structure M for the language of T, such that all
  theorems of T are true in M.

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Interpretation of a theory inside another one

• A model of a theory T is the usual way of
  explaining the mathematica concepts of T.
• An interpretation of a theory T inside a theory U
  defines some model M of T using predicates and
  maps of U, and proving in U that all theorem of T
  are true in the model.
• An interpretation of a theory T inside a theory U
  is the usual way of explaining the mathematical
  concepts of T in term of the mathematical
  concepts of U.

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An example of interpretation

• We may interpre the theory H of Hyperbolic Plane
  inside the theory E of the Euclidian plane.
• The set of elements of H is some set of points of
  the Euclidean Plane. The equivalence relation is
  equality on the Euclidean Plane. Any geometrical
  operation or relation of H is defined through
  some geometrical operation or relation of the
  Euclidian Plane.

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?02-sound models of PA

• A model M of PA is ?02-sound if all closed
  ?02-formula B ?x?N. ?y?N. P(x,y) true in M are
  true in the usual sense
• for all n?N there is some m?N such that P(n,m)
• If M, B are as above, then any intuitionistic
  proof that B is true in M provides some program f
  such that P(x,f(x)) for all x?N.

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Interpreting PA inside HA

• We want to define an interpretation of PA inside
  HA, in order to interpret Excluded Middle as a
  construction in some structure.
• This is no easy task. There are several
  impossibility results we have to turn around (see
  1).
• We also want to get some intuitive explanation of
  the programs we extract from classical proofs.
• Therefore we only consider ?02-sound models M of
  PA.

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The main result

• Let PA?,? be the variant of PA whose set of
  logical connectives is
• ?i?1,n.Ai, ?i?1,n.Ai,
• ?i?P(i), ?i?N.P(i)
• Theorem. There exists a ?02-sound interpretation
  M of PA?,? in HA.
• We may interpret every step of every classical
  proof of PA by some construction in some model M,
  in such a way that a proof of a closed
  ?02-formula defines a program.

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The model M of PA in HA

• M is the set of all iterated limits of
  integers, taken over List(N), the set of lists of
  integers.
• Definition of M generalizes Hayashis Limit
  Computable Mathematics 4, in which limits are
  taken over N, and M is defined in PA itself.
• Limits over List(N) are intended to give a more
  accurate descriptions of computations hidden in
  classical proofs.
• M is defined in HA in order to show we may
  completely describe EM in term of limits.

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Conclusion and Future Work

• Eventually, we switched to a semantical
  interpretation of classical proofs, looking for
  some intuitive understanding of the programs we
  can extract from them.
• This is a work in progress. The model we found
  looks promising, but it should be checked against
  some relevant examples. We have also to implement
  the computations in the model in term of
  backtracking and continuation.
• We are quite confident this is possible.

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4. Appendix. Intepreting HA 1-EM in HA

• We sketch the definition of the interpretation of
  PA in HA in a simple case when EM is restricted
  to 1-EM.
• We define a structure we call N1 (see 3)
• The same construction also produces some model of
  (n1)-EM out of any model of n-EM.
• A model of EM is obtained by defining a
  succession of models of 0-EM, 1-EM, 2-EM, , then
  taking the direct limit of them.

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A notion of Computation State

• Let S List(N) be the set of all finite lists of
  integers, ordered by prefix. We call the elements
  of s computation states.
• Intuitively, s is a list of (codes of) pairs
• ltn1,?x.P1(x)gt, , ltnk,?x.Pk(x)gt
• of some integer ni and some ?01-formulas
  ?x.Pi(x).
• If PPi and Pi(ni) is false, we know that ?x.P(x)
  is false.
• Otherwise we assume that ?x.P(x) is true.

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Dynamic Integers

• We call any LS?N a dynamic integer.
• Intuitively, L(s) is an hypothesis about a value,
  depending on the hypothesis about ?01-formulas
  associated to the state
• s ltn1,?x.P1(x)gt, , ltnk,?x.Pk(x)gt
• If some hypothesis change, then L(s) can change.
  L(s) will be a convergent limit if
• in any correct computation of L,
• eventually L(s) stops changing

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A notion of Agent

• Let Pfin(N) be the set of all finite subsets of
  N.
• We call any FS ? Pfin(N) an agent of the
  computation.
• Intuitively, F(s) is a set of elements of the
  same kind of those which are in the state s
• F(s) ltn1,?x.P1(x)gt, , ltnk,?x.Pk(x)gt
• The agent F requires to add the set F(s) above to
  the state s of the computation.

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A notion of Computation

• Let ? be any recursive weakly increasing
  successions s0 ? s1 ? s2 ? in S
• We say that ? is a computation of an agent F, or
  ?F for short, if, eventually, any element a of
  F(si) either drops out of F(si), or it is added
  to si.
• Formally ?F if, for all a,n?N, there is some
  m?n such that either a?F(sm), or a?sm.
• Remark that this definition describes a notion of
  parallel asyncronous computation.

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Convergence and Equivalence of Dynamic Integers

• Let FS ? Pfin(N) be an agent, L, MS?N be
  dynamic integers. Abbreviate (L is convergent),
  (L, M are equivalent), with L?, (LM). Then we
  define
• F is a construction of L?, or FL? for short, if
  for all computations s0 ? s1 ? s2 ? of F, there
  is some n?N such that L(sn)L(sn1).
• F is a construction of (LM) , or FLM for
  short, if for all computations s0 ? s1 ? s2 ?
  of F, there is some n?N such that L(sn) M(sn).

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The Tarski structure N1

• Let FS ? Pfin(N), and L, MS?N. For any n?N,
  define nºS?N by nº(s) n for all s?S.
• L? if FL? for some F
• (LM) if F (LM) for some F
• N1 is the set of all convergent dynamic
  integers, with equivalence relation . Maps of N1
  are all computable maps fN1 ?N1 such that
• f(L)(s) f(L(s)º)(s) for all s?S.

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A construction for 1-EM in N1

• A characteristic map in N1 for a ?01-formula
  ?y?N.P(x,y) is any map f of N1 such that, for all
  x?N1, f(x) Trueº if and only if Pº(x,y) is true
  for some y?N1.
• Maps of N1 are computable, yet they include
  characteristic maps for all ?01-formulas, and
  therefore some construction for 1-EM.
• Characteristic maps in N of ?01-formulas,
  usually, are not computable. This is no
  contradiction. Characteristic maps in N1 are such
  only up to .

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Bibliography

• 1 S. Berardi, Intuitionistic Completeness for
  First Order Classical Logic, J.S.L., vol 63,
  Number 1, March 1999.
• 2 S. Berardi, Classical Logic as Limit
  Completion, MSCS, 15, 2004 (to appear).
• 3 S. Berardi, A Model for ?02-maps within
  Intuitionistic Arithmetic, Techical Report, Turin
  University, 2004.

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Bibliography

• 4 S. Hayashi and M. Nakata, Towards Limit
  Computable Mathematics, Types for Proofs and
  Programs, Springer LNCS, 125--144, 2001.
• 5 Y. Akama, S. Berardi, S. Hayashi, U.
  Kohlenbach, An Arithmetical Hierarchy of the Law
  of Excluded Middle and Related Principles, pp.
  192-201, 19th IEEE Symposium on Logic in Computer
  Science (LICS 2004), 14-17 July 2004, Turku,
  Finland, Proceedings.