Rv-systems Interactive systems with registers and voices Gheorghe Stefanescu Faculty of Mathematics and Computer Science University of Bucharest Pisa, Italy, 4-th June, 2007 Slide 1/88 Rv-systems / Stefanescu / Pisa, 2007
Rv-systems Contents: • Generalities • A glimpse on AGAPIA programming • Finite interactive systems ← [nfa] • Rv-programs ← [flowchart programs] • Structured rv-programs ← [while programs] • Compiling srv-programs • Floyd-Hoare logics for (s)rv-programs • Miscellaneous • Conclusions Slide 2/88 Rv-systems / Stefanescu / Pisa, 2007
History History • space-time duality “thesis” – Stefanescu, Network algebra , Springer 2000 • finite interactive systems – Stefanescu, Marktoberdorf Summer School 2001 • rv-systems (interactive systems with registers and voices) – Stefanescu, NUS, Singapore, summer 2004 • structured rv-systems – Stefanescu, Dragoi, fall 2006 Slide 3/88 Rv-systems / Stefanescu / Pisa, 2007
ST-Dual picture space High level: x:Int, y:Array[1..10] of Int; ST-Dual picture time tape High level: temporal data memory state stream job requirement Software new job component new state Slide 4/88 Rv-systems / Stefanescu / Pisa, 2007
Processes and transactions Processes and transactions Proc 1 Proc 2 Proc 3 Trans 1 Trans 2 Trans 3 Slide 5/88 Rv-systems / Stefanescu / Pisa, 2007
High level temporal structures sLinkedList 2001 2002 2003 data with usual ( spatial ) 2003 2001 2002 representation: tLinkedList sArray I will be sBool, sInt, sArray, next week sLinkedList, etc. sInt tArray and their time dual 7 3 5 3 on Moon I will be (i.e., data with temporal sBool representation ): 1 tInt next week on Moon tBool, tInt, tArray, 7 3 tLinkedList, etc. tBool 5 3 1 Slide 6/88 Rv-systems / Stefanescu / Pisa, 2007
..High level temporal structures Three allocations of a temporal linked list on a stream: The 1st starts at time t = 10 and is time : 910111213141516171819202122232425262728293031323334353637 .. I w i l l b e o n M o o n n e x t w e e k .. .. 1112131415161718192021222324252627282930313233343536 ⊥ .. The 2nd allocation starts at time t = 19 and is time : 910111213141516171819202122232425262728293031323334353637 .. n e x t w e e k I w i l l b e o n M o o n .. .. 1112131415161718 ⊥ 202122232425262728293031323334353610 .. The 3rd allocation starts at time t = 21 and is time : 910111213141516171819202122232425262728293031323334353637 b e e e e I i k l l M o o o n n n t w w x .. .. .. 34162726323517123620231024 ⊥ 25112830293113141815221933 .. Slide 7/88 Rv-systems / Stefanescu / Pisa, 2007
Space-time converters; computation in space ... n x y − in time space-to-time ... ... ... and ... ... y time-to-space ... converters 2 y space−to−time time−to−space may be used to ... change n x y − computed in space n y computation in time ... ... into a y computation in space paradigm ... ... y 2 y ... n y n y ... Slide 8/88 Rv-systems / Stefanescu / Pisa, 2007
Rv-systems Contents: • Generalities • A glimpse on AGAPIA programming • Finite interactive systems ← [nfa] • Rv-programs ← [flowchart programs] • Structured rv-programs ← [while programs] • Compiling srv-programs • Floyd-Hoare logics for (s)rv-programs • Miscellaneous • Conclusions Slide 9/88 Rv-systems / Stefanescu / Pisa, 2007
Srv-programs for perfect numbers A specification for perfect numbers: 3 components C x , C y , C z where: • C x : read n from north and write n ⌢ ⌊ n / 2 ⌋ ⌢ ( ⌊ n / 2 ⌋− 1 ) ⌢ ... ⌢ 2 ⌢ 1 on east; • C y : read n ⌢ ⌊ n / 2 ⌋ ⌢ ( ⌊ n / 2 ⌋− 1 ) ⌢ ... ⌢ 2 ⌢ 1 from west and write n ⌢ φ ( ⌊ n / 2 ⌋ ) ⌢ ... ⌢ φ ( 2 ) ⌢ φ ( 1 ) on east [ φ ( k ) = “if k divides n then k else 0”]; read n ⌢ φ ( ⌊ n / 2 ⌋ ) ⌢ ... ⌢ φ ( 2 ) ⌢ φ ( 1 ) from west and • C z : subtract from the first the other numbers. These components are composed horizontally . The global input-output specification: if the input number in C x is n, then the output number in C z is 0 iff n is perfect . Slide 10/88 Rv-systems / Stefanescu / Pisa, 2007
..Srv-programs for perfect numbers Two scenarios for perfect numbers: x=6 X tx=6 Y tx=6 Z x=5 X tx=5 Y tx=5 Z x=3 y=6 z=6 tx=3 V tx=3W x=2 y=5 z=5 U tx=2 V tx=0W x=2 y=6 z=3 U tx=2 V tx=2W x=1 y=5 z=5 U tx=1 V tx=1W x=1 y=6 z=1 U tx=1 V tx=1W x=0 y=4 z=4 U x=0 y=6 z=0 (a) (b) Types are denoted as � west | north � → � east | south � Our (s)rv-scenarios are similar with the tiles of Bruni-Gadducci-Montanari, et.al. Slide 11/88 Rv-systems / Stefanescu / Pisa, 2007
..Srv-programs for perfect numbers The 1st AGAPIA program Perfect1 (construction by rows): (X # Y # Z) % while t(x>0) { U # V # W } Its type is Perfect1 : � nil | sn ; nil ; nil � → � nil | sn ; sn ; sn � . Modules: X:: module { listen nil; }{ read x:sn; } { tx:tn; tx=x; x=x/2; }{ speak tx; }{ write x; } Y:: module { listen tx:tn; }{ read nil; } { y:sn; y=tx; }{ speak tx; }{ write y; } Z:: module { listen tx:tn; }{ read nil; } { z:sn; z=tx; }{ speak nil; }{ write z; } U:: module { listen nil; }{ read x:sn; } { tx:tn; tx=x; x=x-1; }{ speak tx; }{ write x; } V:: module { listen tx:tn; }{ read y:sn; } { if(y%tx != 0) tx=0; }{ speak tx; }{ write y; } W:: module { listen tx:tn; }{ read z:sn } { z=z-tx; }{ speak nil; }{ write z; } Slide 12/88 Rv-systems / Stefanescu / Pisa, 2007
..Srv-programs for perfect numbers The 2nd AGAPIA program Perfect2 (construction by columns): (X % while t(x>0) { U } % U1) # (Y % while t(tx>-1) { V } % V1) # (Z % while t(tx>-1) { W } % W1) Its type is Perfect2 : � nil | sn ; nil ; nil � → � nil | nil ; nil ; sn � . New modules: U1:: module { listen nil; }{ read x:sn; } { tx:tn; tx=-1; }{ speak tx; }{ write nil; } V1:: module { listen tx:tn; }{ read y:sn; } { null; }{ speak tx; }{ write nil; } W1:: module { listen tx:tn; }{ read z:sn } { null; }{ speak nil; }{ write z; } Slide 13/88 Rv-systems / Stefanescu / Pisa, 2007
Rv-systems Contents: • Generalities • A glimpse on AGAPIA programming • Finite interactive systems ← [nfa] • Rv-programs ← [flowchart programs] • Structured rv-programs ← [while programs] • Compiling srv-programs • Floyd-Hoare logics for (s)rv-programs • Miscellaneous • Conclusions Slide 14/88 Rv-systems / Stefanescu / Pisa, 2007
Grids (or planar words) A grid (or planar word ) is • a rectangular two-dimensional area • filled in with letters from a given alphabet Example: aabbabb (not used here: aabb... ) abbcdbb ..bc..b bbabbca bbabbca ccccaaa ..c...a A grid p has a north (resp. south, west, east ) border denoted as n ( p ) ( resp. s ( p ) , w ( p ) , e ( p )) Notice: The requirement to have a rectangular area may be weakened, e.g., one may require to have a connected area, not a rectangular one. Slide 15/88 Rv-systems / Stefanescu / Pisa, 2007
..Grids (or planar words) Causality in a grid/scenario: a b c d e f g h Action vs. inter-action: a a b b X Y X Y Z Z a b a b (a) (b) • a two-ways interaction [in (a)] • ... and its grid/scenario representation [in (b)] Slide 16/88 Rv-systems / Stefanescu / Pisa, 2007
The flattening operator The flattening operator ♭ : LangGrids ( V ) → LangWords ( V ) maps sets of grids to sets of strings representing their topological sorting. Example: —start with efgh ; there is one minimal element a ; after its abcd deletion we get efgh ; bcd —the minimal elements are b and e ; suppose we choose b ; what remains is efgh ; and so on; cd —finally a usual word, say abecfgdh , is obtained. Actually, ♭ ( efgh ) = abcd { abcdefgh , abcedfgh , abcefdgh , abcefgdh , abecdfgh , abecfdgh , abecfgdh , abefcdgh , abefcgdh , aebcdfgh , aebcfdgh , aebcfgdh , aebfcdgh , aebfcgdh } Slide 17/88 Rv-systems / Stefanescu / Pisa, 2007
Composition and identities on grids • horizontal composition v ⊲ w v w —it is defined only if e ( v ) = w ( w ) — v ⊲ w is the word obtained putting v on the left of w • vertical composition v · w v —it is defined only if s ( v ) = n ( w ) w — v · w is the word obtained putting v on top of w • vertical identity ε k : —with w ( ε k ) = e ( ε k ) = 0 and n ( ε k ) = s ( ε k ) = k .. • horizontal identity λ k : . —with w ( λ k ) = e ( λ k ) = k and n ( λ k ) = s ( λ k ) = 0 . Slide 18/88 Rv-systems / Stefanescu / Pisa, 2007
Two-dimensional regular expressions Signature: two sets of regular algebra operators, sharing the additive part + , 0 , · , ⋆ , | , ⊲, † , − — (+ , 0 , · , ⋆ , | ) - a Kleene signature for the vertical dimension — (+ , 0 ,⊲, † , − ) - a Kleene signature for the horizontal dimension Two-dimensional regular expressions (denoted 2 RegExp ( V ) ): E :: = a ( ∈ V ) | 0 | E + E | E ∩ E | E · E | E ⋆ | | | E ⊲ E | E † | − Theorem: 2 RegExp ( V ) enriched with letter-to-letter homomorphisms are equivalent with FIS ’s (finite interactive systems). Slide 19/88 Rv-systems / Stefanescu / Pisa, 2007
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