Isomorphic Data Type Transformations Alessandro Coglio Stephen - PowerPoint PPT Presentation

Isomorphic Data Type Transformations Alessandro Coglio Stephen Westfold KESTREL INSTITUTE

Isomorphic data type transformations are useful in program synthesis. some of these may be 𝑡 ! requirements specification isomorphic transformations 𝑡 ! e.g. stepwise refinements . . . intermediate specifications § represent finite sets as repetition-free ordered lists § turn unbounded integers 𝑡 " into bounded integers code generation (under preconditions) (optional in ACL2) 𝑞 implementation § add redundant record components for caching § change loop direction § ...

Isomorphic data type transformations are useful in program synthesis. They are also useful in program analysis. some of these may be isomorphic transformations, " 𝑡 $ which are inherently reversible higher-level representations, e.g. anti-refinements, . . . which may be easier to verify via “inverses” of the transformations § represent repetition-free for refinements " 𝑡 # ordered lists as finite sets § turn bounded integers into " 𝑡 ! code representation unbounded integers (under preconditions) code lifting § remove redundant record 𝑞′ existing program components for caching § change loop direction § ...

Isomorphic data type transformations are useful in program synthesis. They are also useful in program analysis, as well as in analysis-by-synthesis. 𝑡 ! " 𝑡 ! 𝑡 $ top-down . . . derivation 𝑡 # . . . 𝑡 % equal or trivially . . . end-to-end proof " equivalent 𝑡 $ " 𝑡 # that 𝑞′ satisfies 𝑡 ! . . . " 𝑡 % 𝑡 ! bottom-up anti-derivation " 𝑡 ! 𝑞 𝑞′ 𝑞′

Consider two isomorphic sets (data types) 𝑌 and 𝑌 ! with 𝜊 ∶ 𝑌 ⟶ 𝑌′ and 𝜊 "# ∶ 𝑌′ ⟶ 𝑌 . 𝑌 𝜊 "# ∘ 𝜊 = 𝑗𝑒 $ 𝜊 "# 𝜊 𝜊 ∘ 𝜊 "# = 𝑗𝑒 $! 𝑌′

Consider two isomorphic sets (data types) 𝑌 and 𝑌 ! with 𝜊 ∶ 𝑌 ⟶ 𝑌′ and 𝜊 "# ∶ 𝑌′ ⟶ 𝑌 . Consider two isomorphic sets (data types) 𝑍 and 𝑍′ with 𝜑 ∶ 𝑍 ⟶ 𝑍′ and 𝜑 "# ∶ 𝑍′ ⟶ 𝑍 . 𝑌 𝑍 𝜊 "# ∘ 𝜊 = 𝑗𝑒 $ 𝜑 "# ∘ 𝜑 = 𝑗𝑒 % 𝜊 "# 𝜑 "# 𝜊 𝜑 𝜊 ∘ 𝜊 "# = 𝑗𝑒 $! 𝜑 ∘ 𝜑 "# = 𝑗𝑒 %! 𝑌′ 𝑍′

Consider two isomorphic sets (data types) 𝑌 and 𝑌 ! with 𝜊 ∶ 𝑌 ⟶ 𝑌′ and 𝜊 "# ∶ 𝑌′ ⟶ 𝑌 . Consider two isomorphic sets (data types) 𝑍 and 𝑍′ with 𝜑 ∶ 𝑍 ⟶ 𝑍′ and 𝜑 "# ∶ 𝑍′ ⟶ 𝑍 . Consider a function 𝑔 ∶ 𝑌 ⟶ 𝑍 , a computation from inputs of type 𝑌 to outputs of type 𝑍 . 𝑌 𝑍 𝑔 𝜊 "# ∘ 𝜊 = 𝑗𝑒 $ 𝜑 "# ∘ 𝜑 = 𝑗𝑒 % 𝜊 "# 𝜑 "# 𝜊 𝜑 𝜊 ∘ 𝜊 "# = 𝑗𝑒 $! 𝜑 ∘ 𝜑 "# = 𝑗𝑒 %! 𝑌′ 𝑍′

Consider two isomorphic sets (data types) 𝑌 and 𝑌 ! with 𝜊 ∶ 𝑌 ⟶ 𝑌′ and 𝜊 "# ∶ 𝑌′ ⟶ 𝑌 . Consider two isomorphic sets (data types) 𝑍 and 𝑍′ with 𝜑 ∶ 𝑍 ⟶ 𝑍′ and 𝜑 "# ∶ 𝑍′ ⟶ 𝑍 . Consider a function 𝑔 ∶ 𝑌 ⟶ 𝑍 , a computation from inputs of type 𝑌 to outputs of type 𝑍 . We can mechanically construct a function 𝑔 ! ∶ 𝑌′ ⟶ 𝑍′ that makes the diagram commute. 𝑌 𝑍 𝑔 𝜊 "# ∘ 𝜊 = 𝑗𝑒 $ 𝜑 "# ∘ 𝜑 = 𝑗𝑒 % 𝜊 "# 𝜑 "# 𝜊 𝜑 𝜊 ∘ 𝜊 "# = 𝑗𝑒 $! 𝜑 ∘ 𝜑 "# = 𝑗𝑒 %! 𝑔′ 𝑌′ 𝑍′ 𝑔 = 𝜑 "# ∘ 𝑔′ ∘ 𝜊 𝑔′ = 𝜑 ∘ 𝑔 ∘ 𝜊 "# ⟺

Consider two isomorphic sets (data types) 𝑌 and 𝑌 ! with 𝜊 ∶ 𝑌 ⟶ 𝑌′ and 𝜊 "# ∶ 𝑌′ ⟶ 𝑌 . Consider two isomorphic sets (data types) 𝑍 and 𝑍′ with 𝜑 ∶ 𝑍 ⟶ 𝑍′ and 𝜑 "# ∶ 𝑍′ ⟶ 𝑍 . Consider a function 𝑔 ∶ 𝑌 ⟶ 𝑍 , a computation from inputs of type 𝑌 to outputs of type 𝑍 . We can mechanically construct a function 𝑔 ! ∶ 𝑌′ ⟶ 𝑍′ that makes the diagram commute. 𝑌 𝑍 𝑔 we could just define 𝑔′ like this, but that is not very interesting 𝜊 "# 𝜑 "# 𝜊 𝜑 𝑔′ ≡ 𝜑 ∘ 𝑔 ∘ 𝜊 "# 𝑔′ 𝑌′ 𝑍′ 𝑔′ = 𝜑 ∘ 𝑔 ∘ 𝜊 "#

Consider two isomorphic sets (data types) 𝑌 and 𝑌 ! with 𝜊 ∶ 𝑌 ⟶ 𝑌′ and 𝜊 "# ∶ 𝑌′ ⟶ 𝑌 . Consider two isomorphic sets (data types) 𝑍 and 𝑍′ with 𝜑 ∶ 𝑍 ⟶ 𝑍′ and 𝜑 "# ∶ 𝑍′ ⟶ 𝑍 . Consider a function 𝑔 ∶ 𝑌 ⟶ 𝑍 , a computation from inputs of type 𝑌 to outputs of type 𝑍 . We can mechanically construct a function 𝑔 ! ∶ 𝑌′ ⟶ 𝑍′ that makes the diagram commute. 𝑌 𝑍 𝑔 𝑦 ≡ 𝐣𝐠 𝑏 𝑦 representative 𝑔 𝐮𝐢𝐟𝐨 𝑐 𝑦 recursive 𝐟𝐦𝐭𝐟 𝑑(𝑦, 𝑔(𝑒(𝑦))) definition 𝑏 ⊆ 𝑌 𝑐 ∶ 𝑌 ⟶ 𝑍 𝑑 ∶ 𝑌 × 𝑍 ⟶ 𝑍 𝑒 ∶ 𝑌 ⟶ 𝑌 𝑔 terminates ⊢ ¬𝑏 𝑦 ⟹ 𝜈(𝑒(𝑦)) ≺ 𝜈(𝑦) 𝜊 "# 𝜑 "# 𝜊 𝜑 keep the same structure and add the conversions 𝑔′ 𝑦 " ≡ 𝐣𝐠 𝑏(𝜊 &# 𝑦 " ) 𝐮𝐢𝐟𝐨 𝜑(𝑐(𝜊 &# 𝑦 " )) 𝑔′ 𝐟𝐦𝐭𝐟 𝜑(𝑑(𝜊 &# 𝑦 " , 𝜑 &# 𝑔′(𝜊 𝑒(𝜊 &# 𝑦 " ) ) )) 𝑌′ 𝑍′ 𝑔′ = 𝜑 ∘ 𝑔 ∘ 𝜊 "#

Consider two isomorphic sets (data types) 𝑌 and 𝑌 ! with 𝜊 ∶ 𝑌 ⟶ 𝑌′ and 𝜊 "# ∶ 𝑌′ ⟶ 𝑌 . Consider two isomorphic sets (data types) 𝑍 and 𝑍′ with 𝜑 ∶ 𝑍 ⟶ 𝑍′ and 𝜑 "# ∶ 𝑍′ ⟶ 𝑍 . Consider a function 𝑔 ∶ 𝑌 ⟶ 𝑍 , a computation from inputs of type 𝑌 to outputs of type 𝑍 . We can mechanically construct a function 𝑔 ! ∶ 𝑌′ ⟶ 𝑍′ that makes the diagram commute. 𝑌 𝑍 𝑔 𝑦 ≡ 𝐣𝐠 𝑏 𝑦 representative 𝑔 𝐮𝐢𝐟𝐨 𝑐 𝑦 recursive 𝐟𝐦𝐭𝐟 𝑑(𝑦, 𝑔(𝑒(𝑦))) definition 𝑏 ⊆ 𝑌 𝑐 ∶ 𝑌 ⟶ 𝑍 𝑑 ∶ 𝑌 × 𝑍 ⟶ 𝑍 𝑒 ∶ 𝑌 ⟶ 𝑌 𝑔 terminates ⊢ ¬𝑏 𝑦 ⟹ 𝜈(𝑒(𝑦)) ≺ 𝜈(𝑦) 𝜊 "# 𝜑 "# 𝜊 𝜑 keep the same structure automatic and add the conversions 𝑔′ 𝑦 " ≡ 𝐣𝐠 𝑏(𝜊 &# 𝑦 " ) 𝐮𝐢𝐟𝐨 𝜑(𝑐(𝜊 &# 𝑦 " )) 𝑔′ 𝐟𝐦𝐭𝐟 𝜑(𝑑(𝜊 &# 𝑦 " , 𝜑 &# 𝑔′(𝜊 𝑒(𝜊 &# 𝑦 " ) ) )) 𝑌′ 𝑍′ n o i t 𝜈 " ≡ 𝜈 ∘ 𝜊 &# c u 𝑔′ terminates because 𝑔 does d n i y b ⊢ 𝑔′ = 𝜑 ∘ 𝑔 ∘ 𝜊 "#

𝑔 𝑦 ≡ 𝐣𝐠 𝑏 𝑦 𝐮𝐢𝐟𝐨 𝑐 𝑦 𝐟𝐦𝐭𝐟 𝑑(𝑦, 𝑔(𝑒(𝑦))) 𝑌 𝑍 𝑔 keep the same structure automatic and add the conversions 𝑔′ 𝑦 " ≡ 𝐣𝐠 𝑏(𝜊 &# 𝑦 " ) 𝜊 "# 𝜑 "# 𝜊 𝜑 𝐮𝐢𝐟𝐨 𝜑(𝑐(𝜊 &# 𝑦 " )) 𝐟𝐦𝐭𝐟 𝜑(𝑑(𝜊 &# 𝑦 " , 𝜑 &# 𝑔′(𝜊 𝑒(𝜊 &# 𝑦 " ) ) )) expand the definitions user-guided and rewrite/simplify 𝑔′ 𝑌′ 𝑍′ 𝑔′′ 𝑦 " ≡ 𝐣𝐠 𝑏′ 𝑦′ goal: no trace of 𝐮𝐢𝐟𝐨 𝑐′ 𝑦′ ⊢ 𝑔′ = 𝜑 ∘ 𝑔 ∘ 𝜊 "# 𝑌 , 𝑍 , 𝜊 , 𝜊 &# , 𝜑 , 𝜑 &# 𝐟𝐦𝐭𝐟 𝑑′(𝑦′, 𝑔′′(𝑒′(𝑦′))) ⊢ 𝑔 "" = 𝑔′

This is a general method: Consider a function 𝑕 that calls 𝑔 , 𝑔 # , 𝑔 0 , etc. automatically create an isomorphic version We can apply the same general method to 𝑕 . and semi-automatically rewrite/simplify it. If 𝑕 manipulates the data being transformed only through 𝑔 , 𝑔 # , 𝑔 0 , etc., we can automate We can do it for 𝑔 , 𝑔 # , 𝑔 0 , etc., obtaining 𝑔′ , 𝑔′′ , 𝑔 # ′ , 𝑔 # ′′ , 𝑔 0 ′ , 𝑔 0 ′′ , etc., the rewriting/simplification step as well. 𝑔 𝑦 ≡ … 𝑕 … ≡ … 𝑔 … … keep the same structure keep the same structure and add the conversions and add the conversions ⊢ 𝑔 = 𝜑 "# ∘ 𝑔′ ∘ 𝜊 ≡ … 𝜑(𝑔(𝜊 &# … )) … 𝑔′ 𝑦′ ≡ … 𝑕′ … expand the definitions expand the definitions and rewrite/simplify and rewrite/simplify 𝑔′′ 𝑦 " ≡ … 𝑕′′ … ≡ …

This is a general method: Consider a function 𝑕 that calls 𝑔 , 𝑔 # , 𝑔 0 , etc. automatically create an isomorphic version We can apply the same general method to 𝑕 . and semi-automatically rewrite/simplify it. If 𝑕 manipulates the data being transformed only through 𝑔 , 𝑔 # , 𝑔 0 , etc., we can automate We can do it for 𝑔 , 𝑔 # , 𝑔 0 , etc., obtaining 𝑔′ , 𝑔′′ , 𝑔 # ′ , 𝑔 # ′′ , 𝑔 0 ′ , 𝑔 0 ′′ , etc., the rewriting/simplification step as well. And we can do everything in one step. 𝑔 𝑦 ≡ … 𝑕 … ≡ … 𝑔 … … keep the same structure and add the conversions keep the same structure and replace 𝑔 with 𝑔′ etc. 𝑔′ 𝑦′ ≡ … expand the definitions 𝑕′ … ≡ … 𝑔′ … … and rewrite/simplify 𝑔′′ 𝑦 " ≡ …