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

Intuitive motivations: two analogies P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 2 / 1

Intuitive motivations: two analogies Analogy 1: Modus ponens / transitivity rule for equality: A → B t = r r = s A ≈ t = s B P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 2 / 1

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 = s t = r r = s t = t [ refl ] s = t [ symm ] [ trs ] t = s t i = s i f n ( t 1 , . . . , t n ) = f n ( t 1 . . . , t i − 1 , s i , t i + 1 , . . . , t n ) [ congr: 1 ≤ i ≤ n ] Birkhoff’s completeness theorem for equational logic (1935): ⊢ E t = s ⇔ E | = t = s P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 3 / 1

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 P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 4 / 1

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 ? P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 5 / 1

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) P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 6 / 1

(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. P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 7 / 1

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). . . . P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 8 / 1

Overview ✞ ☎ synthetic (CL-like or λ ) proof-systems (“S-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”) ✝ ✆ 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 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 P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 9 / 1

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 : CL ext (& generalizations) P .M. 2009, A solution to Curry and Hindley’s problem on combinatory strong reduction , Arch. Math. Logic 48, 159-184. P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 10 / 1

Synthetic systems CL Axiom schemas: K ts = t S tsr = tr ( sr ) [ I t = t ] Inference rules: ̺ (reflexivity) σ (symmetry) τ (transitivity) t = s t = s (app-congruence rules) rt = rs µ tr = sr ν CL ext tx = sx := CL + { x / ∈ V ( ts ) } Ext t = s P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 11 / 1

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 � k F , d F � s.t.: k F , the index of F under X , is a non negative integer; d F , the definition of F under X , is a term with V ( d F ) ⊆ { v 1 , . . . , v k F } . Intuitively, X fixes an axiom schema for each primitive combinator F ∈ X : F t 1 . . . t k F = d F [ v 1 / t 1 , . . . , v k F / t k F ] ( AX F ) X CL [ X ] / CL ext [ X ] are now defined exactly as CL / CL ext , except that the axiom schemas for the combinators K , S (I) are replaced by the set { ( AX F ) X | F ∈ X } of axiom schemas induced by X . P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 12 / 1

Example 1 The familiar pair { K , S } of primitive combinators of CL corresponds to the combinatory system C such that: C = { K , S } and k K = 2 d K = v 1 and k S = 3 d S = v 1 v 3 ( v 2 v 3 ) Example 2 But also the following is a perfectly legitimate (maybe odd!) combinatory system: X = { F , G , H } and d F = FF ( HG )( GH ) k F = 0 and d G = v 2 GH v 1 k G = 2 and d H = v 1 v 1 (( H v 2 )( F v 1 ))( v 4 H ) k H = 4 P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 13 / 1

λβ Axiom schema: ( λ x . t ) r = t [ x / r ] ( β -conversion) Inference rules: ̺ , σ , τ , µ , ν , plus t = s (abstr-congruence rule) λ x . t = λ x . s ξ λβη := λβ + λ x . tx = t { x / ∈ V ( t ) } ( η -conversion) or, equivalently, λβ + Ext P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 14 / 1

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. P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 15 / 1

Example: introduction rules for the combinator S S tsr = tr ( sr ) [ AX S ] � � � � � � tr ( sr ) p 1 . . . p n = q q = tr ( sr ) p 1 . . . p n S tsrp 1 . . . p n = q [ S l ] [ S r ] q = S tsrp 1 . . . p n where n ≥ 0 , i.e.: the “side terms” p 1 , . . . , p n may be missing P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 16 / 1

In general: introduction rules for a primitive combinator F F t 1 . . . t k F = d F [ t 1 , . . . , t k F ] ( AX F ) X � � � � � � d F [ t 1 , . . . , t k F ] p 1 . . . p n = s s = d F [ t 1 , . . . , t k F ] p 1 . . . p n [ F l ] X [ F r ] X F t 1 . . . t k F p 1 . . . p n = s s = F t 1 . . . t k F p 1 . . . p n where n ≥ 0 . (We write t [ s 1 , . . . , s n ] short for t [ v 1 / s 1 , . . . , v n / s n ] ) P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 17 / 1

β -introduction rules ( λ x . t ) r = t [ x / r ] [ β − conv ] � � � � � � t [ x / r ] p 1 . . . p n = q q = t [ x / r ] p 1 . . . p n ( λ x . t ) rp 1 . . . p n = q [ β l ] [ β r ] q = ( λ x . t ) rp 1 . . . p n where n ≥ 0 , i.e.: the “side terms” p 1 , . . . , p n may be missing P . Minari (UNIFI) Transitivity Elimination JÄGERFEST 2013 18 / 1