Lecture 5: Termination of proof reduction in Deduction modulo theories

Teacher: Gilles Dowek

Theories:

  • Arithmetic
  • Simple Type Theory (implemented in HOL Light (Amazon), Isabelle/HOL (Cambridge), HOL4 (Munich))
  • Dependent Type Theory (Coq, Agda, Lean, …)

Right now: termination of proof reduction in predicate logic

Oddly: termination of $λ$-calculus: similar technique here → because everything can be expressed in a unifying proof langugage (BHK interpretation, interpretation of proof: Curry-de Bruijn-Howard)

Derivation tree:

  • one way to check that a proof is well-formed: go through the derivation tree and check each node ⇒ at each step: “which rule may apply”? ⟶ so you have to check all of them
  • workaround: label the nodes of the derivation tree with the names of the rules, and check (only one rule to check at each step)
  • if you just write the name of the rules, and not the conclusions: you can reconstruct the full tree
\[\infer[∧ \text{-intro}]{⊢ ⊤ ∧ ⊤}{ \infer[⊤ \text{-intro}]{⊢ ⊤}{\phantom{⊢ ⊤}} & \infer[⊤ \text{-intro}]{⊢ ⊤}{\phantom{⊢ ⊤}}}\]

or

\[\infer{⊢ ⊤ ∧ ⊤}{ ⊢ ⊤ & ⊢ ⊤}\]

or you could just specify the names of the rules:

\[∧\text{-intro}(⊤\text{-intro}, ⊤\text{-intro})\]

this is a proof term

Shorthand for the one above:

\[⟨ \underbrace{I}_{⊤\text{-intro}}, I⟩\]

Problem: with the axiom sequents! The name of the rule is not enough anymore! So the relation associating a conclusion to premises must be a function (which is not the case for $axiom$) ⟶ replace axiom by functions $axiom_{P,Q ⊢ P}$ (disambiguation of the axiom rule so that we have a family of functions).

Example of proof term with axiom rules:

\[⟨ ax_{P,Q ⊢ P}, ax_{P,Q ⊢ Q} ⟩\]

But this is not the way we will represent proofs. We will forget about the contexts: in ND, contexts are almost always the same (except for bookkeeping with $⇒$-intro, etc…):

\[⟨ ax_{P,Q ⊢ P}, ax_{P,Q ⊢ Q} ⟩ \leadsto ⟨α, β⟩\]

where you know you’re in the context $α ≝ P, β ≝ Q$

But some rules, as $⇒$-intro, extend context ⟶ we add named hypotheses to our proof: $λβ:B \, π$

Cut-elimination for $⇒$ for example: from $π$ proof of $Γ, A ⊢ B$, remove the hypothesis $A$ in all sequents, replace the axiom rules on this proposition by $π’$ of $Γ ⊢ A$ ⟶ substitute $π’$ for $α$ (associated to $A$) in $π$

Rules

  • $∧$-intro: $⟨π, π’⟩$
  • $∧$-elim 1: $fst(π)$
  • $∧$-elim 2: $snd(π)$
  • $⇒$-intro: $λα:A \, π$
  • $⇒$-elim: $app(π, π’)$, also written $(π \, π’)$
  • etc…

Proof reduction rules

\[(λα: A\, π_1) π_2 ⟶ (π_2/α) π_1\]

Proof reduction is $β$-reduction! Actually, not really, you also have:

\[fst(⟨π_1, π_2⟩) ⟶ π_1\\ snd(⟨π_1, π_2⟩) ⟶ π_2\]

With implication only: it’s exactly ST $λ$-calculus (minimal intuitionistic logic).

Brouwer-Heyting-Kolmogorov (BHK) interpretation

  • Brouwer/Heyting: advisor/student
  • Kolmogorov: discovered it independently

BHK interpretation:

  • How do you build a proof of $A∧B$?

    • build a proof of $A$ and a proof of $B$

  • How do you use a proof of $A∧B$?

    • to build a proof of $A$ or a proof of $B$

Balance between intro/elim: cut-elimination: building and using amounts to do nothing. Same issue in social science: what’s a knife, primarily? (something which is built with a blade and a handler or something the purpose of which is to cut things?)

  • How do you build an ordered pair of a proof of $A$ and a proof of $B$?

    • build a proof of $A$ and a proof of $B$

  • How do you use an ordered pair of a proof of $A$ and a proof of $B$?

    • to build a proof of $A$ or a proof of $B$

A proof of $A ∧ B$ is built and used like an ordered pair ⟶ so you may as well say that they are the same (if building and using are done in the same way, you have the same objects)

Implication: trickier, but similarly: a proof of $A ⇒ B$ is an algorithm mapping proofs of $A$ to proofs of $B$

$A ⇒ B$ is an “idle” object that “waits” for a proof of $A$ to produce a proof of $B$ ⟶ it’s an algorithm

And analogously for other connectives.

Curry-de Bruijn-Howard correspondence

Types put on proof terms:

$Φ(A)$: type of proofs of $A$

\[Φ(A ⇒ B) = Φ(A) → Φ(B)\]

$Φ$ is an isomorphism between propositions and types (Curry-Howard isomorphism)

Enriched ND (with typing rules):

\[\cfrac{Γ ⊢ π: A ⇒ B \qquad Γ ⊢ π': A}{Γ ⊢ (π \, π'): B}\]

Big difference with ND: derivability of $Γ ⊢ π: A$ is decidable: just look at $π$! (it’s not proof search but proof checking, whereas in ND: $Γ ⊢ A$ requires proof finding). Actually, you have a “tiny” copy of the derivation in $π$.

Termination of proof reduction in predicate logic

Like for ST $λ$-calculus, we will prove: if $Γ ⊢ π$, then $π ∈ R_A$, and $π$ strongly terminates (ST).

$π ∈ R_{∀x.\, A}$ if $π$ ST and when it reduces to $λx \, π_1$, then for all term $t ∈ R_{A}$, $(π \, t) ∈ R_{(t/x)A}$. Not the same problem as in System F, as $t$ is just a term (it’s weight can be regarded as $0$).

Termination of proof-reduction in some theories

Cornerstone of the course

Main tool: the notion of model

\[\infer{⊢ Ω: Q}{ \infer{⊢ λα. (αα): P ⇒ Q}{ \infer{P ⊢ αα: Q}{ \infer{α: P ⊢ α: P ⇒ Q}{\phantom{α: P ⊢ α: P ⇒ Q}} & \infer{α: P ⊢ α: P}{\phantom{α: P ⊢ α: P}} } } & \infer{⊢ λα. (αα): P}{ \qquad // \qquad } }\]

$R_A ∈ 𝒞$, where $𝒞$ is the set of candidates

Define a function $R$ mapping every proposition $A$ to a candidate $R_A$ s.t.

  1. $R_{A∧B} = R_A \tilde ∧ R_B, \; R_{A⇒B} = R_A \tilde ⇒ R_B$,etc.
  2. if $A≡B$, then $R_A = R_B$

and prove that all proofs of $A$ are in $R_A$, hence strongly terminate.

Examples:

  • for $P ⟶ Q ⇒ Q$:

    • $R$ maps $Q$ to $\tilde ⊤$, $P$ to $\tilde ⊤ \tilde ⇒ \tilde ⊤$

  • doesn’t work for $P ⟶ P ⇒ Q$, as $P$ appears in negative position

In a Heyting algebra:

\[\tilde ⊤ = \tilde ⊤ \tilde ⇒ \tilde ⊤\]

because:

  • $\tilde ⊤ ≥ \tilde ⊤ \tilde ⇒ \tilde ⊤$ (maximality)
  • $\tilde ⊤ ≤ \tilde ⊤ \tilde ⇒ \tilde ⊤ ⟺ \tilde ⊤ \tilde ∧ \tilde ⊤ ≤ \tilde ⊤$ (maximality again)

and we conclude by antisymmetry.

But this doesn’t hold for reductibility candidates: $\tilde ⊤ ≠ \tilde ⊤ \tilde ⇒ \tilde ⊤$, because $δ ≝ λα:A. (α \, α)$ is in $\tilde ⊤$ ($δ$ is ST) but not in $\tilde ⊤ \tilde ⇒ \tilde ⊤$! ($δ$ is ST but $(δ \, δ)$ is not)

So reductibility candidates form a pre-Heyting algebra.

Completeness property: to build models with fixpoints, as with

\[N(x) ⟶ ∀x, c. (0 ∈ c ⇒ ∀x (N(x) ⇒ x ∈ c ⇒ S(x) ∈ c)) ⇒ x ∈ c\]

  • In ST $λ$-calculus: all you have is constant functions and functions that add a constant number to its argument.

    • $λx, y. SS(0)$
    • $λx, y. SS(x)$
    • but not $x ⟼ 2×x$, etc…
    • lacking: induction. If you add it, you get Gödel System T.

  • Arithmetic: much more powerful programming language (you have induction in arithmetic!)

    • $π: ∀x, ∃y. (x = 2 × y \; ∨ \; x = 2 × y + 1)$
\[N(y) ⟶ ∀c (0 ∈ c ⇒ ∀x (N(x) ⇒ x ∈ c ⇒ S(x) ∈ c) ⇒ y ∈ c)\]

so

\[α \, c \, π \, π': y ∈ c\\ λy, α. (α \, c \, π \, π'): ∀y(N(y) ⇒ y ∈ c)\]

In Arithmetic: we can encode every provably total computable function (example of non provably total computable function: an interpretor of arithmetic)

Proofs are programs and proof-reduction is an interpreter.

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