Transitivity Elimination: Where and Why Pierluigi Minari Section of - - PowerPoint PPT Presentation

Transitivity Elimination: Where and Why

Pierluigi Minari

Section of Philosophy, DILEF, University of Florence minari@unifi.it

Advances in Proof Theory 2013 Bern, December 13-14, 2013

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Intuitive motivations: two analogies

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Intuitive motivations: two analogies

Analogy 1: Modus ponens / transitivity rule for equality: A → B A B ≈ t = r r = s t = s

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Standard presentation of an equational proof system E:

certain specific axioms (a given set E of equation schemas) the usual inference rules for equality: t = t [refl] t = s s = t [symm] t = r r = s t = s

[trs]

ti = si f n(t1, . . . , tn) = f n(t1 . . . , ti−1, si, ti+1, . . . , tn) [congr: 1≤i≤n]

Birkhoff’s completeness theorem for equational logic (1935): ⊢E t = s ⇔ E | = t = s

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The transitivity rule (which cannot be dispensed with, except that in trivial cases) has an inherently synthetic character in combining derivations, like modus ponens in Hilbert-style proof systems Potential loss of relevant information along formal derivations (no kind of “subterm property” available!) As a consequence, naive proof-theoretic arguments are usually inapplicable (e.g.: syntactic consistency proofs by induction on the length of derivations) In general, derivations lack any significant mathematical structure All in all, ‘synthetic’ equational calculi do not lend themselves directly to proof-theoretical analysis

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Question

Are there significant cases in which it is both possible and useful to turn a ‘synthetic’ equational proof system into an equivalent ‘analytic’ proof system, one in which the transitivity rule is provably redundant?

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Analogy 2: Cut-elimination / Church-Rosser: (syntactic) proofs of cut-elimination for Gentzen-style calculi ≈? some proofs of the Church-Rosser theorem for (weak, β-, βη-) reduction (e.g., proofs à la Tait & Martin-Löf, using parallel reduction)

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(Partial) answer: where

Equational theories of type-free operations, in particular: combinatory logic (more generally: arbitrary “combinatory systems”) and lambda-calculus can be presented through an ‘analytic’ proof system.

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Answer: why

Conceptual interest: analyticity at work in an equational environment (subterm property); direct consistency proofs. New (and short) proofs of well-known key results concerning reductions (like Confluence, Standardization, Leftmost reduction, η-Postponement) can be given in a unified framework by purely proof-theoretical methods. Decidability of (pure, linear and recursive) fragments of CL with extensionality, like e.g. BCK + ext. Positive solution of Curry’s problem (1958) on combinatory strong reduction (a “Methodenreinheit” issue). . . .

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Overview

✞ ✝ ☎ ✆

synthetic (CL-like or λ) proof-systems (“S-systems”)

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Overview

✞ ✝ ☎ ✆

synthetic (CL-like or λ) proof-systems (“S-systems”) ⇓

✞ ✝ ☎ ✆

equivalent (candidate) analytic proof-systems (“A-systems”)

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Overview

✞ ✝ ☎ ✆

synthetic (CL-like or λ) proof-systems (“S-systems”) ⇓

✞ ✝ ☎ ✆

equivalent (candidate) analytic proof-systems (“A-systems”) ⇓ (effective) transitivity elimination for A-systems

P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 9 / 1

Overview

✞ ✝ ☎ ✆

synthetic (CL-like or λ) proof-systems (“S-systems”) ⇓

✞ ✝ ☎ ✆

equivalent (candidate) analytic proof-systems (“A-systems”) ⇓ (effective) transitivity elimination for A-systems ⇒ consistency Church-Rosser

P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 9 / 1

Overview

✞ ✝ ☎ ✆

synthetic (CL-like or λ) proof-systems (“S-systems”) ⇓

✞ ✝ ☎ ✆

equivalent (candidate) analytic proof-systems (“A-systems”) ⇓ (effective) transitivity elimination for A-systems ⇒ consistency Church-Rosser ⇓ “normalizability” of transitivity-free derivations

P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 9 / 1

Overview

✞ ✝ ☎ ✆

synthetic (CL-like or λ) proof-systems (“S-systems”) ⇓

✞ ✝ ☎ ✆

equivalent (candidate) analytic proof-systems (“A-systems”) ⇓ (effective) transitivity elimination for A-systems ⇒ consistency Church-Rosser ⇓ “normalizability” of transitivity-free derivations ⇓ applications to combinatory/ lambda reductions

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Combinatory Logic: CL (& generalizations)

P .M. 2004, Analytic combinatory calculi and the elimination of transitivity, Arch. Math. Logic 43, 159-191.

Lambda-Calculus: λβ, λβη

P .M. 2005, Proof-theoretical methods in combinatory logic and λ-calculus, in: S. Cooper et

• al. (Eds), CiE 2005: New Computational Paradigms, Amsterdam, 148-157.

P .M. 2007, Analytic proof systems for λ-calculus: the elimination of transitivity, and why it matters, Arch. Math. Logic 46, 385-424.

Extensional Combinatory Logic: CLext (& generalizations)

P .M. 2009, A solution to Curry and Hindley’s problem on combinatory strong reduction,

• Arch. Math. Logic 48, 159-184.

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Synthetic systems

CL

Axiom schemas: Kts = t Stsr = tr(sr) [ It = t ] Inference rules: ̺ (reflexivity) σ (symmetry) τ (transitivity) t = s rt = rs µ t = s tr = sr ν

(app-congruence rules)

CLext

:= CL + tx = sx t = s

Ext {x/ ∈V(ts)}

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CL generalized

A combinatory system X is a map, defined on a non-empty set X = dom(X) of primitive combinators (F, G . . . ), which associates to each F ∈ X a pair kF, dF s.t.: kF, the index of F under X, is a non negative integer; dF, the definition of F under X, is a term with V(dF) ⊆ {v1, . . . , vkF}. Intuitively, X fixes an axiom schema for each primitive combinator F ∈ X: Ft1 . . . tkF = dF[v1/t1, . . . , vkF/tkF] (AX F)X CL[X] / CLext[X] are now defined exactly as CL / CLext, except that the axiom schemas for the combinators K, S (I) are replaced by the set {(AX F)X | F ∈ X}

• f axiom schemas induced by X.

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Example 1

The familiar pair {K, S} of primitive combinators of CL corresponds to the combinatory system C such that: C = {K, S} kK = 2 and dK = v1 kS = 3 and dS = v1v3(v2v3)

Example 2

But also the following is a perfectly legitimate (maybe odd!) combinatory system: X = {F, G, H} kF = 0 and dF = FF(HG)(GH) kG = 2 and dG = v2GHv1 kH = 4 and dH = v1v1((Hv2)(Fv1))(v4H)

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λβ

Axiom schema: (λx.t)r = t[x/r]

(β-conversion)

Inference rules: ̺, σ, τ, µ, ν, plus t = s λx.t = λx.s ξ

(abstr-congruence rule)

λβη

:= λβ + λx.tx = t {x/

∈V(t)}

(η-conversion)

• r, equivalently, λβ + Ext

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Analytic “A”-systems: main features

1

combinatory axiom schemas / β-conversion schema are replaced by introduction rules (to the left / to the right), as follows.

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Example: introduction rules for the combinator S

Stsr = tr(sr)

[AX S]

• tr(sr)p1 . . . pn = q

Stsrp1 . . . pn = q [Sl] q = tr(sr)p1 . . . pn q = Stsrp1 . . . pn

[Sr]

where n ≥ 0, i.e.: the “side terms” p1, . . . , pn may be missing

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In general: introduction rules for a primitive combinator F

Ft1 . . . tkF = dF[t1, . . . , tkF] (AX F)X

• dF[t1, . . . , tkF]p1 . . . pn = s

Ft1 . . . tkFp1 . . . pn = s

[Fl]X

s = dF[t1, . . . , tkF]p1 . . . pn s = Ft1 . . . tkFp1 . . . pn

[Fr]X

where n ≥ 0. (We write t[s1, . . . , sn] short for t[v1/s1, . . . , vn/sn])

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β-introduction rules

(λx.t)r = t[x/r]

[β−conv]

• t[x/r]p1 . . . pn = q

(λx.t)rp1 . . . pn = q [βl] q = t[x/r]p1 . . . pn q = (λx.t)rp1 . . . pn

[βr]

where n ≥ 0, i.e.: the “side terms” p1, . . . , pn may be missing

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Analytic “A”-systems: main features (contd)

1

combinatory axiom schemas / β-conversion schema are turned into the corresponding left/right introduction rules, as shown.

2

symmetry rule ◮ dropped

3

reflexivity (0-premises) rule ◮ restricted to atomic terms (̺′)

4

app-congruence (= monotony) rule(s) ◮ taken in the parallel version t = s p = q tp = sq

App

5

extensionality (if any) ◮ always taken in the rule-version tx = sx t = s

Ext {x/ ∈V(ts)}

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Analytic “A”-systems (summarizing)

A...[. . .] :=        ̺′, App, τ / and ξ “structural” rules Fl, Fr / or βl, βr introduction rules

• Ext

extensionality rule

• Synthetic systems
• Equivalent analytic systems

CL

• A[C]

CL[X]

• A[X]

CLext / CLext[X]

• Aext[C] / Aext[X]

λβ

• A[β]

λβη

• Aext[β]

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Main Result

τ-elimination

A-systems admit (effective) transitivity elimination Proofs (listed in order of increasing complexity): A[X] (X arbitrary) Aext[X] (X linear) A[β] and Aext[β] Aext[X] (X arbitrary)

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Immediate applications of τ-elimination

Fact

Transitivity-free derivations D ⊢ t = s trivially enjoy a sort of subterm property. Namely: for A-Systems without Ext: if D contains an application of a left (right) F-intro, resp. of a left (right) β-intro, then t (s) contains a F-redex, resp. a β-redex. for A-systems with Ext: if D contains an application of a left (right) F-intro, resp. of a left (right) β-intro, then t (s) contains an occurrence of F, resp. an

• ccurrence of λ.

Corollary 1 [Consistency]

For every analytic system A: A x = y (with x distinct from y) A (hence the corresponding synthetic system) is consistent

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Corollary 2 [Church-Rosser]

The reductions ։X

(weak X-combinatory reduction)

։β , ։βη

(β- and βη-reduction)

> −

(strong combinatory reduction)

are confluent Proof. From any given τ-free derivation D ⊢ t = s in an analytic system A, we can extract a term r such that t ։ r և s by straightforward induction on the length of D (where ։ is the appropriate reduction, depending on A).

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Further applications

Normal form(s) of τ-free derivations

τ-free derivations have nice structural properties and can be shown to normalize to suitable normal forms. By exploiting this feature, new very short demonstrations of well known results concerning reductions can be given, including: Standardization Leftmost reduction (in particular for λβη-reduction) η-Postponement, . . . On can also prove, e.g. the decidability of a natural class of fragments of CLext

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τ-elimination

A[X] systems — Proof strategy

We show how to eliminate a topmost application of τ : D1 ⊢

− t = s , D2 ⊢ − s = r

◮ D∗ ⊢

− t = r

The proof runs by ω3-induction: main: h′(D1) + h′(D2) secondary: s(D1) + s(D2) ternary: s This strategy doesn’t work when the extensionality rule is present, coupled with non linear combinators.

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Aext[X] systems — Proof strategy

We show that the following deep transitivity rule t = s Φ[ [s] ] = r Φ[ [t] ] = r

τ ∗

is eliminable. The proof consists of four main steps (in this order):

1

deep F-inversion

2

left τ-elimination

3

deep F-introduction

4

elimination of a topmost occurrence of [τ ∗]

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Step 1: deep F-inversion Lemma

The following deep combinatory inversion rules are τ-free admissible for any F ∈ X: Φ[ [Ft1 . . . tk] ] = s Φ[ [dF[t1, . . . , tk]] ] = s [Finv

l ]

s = Φ[ [Ft1 . . . tk] ] s = Φ[ [dF[t1, . . . , tk]] ] [Finv

r ]

Moreover, [Finv

l ] and [Finv r ] preserve right-handedness, resp.

left-handedness.

Proof.

“Marking” technique . . .

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Step 2: left τ-elimination

To any given pair D1 ⊢

− L t = s

and D2 ⊢

− s = r

• f τ-free derivations, such that D1 is a left derivation, we can effectively

associate a τ-free derivation D∗ ⊢

− t = r

which is a left derivation provided D2 is such.

Proof.

Main induction on s(D2), secondary induction on s(D1), ternary induction on s, using deep F-inversion.

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Step 3: deep F-introduction

The following deep combinatory introduction rules are τ-free admissible for any F ∈ X: Φ[ [dF[t1, . . . , tk]] ] = s Φ[ [Ft1 . . . tk] ] = s

[F+

l ]

s = Φ[ [dF[t1, . . . , tk]] ] s = Φ[ [Ft1 . . . tk] ]

[F+

r ]

Moreover, [F+

l ] and [F+ r ] preserve left-handedness, resp.

right-handedness.

Proof.

By left τ-elimination.

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Final step: main elimination Lemma

To each pair of τ-free derivations D1 ⊢

− t = s

and D2 ⊢

− Φ[

[s] ] = r we can effectively associate a τ-free derivation D∗ ⊢

− Φ[

[t] ] = r

Proof.

We use deep F introduction and inversion. The proof runs by ω3-induction main: s(D1) secondary: s ternary: h(D2) taking main cases according to the last inference R of D1.

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Combinatory strong reduction

Primitive combinators: I , K , S t t ρ It t I Kts t K Stsr tr(sr) S t s rt rs µ t s tr sr ν t r r s t s

τ

t s λ∗x.t λ∗x.s ξ Abstraction is defined according to the strong algorithm.

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Curry’s problem concerning > −

• H. B. Curry and R. Feys, Combinatory Logic, Vol. I, 1958

List of “Unsolved problems” in § 6 F.5 “c. Is it possible to prove the Church-Rosser property directly for strong reduction, without having recourse to transformations between that theory and the theory of λ-conversion? . . . ”

Remark

A solution was advanced by K. Loewen in 1968. His proof, however, doesn’t work because of a serious mistake — as pointed out in Hindley’s MR review (1970).

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Hindley’s statement of the problem (2006)

— Problem #1, TLCA List of Open Problems, http://tlca.di.unito.it/opltlca/

Submitted by Roger Hindley Date: Known since 1958!

• Statement. Is there a direct proof of the confluence of βη-strong reduction?

Problem Origin. First posed by Haskell Curry and Roger Hindley.

The βη-strong reduction is the combinatory analogue of βη-reduction in λ-calculus. It is confluent. Its only known confluence-proof is very easy, [Curry and Feys, 1958, 6F, p. 221 Theorem 3], but it depends on the having already proved the confluence of λβη-reduction. Thus the theory of combinators is not self-contained at present. Is there a confluence proof independent of λ-calculus?

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Hindley’s statement of the problem (2006)

— Problem #1, TLCA List of Open Problems, http://tlca.di.unito.it/opltlca/

Submitted by Roger Hindley Date: Known since 1958!

• Statement. Is there a direct proof of the confluence of βη-strong reduction?

Problem Origin. First posed by Haskell Curry and Roger Hindley.

The βη-strong reduction is the combinatory analogue of βη-reduction in λ-calculus. It is confluent. Its only known confluence-proof is very easy, [Curry and Feys, 1958, 6F, p. 221 Theorem 3], but it depends on the having already proved the confluence of λβη-reduction. Thus the theory of combinators is not self-contained at present. Is there a confluence proof independent of λ-calculus? Our confluence proof for is independent of λ-calculus!

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