Degrees of parallelism and Boolean algebras Guillaume Geoffroy I2M, Aix-Marseille Université 12 June 2019 1 / 9
Denotational models Programming Denotational language model
Denotational models Programming Denotational language model program t denotation � t �
Denotational models Programming Denotational language model program t denotation � t � � (fun x → t ) u � = � t [ x := u ] �
Denotational models Programming Denotational language model program t denotation � t � � (fun x → t ) u � = � t [ x := u ] � etc. 2 / 9
Left-first or fun x y if ( x ) then true → else y 3 / 9
Left-first or fun x y if ( x ) then true → else y x y true false true true true false true false 3 / 9
Left-first or fun x y if ( x ) then true → else y x y true false true true true false true false ? true 3 / 9
Left-first or fun x y if ( x ) then true → else y x y true false true true true false true false ? true ? 3 / 9
Left-first or fun x y if ( x ) then true → else y x y true false ? true true true ? false true false ? ? true ? ? 3 / 9
Right-first or fun x y if ( y ) true → else x x y true false ? true true true true false true false ? ? ? ? ? 4 / 9
Parallel (symmetric) or x y true false ? true true true true false true false ? ? true ? ? 5 / 9
Parallel (symmetric) or fun x y ??? → x y true false ? true true true true false true false ? ? true ? ? 5 / 9
Voting function true true ? true ? true true true true ? true true false false ? false ? false false false false ? false false 6 / 9
Voting and parallel or can simulate each other 7 / 9
Voting and parallel or can simulate each other ← → � − → � � ← − � or = fun x y → vote or x y or x y 7 / 9
Voting and parallel or can simulate each other ← → � − → � � ← − � or = fun x y → vote or x y or x y true 7 / 9
Voting and parallel or can simulate each other ← → � − → � � ← − � or = fun x y → vote or x y or x y true ( ← and x y ) ( ← → → ( ← → � ← � �� vote = fun x y z → ← → → or or and y z ) and z x ) 7 / 9
Degrees of parallelism ← → vote or vote 4 G 3 Sequential 8 / 9
Degrees of parallelism: the wild west ← → vote or vote 4 G 3 Sequential 8 / 9
Degrees of parallelism and Boolean algebras Formulas in the language Very particular of Boolean algebras denotational models
Degrees of parallelism and Boolean algebras Formulas in the language Very particular of Boolean algebras denotational models formula F d denotation d
Degrees of parallelism and Boolean algebras Formulas in the language Very particular of Boolean algebras denotational models formula F d denotation d d simulates e iff
Degrees of parallelism and Boolean algebras Formulas in the language Very particular of Boolean algebras denotational models formula F d denotation d F d entails F e d simulates e iff 9 / 9
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