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June 17

LR (0) Conjunctive Grammars and Deterministic Synchronized Alternating Pushdown Automata Tamar Aizikowitz and Michael Kaminski Technion – Israel Institute of Technology. June 17. CSR 2011, St. Petersburg, Russia. Context-Free Languages.

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June 17

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  1. LR(0) Conjunctive Grammars and Deterministic SynchronizedAlternating Pushdown AutomataTamar Aizikowitz and Michael KaminskiTechnion – Israel Institute of Technology June 17 CSR 2011, St. Petersburg, Russia

  2. Context-Free Languages Combine expressiveness with polynomial parsing, which is appealing for practical applications. One of the most widely used language class in Computer Science. At the theoretical basis of Programming Languages, Computational Linguistics, Formal Verification, Computational Biology, and more.

  3. Extended Models Goal: Models of computation that generate a slightly stronger language class without sacrificing polynomial parsing. Why? Such models seem to have great potential for practical applications. In fact, several fields (e.g.,Computational Linguistics) have already voiced their need for a stronger language class.

  4. Conjunctive Grammars • Conjunctive Grammars (CG) [Okhotin, 2001] are an extension of context-free grammars. • Add explicit intersection rules S (A & B)⋯ (w & w) w • Semantics:L(A & B) = L(A) ⋂ L(B) • Since context-free languages are not closed under intersection, CG is a wider class of languages • Retain polynomial parsing and therefore can be useful for practical applications

  5. Synchronized Alternating PDA Synchronized Alternating Pushdown Automata (SAPDA) [Aizikowitz & Kaminski, 2008] extend PDA. Stack modeled as tree in which all braches must accept. A limited form of synchronization createslocalized parallel computations. Firstautomaton counterpart shown for Conjunctive Grammars.

  6. Outline • Model Definitions • Equivalence Results • Linear-time LR(0) Parsing • Summary

  7. Model Definitions Conjunctive Grammars Synchronized Alternating Pushdown Automata

  8. Conjunctive Grammars non-terminals terminals derivation rules start symbol The case of all n=1 isan ordinary CFG • A CG is a quadruple: G=(V , Σ , P , S ) • Rules:A → (α1&⋯&αn), where A∊V,αi∊(V⋃Σ)* • Examples: A→ (aAB & Bc & aD) ; A→abC • Derivation steps: • s1As2s1(α1&⋯&αn)s2,where A→(α1&⋯&αn)∊P • s1(w&⋯&w)s2s1 w s2, where w∊ Σ*

  9. Grammar Language Language:L(G) = {w∊Σ*| S*w}, i.e., L(G) consists of all terminal words w derivable from the start symbol S. Note: ( , ) , and & are not terminal symbols. Therefore, all conjunctions must be collapsed in order to derive a terminal word. Semantics:(A & B) *w if and only if A *wand B *w. Therefore, L(A & B) = L(A) ⋂ L(B).

  10. Example: Multiple Agreement • Example: a CG for the multiple-agreement language {anbncn | n∊ℕ}: • C→Cc | D ; D→aDb | ε:L(C) = {anbncm | n,m∊ℕ} • A→aA | E ; E→bEc | ε:L(A) = {ambncn | n,m∊ℕ} • S→ (C & A) :L(S ) = L(C) ⋂ L(A) • S (C&A)  (Cc&A)  (Dc&A)  (aDbc&A)   (abc&A) ⋯ (abc&abc) abc

  11. Synchronized Alternating Pushdown Automata (SAPDA) non-deterministic model, i.e., many possible transitions • An extension of classical PDA. • Transitions are made to conjunctions of ( state , stack-word ) pairs, e.g., δ(q , σ, X) = { (p1 , XX)∧(p2 ,Y) , (p3 , Z) } • Note:if all conjunctions are of one pair only, the automaton is an “ordinary” PDA.

  12. SAPDA Stack Tree The stack of an SAPDA is a tree.A transition to n pairs splits the current branch into n branches. Branches are processed independently. Empty sibling branches can be collapsed if they are synchronized, i.e., are in the same state and have read the same portion of the input. p D q q q q A C ε ε A q B (q,A)(p,DC)δ(q,σ,A) collapse B B B

  13. SAPDA Definition states terminals stack symbols initial state transition function initial stack symbol An SAPDA is a sextuple A = ( Q , Σ , Γ , q0 , δ , ⊥ ) Transition function: δ(q,σ,X) ⊆ {(q1,α1)∧⋯∧(qn,αn) | qi∊Q, αi∊Γ*, n∊ℕ}

  14. SAPDA Computation and Language have the same state and remaining input q0 ⊥ q ε • Computation: • Each step, a transition is applied to one stack-branch • If a stack-branch is empty, it cannot be selected • Synchronous empty sibling branches are collapsed • Initial Configuration: • Accepting configuration: • Language:L(A)={w∊Σ*| ∃q∊Q, (q0,w,⊥)⊢*(q,ε,ε)} • Note: all branches must empty, i.e., must “agree”.

  15. Equivalence Results SAPDA and Conjunctive Grammars

  16. Equivalence of SAPDA and CG • Theorem 1.A language is generated by an CG if and only if it is accepted by an SAPDA. • The equivalence is analogous to the classical equivalence between CFG and PDA. • The proofs of the equivalence are extended versions of the classical proofs. • Corollary:Single-state SAPDA and multi-state SAPDA are equivalent (like ordinary PDA).

  17. Linear-time LR(0) Parsing Deterministic SAPDA and LR(0) CG Linear-time Implementation of DSAPDA and LR(0) Parsing

  18. Deterministic Context-free Languages DCFL are a sub-family of context-free languages, accepted by Deterministic PDA (DPDA). LR(k) languages were introduced by Knuth, and shown to be equivalent to DPDA, [Knuth, 1965]. Knuth developed a linear-time parsing algorithm for DCFL  the basis of Compilation Theory. Knuth also proved that LR(0) grammars are equivalent to DPDA which accept by empty stack.

  19. Deterministic SAPDA Deterministic SAPDA are defined according to the classical notion of a deterministic model. That is, an SAPDA is deterministic if it has only one possible step from any given configuration. Remark:A deterministic SAPDA has exactly one computation on any given input word w.

  20. LR(0) Conjunctive Grammars To define LR(0) conjunctive grammars, we extend the notions of viable prefixes, valid items ([A→]), etc… A conjunctive grammar is LR(0) if a conflict-free item automaton can be constructed for it. Main issue: How to build the item automaton so that it supports conjunctive rules... Solution: Read the paper 

  21. Equivalence Results Theorem 2.A language is generated by an LR(0) CG if and only if it is accepted by a DSAPDA. The equivalence is analogous to the classical equivalence between LR(0) CFG and DPDA. The proofs of the equivalence are extended versions of the classical proofs.

  22. The DSAPDA Membership Problem By using a naive implementation of a DSAPDA, we have an exponential time solution for the membership problem. Each branch is linear, but there can be an exponential number of branches.

  23. Do we need so many branches?

  24. DSAPDA Efficient Implementation There are at most M = |Q|  |Γ| different combinations for branch head configurations. Thus, at any given time, we need at most M heads.

  25. Linear-time Parsing • Based on the efficient implementation of a DSAPDA, we can build an LR(0) parser. • The parser runs in linear time for the boolean closure of LR(0) Conjunctive Languages • This class strictly Includes the boolean closure of classical LR(0) languages. • { ai1 bai2 b2 ⋯ ainbn$ bai1 bai2 ⋯ bain$ | n ≥ 1  ij≥ 1 } • That is, we have linear-time parsing for a wider class of languages.

  26. Summary • Conjunctive Languages are interesting because: • They are a strong, rich class of languages. • They are polynomially parsable. • Their models of computation are intuitive and easy to understand; highly resemble classical CFG and PDA. • SAPDA are the first automaton model presented for Conjunctive Languages. • They are a natural extension of PDA. • They lend new intuition on Conjunctive Languages. • They are the basis of a linear parsing algorithm for a wide class of languages.

  27. Thank you.

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