1. Conjunctive Grammars and Alternating Automata Tamar Aizikowitz and Michael Kaminski Technion – Israel Institute of Technology WoLLIC 2008 Heriot-Watt University

2. 2 of 25 Introduction: ND Computational Models Non-deterministic computational models have existential acceptance conditions At least one computation must accept E.g., FSA, PDA, CFG, TM Languages accepted have disjunctive quality A word must meet one of many possible conditions (computations)  Language class closed under union

3. 3 of 25 Introduction: Co-ND Computational Models Dual computational models have universal acceptance conditions All computations must accept E.g., Universal TM ; accepts class Co-NP Languages accepted have conjunctive quality A word must meet all conditions (computations)  Language class closed under intersection

4. 4 of 25 Introduction: Combined Computational Models Several models combining existential and universal computations have been explored. We explore extensions of models for Context Free Languages. Specifically: Conjunctive Grammars Alternating Pushdown Automata

5. 5 of 25 Introduction: Conjunctive Grammars ( CG ) Introduced by Alexander Okhotin in 2001 * Extension of Context Free Grammars Add power of explicit intersection  CG- s generate a larger class of languages * Okhotin A., Conjunctive Grammars, Journal of Automata, Languages and Combinatorics 6(4) (2001) 519-535 Reminder: Context free languages are not closed under intersection…

6. 6 of 25 Polynomial TimeCG LanguagesFinite ∩ CF Introduction: Conjunctive Grammars ( CG ) Conjunctive Grammars generate: Context Free languages Finite ∩ of CF languages Some more languages Generated languages are polynomial No known non-trivial technique to prove a language cannot be generated by a CG  Exact placing in the Chomsky Hierarchy not known Context Free

7. 7 of 25 Introduction: Alternating Automata Alternating Automata were introduced by Chandra et.al. in 1981 * Computations alternate between existential and universal acceptance modes Well known examples: Alternating Finite State Automata (Verification) Alternating Turing Machines (Complexity Theory) Both equivalent to non-alternating counterparts * Chandra, A.K., Kozen, D.C., Stockmeyer, L.J., Alternation. Journal of the ACM 28(1) (1981) 114-133

8. 8 of 25 Introduction: Alternating Pushdown Automata ( APDA ) Further explored by Ladner et.al. in 1984 * Add conjunction to computations  Not equivalent to standard PDA model Accept exactly the Exp. Time Languages  Not equivalent to the CG model * Ladner, R.E., Lipton, R.J., Stockmeyer, L.J., Alternating pushdown and stack automata. SIAM Journal on Computing 13(1) (1984) 135-155

9. 9 of 25 Introduction: Synchronized APDA ( SAPDA ) We introduce a new model: Synchronized Alternating Pushdown Automata Equivalent to Conjunctive Grammar model In fact, this is the first class of automata suggested for Conjunctive Grammars

10. 10 of 25 Outline Conjunctive Grammars Synchronized Alternating PDA Equivalence Results: CG ~ SAPDA Future Work

11. 11 of 25 Conjunctive Grammars: Model Definition G = ( V, T, P, S ) V,T,S as in the standard CFG case P contains rules X → (  1 &  &  n ) n = 1 gives standard CFG rules Conjunctive Formulas: {  }   V  T are formulas If  and  are formulas then  is a formula If  1,…,  n are formulas then (  1 &  &  n ) is a formula conjunct

12. 12 of 25 Conjunctive Grammars: Model Definition Derivation: Application: s 1 X s 2  s 1 (  1 &  &  n )s 2 s.t. X → (  1 &  &  n )  P Contraction: s 1 (w &  & w) s 2  s 1 w s 2 Language: L(G) = {w  T * | S  * w} Note: (  &  )  * w iff   * w and   * w

13. 13 of 25 Conjunctive Grammars: Example 1 L = {a n b n c n | n   } Note: L = L 1  L 2 where L 1 = {a n b n c i | n,i   } L 2 = {a i b n c n | n,i   } G = ({S,S 1,S 2,S 3,S 4 }, {a,b,c}, S, P) where P = : S 1 → a S 1 b |  ;S 2 → c S 2 |  S 3 → a S 3 |  ;S 4 → b S 4 c |  S → (S 1 S 2 & S 3 S 4 ) S1S2*L1S1S2*L1 S3S4*L2S3S4*L2 L1  L2L1  L2

14. 14 of 25 Conjunctive Grammars: Example 1 Derivation of aaabbbccc : S  (S 1 S 2 & S 3 S 4 )  (aS 1 bS 2 & S 3 S 4 )  …  (aaaS 1 bbbS 2 & S 3 S 4 )  (aaabbbS 2 & S 3 S 4 )  (aaabbbcS 2 & S 3 S 4 )  …  (aaabbbccc & S 3 S 4 )  …  (aaabbbccc & aaabbbccc)  aaabbbccc S → (S 1 S 2 & S 3 S 4 )S 1 → aS 1 b S 1 →  S 2 → cS 2 contraction S 2 → cS 2

15. Conjunctive Grammars: Interesting Languages CG s can generate some interesting languages: Multiple agreement: {a n b n c n | n   } Cross agreement: {a n b m c n d m | n,m   } Reduplication: {w$w | w  {a,b} * }  all mildly context-sensitive languages ( MCS ) CG s can also generate “stronger” languages such as: {ba 2 ba 4  ba 2 n b | n   }  not MCS because not semi-linear! 15 of 25

16. 16 of 25 Synchronized Alternating Pushdown Automata ( SAPDA ) Extension of the standard PDA model Transitions are to conjunctions of (state,stack-word) pairs E.g.  (q, , X ) = {( p 1, XX )  ( p 2, Y ), ( p 3, Z ) } If all conjunctions are of one pair then we have the standard PDA model Non-deterministic model = many possible conjunctions

17. Synchronized Alternating Pushdown Automata ( SAPDA ) Stack memory is a tree Each leaf has a separate processing head A conjunctive transition to n pairs splits the current branch to n branches Branches are processed independently Sibling branches must empty synchronously. A B C D A q p 17 of 25

18. 18 of 25 SAPDA: Model Definition A = (Q, , , , q 0,  ) For every (q, , X )  Q  (  {  })    (q, , X )  {(q 1,  1 )    (q n,  n ) | q i  Q,  i   *, n   } Example:  (q, , X ) = {( p 1, Z )  ( p 2, YY )} Y Y Z p1p1 p2p2 X q 

19. Configuration: a labeled tree T where Internal nodes labeled α denoting stack contents Leaves labeled (q,w,α) denoting current state, remaining input and stack contents Initial Configuration: the tree T 0 which is the tree Accepting Configuration: a tree T e s.t. it is the tree for some q  Q 19 of 25 SAPDA: Configurations (q0,w,)(q0,w,) (q,, )(q,, ) BA (q,ba, A)( p,a, DC ) A B C D A q p abba

20. 20 of 25 SAPDA: Computation Computation: Each computation step, a transition is applied to one stack-branch If a stack-branch empties, it cannot be selected If all siblings branches are empty and “synchronized” then they are collapsed Synchronized Collapsing: All siblings… Are empty Have the same state Have the same remaining input Are all labeled (q,w, ɛ )

21. 21 of 25 SAPDA: Language Accepting Computation: The final configuration is an accepting one, i.e. (q, ,  ) for some q  Q. L(A) = {w   * | A has an accepting computation on w} Note: Acceptance by accepting states can also be defined. Both models of acceptance are equivalent.

22. SAPDA: Example L = {w  {a,b,c} * | # a (w) = # b (w) = # c (w)} 22 of 25 a a b c c a b b c δ (q 0, ε,  ) = (q 1,  1 )  (q 2,  2 ) δ (q 1, a,  1 ) = (q 1, a  1 )δ (q 2, a,  2 ) = (q 2,  2 ) δ (q 1, a, a) = (q 1, aa) δ (q 2, a,  2 ) = (q 2,  2 ) δ (q 1, b, a) = (q 1, ε) δ (q 2, b,  2 ) = (q 2, b  2 ) δ (q 1, c, a) = (q 1, a)δ (q 2, c, b) = (q 2, ε) δ (q 1, ε,  1 ) = (q 0, ε)δ (q 2, ε,  2 ) = (q 0, ε) q1q1 q2q2 q0q0 q0q0 q0q0  ε 11 22 a a b ε ε

23. 23 of 25 Equivalence Results Theorem 1: SAPDA  CG Theorem 2: CG  1-state SAPDA ( 1SAPDA ) Corollary 1: SAPDA ~ CG Corollary 2: SAPDA ~ 1SAPDA Proofs: Both are extensions of the classical ones Surprisingly, the grammar-to-automaton translation is the more complicated one…

24. Future Directions Linear CG and 1-turn SAPDA It is well known that LCFG ~ 1-turn PDA Linear CG : one non-terminal in each conjunct 1-turn SAPDA : each stack-branch turns once Initial results point towards: 1-turn SAPDA ~ LCG Finite-turn SAPDA, Deterministic SAPDA … Possibly, find a method to prove a language cannot be accepted by an SAPDA / CG … 24 of 25

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