Logic & Programming

I will give a talk on type isomorphisms on the Deducteam seminar at INRIA Paris — av. de Italie, on February 7, 2014, at 10:00.

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I will give a talk at the seminar of the LCR team, Université Paris 13 (March 7).

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I will give a talk at Kosta Došen's General Proof Theory seminar at the Serbian Academy of Sciences and Arts (February 24).

http://www.mi.sanu.ac.rs/seminars/seminar18.htm

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I set up a web page with materials for the winter school Recent developments in Type Theory, Mathematical Structures of Computation – Lyon 2014.

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My PhD thesis has been approved for defense. You can find it here : http://tel.archives-ouvertes.fr/tel-00529021/en/

Chapter 1 gives a constructive version of a Henkin-style proof of Godel's completeness theorem. Chapter 2 introduces a notion of model similar to Kripke models using which one can give a normalization-by-evaluation style proof of completeness for classical logic. Chapter 3 does the same thing for intuitionistic logic (including disjunction and the existential). Finally, Chapter 4 presents an extension of intuitionistic logic with delimited control operators which still remains intuitionistic (preserves disjunction and existence properties) and can prove the Double Negation Shift and Markov's Principle.

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Published on Feb. 5th, 2010 on https://speleologic.livejournal.com/2595.html

The purpose of this post is a detailed proof, following Coquand[1], of the Open Induction Principle using Decidable-Bar Induction[2].

Open induction (OI) is useful when one can not do a proof by usual well-founded induction, that is, when there is no well-founded relation that can be used. In such cases, if the space one is in is compact, and the property one wants to prove is open, one can use a non-well-founded order to do an induction proof. From the computer science perspective, this means one can build a terminating recursive program for computing a property that does not depend on a structurally decreasing criterium of termination.

In this post we concentrate on Cantor space 2N, but it is, in some sense, enough, because every compact space is an image of Cantor space. A member α of 2N is a map N → 2, i.e. an infinite stream of bits, i.e. an infinite branch of the infinite binary tree. α<β if α is “left of” β as a branch of the infinite tree, that is, α<β if they are bit streams which are equal for a finite initial segment and then at some length n of the segment α(n) is “left” while β(n) is “right”. We write αn for the finite initial segment of α of lenght n. Each node in the infinite binary tree corresponds to some initial segment, and vice-versa. Each such node, i.e. each finite bit string, is called a basic open. An open, or open subspace, of Cantor space is an arbitrary union of basic opens. Equivalently, an open is determined by a Boolean function G on arbitrary lenght, but finite, bit strings, G : 2* → 2, which is monotone: If s, s′:2*, s is an initial segment of s′, and G(s)=⊤, then also G(s′)=⊤.

Axiom 1 (Decidable-Bar Induction) For all unary predicates P and Q on 2* (finite binary strings), such that P is decidable and ∀ s. P s → Q s, if ∀ α∈2N. ∃ n∈ N. P(αn) and ∀ s∈2*. (∀ b∈ 2. Q ⟨ b::s ⟩) → Q s then Q ⟨ ⟩.

Theorem 2 (Open Induction Principle) Let U be an open set of Cantor space 2N and < be a lexicographic ordering on 2N. If ∀ α. (∀ β<α. β∈ U) → α∈ U () then ∀ α. α∈ U. Proof. Let α : 2N be given. U is open in 2N, hence given by a mapping G : 2 → 2 which is monotone w.r.t. initial segments: if s is a prefix of t, G s = ⊤ implies G t = ⊤. We say that G bars s with witness k (notation G|k s) if all k−bit-extensions of s satisfy G.

We define mutually, by (primitive) recursion on finite strings of numbers, two functions f:N* → 2* and acc:N*→ 2:

acc ⟨ ⟩ = ⊤ acc ⟨ 0::s ⟩ = acc(s) acc ⟨ n+1::s ⟩ = acc(s) ∧ G|n ⟨ ⊥::f(s) ⟩

f ⟨ ⟩ = ⟨ ⟩ f ⟨ 0::s ⟩ = ⟨ ⊥::f(s) ⟩ f ⟨ n+1::s ⟩ = ⟨ acc⟨ n+1::s ⟩::f(s) ⟩ We set

P(s) := G(f(s)) Q(s) := acc(s) → ∃ m. G|m f(s) and apply bar induction, which gives us a proof of Q(⟨ ⟩), which, since acc(⟨ ⟩)=⊤, gives an m∈ N such that any bit-string with length m satisfies G. Hence, also αm, therefore α∈ U.

The hypotheses of (D-BI) are satisfied by the following lemmas.

Lemma 3 ∀ α∈NN. ∃ n∈ N. G(f(αn)) Proof. We write f(α) for the natural exension of f to infinite sequences. The lemma states that ∀ α∈NN. f(α)∈ U. Let α be given. Apply (). Let β<f(α), that is, there is some n such that βn = ⟨ ⊥::s ⟩ and ⟨ ⊤::s ⟩ = f(α) n = f(αn) for some s∈2 (of length n−1). By the defining equations of f and app, n>0, s=f(α(n−1)), acc(s) and G|k⟨ ⊥::s ⟩, that is, G|kβn, for some k. Hence, G(β(n+k)) = ⊤ i.e. β∈ U.

Lemma 4 For any s∈N*, if (∀ n∈ N. acc⟨ n::s ⟩ → ∃ m. G|m f⟨ n::s ⟩), and acc(s), then ∃ n. G|n f(s). Proof. Since acc(s), also acc⟨ 0::s ⟩, and the hypothesis gives us an m such that G|mf⟨ 0::s ⟩, that is, G|m⟨ ⊥::f(s) ⟩. This means that acc⟨ m+1::s ⟩, which we use with the hypothesis again to get a k such that G|kf⟨ m+1::s ⟩ i.e. Gk⟨ acc⟨ m+1::s ⟩::f(s) ⟩ i.e. G|k⟨ ⊤::f(s) ⟩. We showed that G bars both ⟨ ⊤::f(s) ⟩ and ⟨ ⊥::f(s) ⟩, which means that G bars f(s) already: G|max(m,k)f(s).

Lemma 5 ∀ s∈N*. G(f(s)) → acc(s) → ∃ m. G|m f(s). Proof. G(f(s)) iff G|0 f(s).

Remark 6 The given lemmas, and the functions f and acc, are more general than needed for (D-BI), since they take input of type N* instead of 2*.

References [1] Thierry Coquand. A note on the open induction principle, 1997. [2] Anne S. Troelstra and Dirk van Dalen. Constructivism in Mathematics: An Introduction I and II, volume 121, 123 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1988.

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Published on Apr. 19th, 2009 on https://speleologic.livejournal.com/1696.html

Some time ago I came upon a mention of Ishihara's proof technique in constructive analysis and a claim:

In [6], Ishihara introduced the following two lemmas in which, surprisingly, we can use completeness to make a decision which at first sight would seem to be impossible with purely constructive techniques (that is, with intuitionistic, rather than classical, logic). (Bridges, van Dalen, Ishihara: Ishihara’s proof technique in constructive analysis)

One does not have to be interested in mathematical analysis to appreciate it, however, it is enough to think about a sequence of computation steps and a measure of “distance” between every two consecutive computation steps. If one can prove that for a sequence the distance gets smaller and smaller as time goes on we say that the computation converges or is Cauchy. When we consider these converging processes of computation as inhabitants of some universe X we can say that X is a complete metric space.

Here I want to give a simpler (more predicative) proof of Bridges-vanDalen-Ishiharas's Proposition 1, which more obviously gives rise to a program than theirs.

Proposition 1. If X is a complete metric space, X=P∪Q, (xn) is a sequence in X that converges to x and we have ∀y:X, x#y ∨ y∉Q, then we have that (∀n. xn ∈ P) or (∃ n. x_n ∈ Q).

The role of x, y as members of X is to stand for a sequence (xn), (yn) together with a proof that the sequences make progress as time progresses, i.e. they are Cauchy/converging. The relation '#' is called apartness and it means that the two computation processes are not 'the same'.

Proof. Start with a sequence (xn). Define the binary sequence λ to be 0 as far as the elements of the sequence (xn) are in P, and to become 1 forever as soon as one of the elements of (x_n) is in Q.

Now define the sequence (yn) to be x as far as λn is 0 and to become xk as soon as λn becomes 1, where k is the smallest natural number at which λ becomes 1, i.e. the k-th element of the sequence (xn) is in Q. The sequence (yn) we defined is very similar to a constant sequence: it is either x forever, or it is x for a finite number of starting elements and then it becomes x_k forever.

In any case (yn) is Cauchy, so we can think of it as an entity y from the universe X. We now use the hypothesis given in Proposition 1 and look at two cases. if x#y then there is k, the smallest number such that λk = 1, which implies xk ∈ Q if y∉Q then we can show that ¬∃n. λn = 1 and, since λ is decidable, ∀n. λn = 0, which implies ∀n. xn ∈ P QED

Apparently, in constructive analysis they use Proposition 1 (and 2, which was not shown) as constructive instances of the general law of excluded middle. Is it useful in logic and computer science?

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