The Beta Cube a 1 , 2 Pablo Nogueira 2 us Gallego Arias 2 Alvaro - PowerPoint PPT Presentation

The Beta Cube ´ ıa 1 , 2 Pablo Nogueira 2 us Gallego Arias 2 Alvaro Garc´ Emilio Jes´ 1 IMDEA Software Institute 2 Babel Research Group, Universidad Polit´ ecnica de Madrid ITU 2011, Kobenham 1 / 29

Context ◮ Reduction strategies for the pure (untyped) lambda calculus. . . 2 / 29

Context ◮ Reduction strategies for the pure (untyped) lambda calculus. . . ◮ Normal Order, Applicative (Standard) Order [Barendregt 1984]. 2 / 29

Context ◮ Reduction strategies for the pure (untyped) lambda calculus. . . ◮ Normal Order, Applicative (Standard) Order [Barendregt 1984]. ◮ Call-by-name, Call-by-value [Plotkin 1975] [Sestoft 2002]. 2 / 29

Context ◮ Reduction strategies for the pure (untyped) lambda calculus. . . ◮ Normal Order, Applicative (Standard) Order [Barendregt 1984]. ◮ Call-by-name, Call-by-value [Plotkin 1975] [Sestoft 2002]. ◮ Head spine (he) [Sestoft 2002] (headNF in [Paulson 1996]). 2 / 29

Context ◮ Reduction strategies for the pure (untyped) lambda calculus. . . ◮ Normal Order, Applicative (Standard) Order [Barendregt 1984]. ◮ Call-by-name, Call-by-value [Plotkin 1975] [Sestoft 2002]. ◮ Head spine (he) [Sestoft 2002] (headNF in [Paulson 1996]). ◮ Hybrid applicative order (ha) [Sestoft 2002]. 2 / 29

Context ◮ Reduction strategies for the pure (untyped) lambda calculus. . . ◮ Normal Order, Applicative (Standard) Order [Barendregt 1984]. ◮ Call-by-name, Call-by-value [Plotkin 1975] [Sestoft 2002]. ◮ Head spine (he) [Sestoft 2002] (headNF in [Paulson 1996]). ◮ Hybrid applicative order (ha) [Sestoft 2002]. ◮ Hybrid normal order (hn) [Sestoft 2002]. 2 / 29

Context ◮ Reduction strategies for the pure (untyped) lambda calculus. . . ◮ Normal Order, Applicative (Standard) Order [Barendregt 1984]. ◮ Call-by-name, Call-by-value [Plotkin 1975] [Sestoft 2002]. ◮ Head spine (he) [Sestoft 2002] (headNF in [Paulson 1996]). ◮ Hybrid applicative order (ha) [Sestoft 2002]. ◮ Hybrid normal order (hn) [Sestoft 2002]. ◮ Head reduction (hr) [Barendregt 1984]. 2 / 29

Context ◮ Reduction strategies for the pure (untyped) lambda calculus. . . ◮ Normal Order, Applicative (Standard) Order [Barendregt 1984]. ◮ Call-by-name, Call-by-value [Plotkin 1975] [Sestoft 2002]. ◮ Head spine (he) [Sestoft 2002] (headNF in [Paulson 1996]). ◮ Hybrid applicative order (ha) [Sestoft 2002]. ◮ Hybrid normal order (hn) [Sestoft 2002]. ◮ Head reduction (hr) [Barendregt 1984]. ◮ . . . 2 / 29

Context ◮ Reduction strategies for the pure (untyped) lambda calculus. . . ◮ Normal Order, Applicative (Standard) Order [Barendregt 1984]. ◮ Call-by-name, Call-by-value [Plotkin 1975] [Sestoft 2002]. ◮ Head spine (he) [Sestoft 2002] (headNF in [Paulson 1996]). ◮ Hybrid applicative order (ha) [Sestoft 2002]. ◮ Hybrid normal order (hn) [Sestoft 2002]. ◮ Head reduction (hr) [Barendregt 1984]. ◮ . . . ◮ . . . defined by sets of big-step rules. 2 / 29

What are strategies useful for? ◮ Program optimization via partial evaluation. 3 / 29

What are strategies useful for? ◮ Program optimization via partial evaluation. ◮ β -equivalence testers for typing rules in dependent types. 3 / 29

What are strategies useful for? ◮ Program optimization via partial evaluation. ◮ β -equivalence testers for typing rules in dependent types. ◮ Interpreting universes in structural generic programming with dependent types. 3 / 29

What are strategies useful for? ◮ Program optimization via partial evaluation. ◮ β -equivalence testers for typing rules in dependent types. ◮ Interpreting universes in structural generic programming with dependent types. ◮ . . . 3 / 29

Pure lambda calculus reduction strategies (big-step) Call-by-name (cbn): cbn x → x cbn λ x . B → λ x . B �� �� �� �� cbn → M ′ ≡ λ x . B cbn [ N / x ] B → S M cbn M N → S �� �� �� �� → M ′ �≡ λ x . B cbn M → M ′ N cbn M N 4 / 29

Pure lambda calculus reduction strategies (big-step) Call-by-name (cbn): Normal order (nor): nor cbn x → x x → x nor → B ′ B nor → λ x . B ′ cbn λ x . B λ x . B → λ x . B �� �� �� �� �� �� �� �� cbn → M ′ ≡ λ x . B cbn → M ′ ≡ λ x . B cbn nor [ N / x ] B → S [ N / x ] B → S M M nor cbn M N → S M N → S �� �� �� �� �� �� �� �� → M ′ �≡ λ x . B → M ′ �≡ λ x . B cbn cbn M ′ nor nor → M ′′ → N ′′ M M N → M ′′ N ′′ nor → M ′ N cbn M N M N Subsidiary Hybrid 4 / 29

Pure lambda calculus reduction strategies (big-step) Call-by-name (cbn): Normal order (nor): nor cbn x → x x → x nor → B ′ B nor → λ x . B ′ cbn λ x . B λ x . B → λ x . B �� �� �� �� �� �� �� �� cbn → M ′ ≡ λ x . B cbn → M ′ ≡ λ x . B cbn nor [ N / x ] B → S [ N / x ] B → S M M nor cbn M N → S M N → S �� �� �� �� �� �� �� �� → M ′ �≡ λ x . B → M ′ �≡ λ x . B cbn cbn M ′ nor nor → M ′′ → N ′′ M M N → M ′′ N ′′ nor → M ′ N cbn M N M N Subsidiary Hybrid Hybrid reduces in more places than subsidiary! 4 / 29

Pure lambda calculus reduction strategies (big-step) Rule Template: → B ′ B la var x abs → λ x . B ′ λ x . B st → x st → M ′ ≡ λ x . B → N ′ [ N ′ / x ] B su red M op 1 N ar 1 → S M N st → S → M ′ �≡ λ x . B M ′ op 2 → M ′′ → N ′ app M op 1 N ar 2 → M ′′ N ′ M N st 5 / 29

Pure lambda calculus reduction strategies (big-step) Rule Template: → B ′ B la var x abs → λ x . B ′ λ x . B st → x st → M ′ ≡ λ x . B → N ′ [ N ′ / x ] B su red M op 1 N ar 1 → S M N st → S → M ′ �≡ λ x . B M ′ op 2 → M ′′ → N ′ app M op 1 N ar 2 → M ′′ N ′ M N st 5 / 29

Pure lambda calculus reduction strategies (big-step) Rule Template (cbn): B id → B var x cbn abs λ x . B cbn → λ x . B → x → M ′ ≡ λ x . B red M cbn N id [ N / x ] B cbn → N → S M N cbn → S → M ′ �≡ λ x . B M ′ id → M ′ app M cbn N id → N → M ′ N M N cbn 5 / 29

Pure lambda calculus reduction strategies (big-step) Rule Template (cbv): B id → B var x cbv abs λ x . B cbv → λ x . B → x → M ′ ≡ λ x . B → N ′ [ N ′ / x ] B cbv red M cbv N cbv → S M N cbv → S → M ′ �≡ λ x . B M ′ id → M ′ → N ′ app M cbv N cbv → M ′ N ′ M N cbv 5 / 29

Pure lambda calculus reduction strategies (big-step) Rule Template (aor): → B ′ B aor var x aor abs → λ x . B ′ λ x . B aor → x → M ′ ≡ λ x . B → N ′ [ N ′ / x ] B aor red M aor N aor → S M N aor → S → M ′ �≡ λ x . B M ′ id → M ′ → N ′ app M aor N aor → M ′ N ′ M N aor 5 / 29

Pure lambda calculus reduction strategies (big-step) Rule Template (nor): → B ′ B nor var x nor abs → λ x . B ′ λ x . B nor → x → M ′ ≡ λ x . B red M cbn N id [ N / x ] B nor → N → S M N nor → S → M ′ �≡ λ x . B M ′ nor → M ′′ → N ′ app M cbn N nor → M ′′ N ′ M N nor Use of op 1 and op 2 to accomodate hybrid strategies! 5 / 29

The Beta Cube Parameters la , ar 1 and ar 2 are either recursive calls or identity. Interpreted as boolean switches: 6 / 29

� � � � � � � � � � The Beta Cube Parameters la , ar 1 and ar 2 are either recursive calls or identity. Interpreted as boolean switches: la AOR CBV � � � � � � � � � � � � ar 1 � � 1 0 1 0 0 1 ( ⊑ ) � � � 1 1 0 0 1 0 � � � � ar 2 � � � � � � HE CBN 6 / 29

� � � � � � � � � � The Beta Cube Parameters la , ar 1 and ar 2 are either recursive calls or identity. Interpreted as boolean switches: la AOR CBV 0 weakness 1 strength � � � � � � � � � � � � ar 1 � � 0 non-strictness 1 strictness 1 0 1 0 0 1 ( ⊑ ) � � � 1 1 0 0 1 0 � � � � ar 2 0 headness 1 non-headness � � � � � � HE CBN 6 / 29

� � � � � � � � � � � � � � The Beta Cube Parameters la , ar 1 and ar 2 are either recursive calls or identity. Interpreted as boolean switches: la AOR CBV 0 weakness 1 strength � � � � � � � � � � � � ar 1 � � 0 non-strictness 1 strictness 1 0 1 0 0 1 ( ⊑ ) � � � 1 1 0 0 1 0 � � � � ar 2 0 headness 1 non-headness � � � � � � HE CBN nf wnf hnf whnf 6 / 29

� � � � � � � � � � Axis of eval var x cbn abs λ x . B cbn → λ x . B → x → M ′ ≡ λ x . B red M cbn [ N / x ] B cbn → S M N cbn → S → M ′ �≡ λ x . B app M cbn → M ′ N M N cbn la strength AOR CBV � � � � � � � � � � ar 1 � � � strictness � 1 0 1 0 0 1 � � � 1 1 0 0 1 0 non-headness ar 2 � � � � � � �� �� �� �� � � � � HE CBN 7 / 29