Games in LTL Fragments Salvatore La Torre Dipartimento di - PowerPoint PPT Presentation

Talk Outline � Overview � Notation and general solution to LTL games � Upper bounds: deteministic generators � Lower bounds � Encoding TMs without “next” and “until” � Expspace-hardness of B(L , , ( � )) � 2Exptime-hardness of L , , , ( � ) � Conclusions

Proving lower bounds � Encode acceptance problem for Turing Machines � Crucial point: C j+1 C j i i-1 i i+1 � Problems: � Zoom to a cell content � Compare cells of consecutive configurations

With “until” and “next” � Zoom to cell i = n(b n …b 1 ): � b n …b 1 a to encode “cell b n …b 1 contains a” � � � (b n � (… � (b 1 � a) …)) to check it � Compare across configurations: � Modulo-2 counter to distinguish among consecutive configurations � Constructs of type 0 U (1 � 1 )

New encoding of computations

New encoding of computations � � only checks for subsequences Es. � (b n � (… � (b 1 � a) …)), (“b n …b 1 a” may not be consecutive) � <a 0 > 0 <a 1 > 1 … <a i > i …<a 2n-1 > 2n-1 (proper sequence)

New encoding of computations � � only checks for subsequences Es. � (b n � (… � (b 1 � a) …)), (“b n …b 1 a” may not be consecutive) � <a 0 > 0 <a 1 > 1 … <a i > i …<a 2n-1 > 2n-1 (proper sequence) p n …p 1 a i q 1 …q n

New encoding of computations � � only checks for subsequences Es. � (b n � (… � (b 1 � a) …)), (“b n …b 1 a” may not be consecutive) � <a 0 > 0 <a 1 > 1 … <a i > i …<a 2n-1 > 2n-1 (proper sequence) p n …p 1 a i q 1 …q n p n …p 1 : binary encoding for i q n …q 1 : binary encoding for 2 n -1-i

New encoding of computations � � only checks for subsequences Es. � (b n � (… � (b 1 � a) …)), (“b n …b 1 a” may not be consecutive) � <a 0 > 0 <a 1 > 1 … <a i > i …<a 2n-1 > 2n-1 (proper sequence) p n …p 1 a i q 1 …q n p n …p 1 : binary encoding for i q n …q 1 : binary encoding for 2 n -1-i (p j � {p j0 ,p j1 }, q j � {q j0 ,q j1 })

Property of proper sequences

Property of proper sequences � For <a i > i = u a i v (u-address, v-address) : � <a 0 > 0 ………<a i-1 > i-1 u is the shortest prefix containing u as a subsequence � v <a i+1 > i+1 ………<a 2n-1 > 2n-1 is the shortest suffix containing v as a subsequence � Therefore: � u a v is a subseq of <a 0 > 0 <a 1 > 1 …<a 2n-1 > 2n-1 iff a=a i

Example: proper sequences � 3-bits encoding of aababbab: 000a111 001a011 010b101 011a001 100b110 101b010 110a100 111b000 � For u=011, v=001 : u=011 000a111 001a011 010b101 011 v=001 01 100b110 101b010 110a100 111b000

Example: proper sequences � 3-bits encoding of aababbab: 000a111 001a011 010b101 011a001 100b110 101b010 110a100 111b000 � For u=011, v=001 : u=011 000a111 001a011 010b101 011 v=001 01 100b110 101b010 110a100 111b000

Example: proper sequences � 3-bits encoding of aababbab: 000a111 001a011 010b101 011a001 100b110 101b010 110a100 111b000 � For u=011, v=001 : u=011 000a111 001a011 010b101 011 v=001 01 100b110 101b010 110a100 111b000

Example: proper sequences � 3-bits encoding of aababbab: 000a111 001a011 010b101 011a001 100b110 101b010 110a100 111b000 � For u=011, v=001 : u=011 000a111 001a011 010b101 011 v=001 01 100b110 101b010 110a100 111b000

Example: proper sequences � 3-bits encoding of aababbab: 000a111 001a011 010b101 011a001 100b110 101b010 110a100 111b000 � For u=011, v=001 : u=011 000a111 001a011 010b101 011 v=001 01 100b110 101b010 110a100 111b000

Example: proper sequences � 3-bits encoding of aababbab: 000a111 001a011 010b101 011a001 100b110 101b010 110a100 111b000 � For u=011, v=001 : u=011 000a111 001a011 010b101 011 v=001 01 100b110 101b010 110a100 111b000 0

Example: proper sequences � 3-bits encoding of aababbab: 000a111 001a011 010b101 011a001 100b110 101b010 110a100 111b000 � For u=011, v=001 : u=011 000a111 001a011 010b101 011 v=001 01 100b110 101b010 110a100 111b000 0

Talk Outline � Overview � Notation and general solution to LTL games � Upper bounds: deteministic generators � Lower bounds � Encoding TMs without “next” and “until” U � Expspace-hardness of B(L , , ( � )) � 2Exptime-hardness of L , , , ( � ) � Conclusions

Results � Th 1. Deciding games is 2Exptime-hard L , , , ( � ) (reduction from Alt. Expspace) � Th 2. Deciding games is Expspace-hard B(L , , ( � )) (reduction from Alt. Exptime)

Schema of our reductions � Protagonist (system) � generates configurations � picks transitions when TM in � -states � Adversary (environment) � picks transitions when TM in � -states � raises objections to check if the sequence of configurations is proper and conforms the behaviour of TM

Expspace-hardness

Expspace-hardness � Protagonist generates sequences of positions <a> i (i refers to configuration # and cell #) � Plays:

Expspace-hardness � Protagonist generates sequences of positions <a> i (i refers to configuration # and cell #) � Plays: u 0 a 0 v 0

Expspace-hardness � Protagonist generates sequences of positions <a> i (i refers to configuration # and cell #) � Plays: u 0 a 0 v 0 ok

Expspace-hardness � Protagonist generates sequences of positions <a> i (i refers to configuration # and cell #) � Plays: obj 1 u 0 a 0 v 0 ok

Expspace-hardness � Protagonist generates sequences of positions <a> i (i refers to configuration # and cell #) � Plays: obj 1 obj 1 u 0 a 0 v 0 ok …… u y a y v y ok…… u’ 0 a’ 0 v’ 0 ………u f a f v f

Expspace-hardness � Protagonist generates sequences of positions <a> i (i refers to configuration # and cell #) � Plays: obj 1 obj 1 u 0 a 0 v 0 ok …… u y a y v y ok…… u’ 0 a’ 0 v’ 0 ………u f a f v f ok ok …… f

Expspace-hardness � Protagonist generates sequences of positions <a> i (i refers to configuration # and cell #) � Plays: obj 1 obj 1 obj 1 u 0 a 0 v 0 ok …… u y a y v y ok…… u’ 0 a’ 0 v’ 0 ………u f a f v f ok ok …… f

Expspace-hardness � Protagonist generates sequences of positions <a> i (i refers to configuration # and cell #) � Plays: obj 1 obj 1 obj 1 u 0 a 0 v 0 ok …… u y a y v y ok…… u’ 0 a’ 0 v’ 0 ………u f a f v f ok ok …… f obj 2

Objection 1

Objection 1 � Generation of proper sequences: � verify n(u j+1 )=n(u j )+1 and n(v j )=2 n -1- n(u j ) … p n …p 1 a j q 1 …q n ……obj 1 r n …r 1 s n …s 1

Objection 1 � Generation of proper sequences: � verify n(u j+1 )=n(u j )+1 and n(v j )=2 n -1- n(u j ) same … p n …p 1 a j q 1 …q n ……obj 1 r n …r 1 s n …s 1

Objection 1 � Generation of proper sequences: � verify n(u j+1 )=n(u j )+1 and n(v j )=2 n -1- n(u j ) same same … p n …p 1 a j q 1 …q n ……obj 1 r n …r 1 s n …s 1

Objection 1 � Generation of proper sequences: � verify n(u j+1 )=n(u j )+1 and n(v j )=2 n -1- n(u j ) same same … p n …p 1 a j q 1 …q n ……obj 1 r n …r 1 s n …s 1 (p j � r j 0 ) (p j � r j 1 ) 0 1

Objection 1 � Generation of proper sequences: � verify n(u j+1 )=n(u j )+1 and n(v j )=2 n -1- n(u j ) same diff same diff … p n …p 1 a j q 1 …q n ……obj 1 r n …r 1 s n …s 1 (q j � r j 1 ) (q j � r j 0 ) 0 1

Formula for proper sequences � obj 1 ( � 2 ] [ (succ(r,s) � 1 ) ( � ’ 1 � ’ 2 ) )

Formula for proper sequences � obj 1 ( � 2 ] [ (succ(r,s) � 1 ) ( � ’ 1 � ’ 2 ) ) � 1 = “p is same as r” � 2 = “p is same as r followed by p is same as s”

Formula for proper sequences � obj 1 ( � 2 ] [ (succ(r,s) � 1 ) ( � ’ 1 � ’ 2 ) ) � ’ 1 = “p is same as r” � ’ 2 = “p is same as r followed by q diff from r”

Formula for proper sequences � obj 1 ( � 2 ] [ (succ(r,s) � 1 ) ( � ’ 1 � ’ 2 ) ) � ’ 1 = “p is same as r” � ’ 2 = “p is same as r followed by q diff from r” � Need only formulas in B(L , , ( � ))

Objection 2 � Verify that sequences are TM outcomes � Adversary picks i-1, i, i+1, and j, and checks if cell i of C j+1 can “follow” cells i-1, i, i+1 of C j B(L , , ( � )) � “Small” formulas from do the job (property of proper sequences is crucial to match cell contents using only nested � ) � TM computes in exptime: � at the end of a computation we can zoom to each position generating polynomially many bits

Results � Th 1. Deciding games is 2Exptime-hard L , , , ( � ) (reduction from Alt. Expspace) � Th 2. Deciding games is Expspace-hard B(L , , ( � )) (reduction from Alt. Exptime)

2Exptime-hardness

2Exptime-hardness � We cannot encode configuration # � We can still use proper sequences to zoom to cells within a configuration � Focus on 2 consecutive configurations at a time (modulo-3 counter incremented every time a new configuration is entered)

Objections � Objection 1 similar to previous case � Objection 2 is allowed at the end of every configuration � To check � from the penultimate configuration use obj 2 along with: j � {0,1,2} ( � (j � � (j+1) ¬ � (j+2))) �

Objections � Objection 1 similar to previous case � Objection 2 is allowed at the end of every configuration � To check � from the penultimate configuration use obj 2 along with: j � {0,1,2} ( � (j � � (j+1) ¬ � (j+2))) �

Objections � Objection 1 similar to previous case � Objection 2 is allowed at the end of every configuration � To check � from the penultimate configuration use obj 2 along with: j � {0,1,2} ( � (j � � (j+1) ¬ � (j+2))) � (This is in L , , , ( � ) )

Complexity Det. Generators Games Size L. Dist. B(L , ( � )) Pspace-complete � (Exp) � (Linear) B(L , , ( � )) Exptime-complete � (Exp) � (Exp) B(L , , ( � )) Expspace-complete � (2Exp) � (Exp) B(L , , , ( � )) Expspace-complete � (2Exp) � (Exp) L , , , ( � ) 2Exptime-complete � (2Exp) � (2Exp) LTL 2Exptime-complete � (2Exp) � (2Exp)

Talk Outline � Overview � Notation and general solution to LTL games � Upper bounds: deteministic generators � Lower bounds � Encoding TMs without “next” and “until” U � Expspace-hardness of B(L , , ( � )) � 2Exptime-hardness of L , , , ( � ) � Conclusions

Fair safety-reachability games � Games with fairness: � “(adv plays fair) (prot plays fair wins) � “(prot plays fair) (adv plays fair wins) , ( � )) F : ( ) B(L ( � ) U L , ( � )) B(L � fair safety-reachability games , ( � )) F games are Pspace-complete B(L �

Fair safety-reachability games � Games with fairness: � “(adv plays fair) (prot plays fair wins) � “(prot plays fair) (adv plays fair wins) , ( � )) F : ( ) B(L ( � ) U L , ( � )) B(L � fair safety-reachability games , ( � )) F games are Pspace-complete B(L � Decision algorithm uses Zielonka solution to Muller L , ( � ) games along with det. generators for

Fair safety-reachability games � Games with fairness: � “(adv plays fair) (prot plays fair wins) � “(prot plays fair) (adv plays fair wins) , ( � )) F : ( ) B(L ( � ) U L , ( � )) B(L � fair safety-reachability games , ( � )) F games are Pspace-complete B(L � Hardness: games with “Streett Rabin” winning conditions are Pspace-hard (from QBF)

More in PSPACE � Persistent strategy: On a play, the player picks always the same move visiting the same location (weaker than memoryless)

More in PSPACE � Persistent strategy: On a play, the player picks always the same move visiting the same location (weaker than memoryless) a b a a c b

More in PSPACE � Persistent strategy: On a play, the player picks always the same move visiting the same location (weaker than memoryless) a persistent b a not memoryless a c b

Complexity of L , , ( � ) � Theorem: [Marcinkowski -Truderung CSL‘02] L , , ( � ) For specs in , protagonist has a winning strategy iff can win against an adversary that uses only persistent strategies games are in PSPACE L , , ( � ) �

LTL fragments L , ( � ) L , , ( � ) , ( � )) F B(L B(L , , ( � )) B(L , , ( � )) B(L , , , ( � )) L , , , ( � ) e t e l p m o c - LTL e m i t p x E 2