Recognizing regular tree languages with static information

Transcription

1 Recognizing regular tree languages with static information Alain Frisch (ENS Paris) PLAN-X 2004 p.1/22

2 Motivation Efficient compilation of patterns in XDuce/CDuce/... E.g.: type A = <a>[ A* ] type B = <b>[ B* ] let f ((A B) -> Int) A -> 0 B -> 1 let g ((A B) -> Int) <a>_ -> 0 _ -> 1 Encourage more declarative style in patterns. PLAN-X 2004 p.2/22

3 The separation problem : domain of values Given returns s.t. Two-stage process: the comes after (run-time) Given s.t. We want (pairwise disjoint) and, come first (compile-time), and, computes a function to be efficient, even if computing it is expensive. Parameters: the domain Note: system., the class of possible. is the static information provided by the type PLAN-X 2004 p.3/22

4 Examples : integers; : unions of intervals. Decision tree (dichotomy) : strings; Trie. : finite sets. Can ignore suffixes. E.g.: : strings; : regular languages. Deterministic automaton. : XML documents; languages.??? : XML types (schema) = regular tree PLAN-X 2004 p.4/22

5 Expected properties Traverse each node of the tree only a finite bounded number of times. Ignore subtrees that can be ignored because of the static information and/or because they are irrelevant. Independance w.r.t. the syntax of types (only the denoted sets matter). PLAN-X 2004 p.5/22

6 Classical solutions Backtracking tree automata Don t guarantee linear time recognition. Bottom-up deterministic tree automata. Only consider downward context (subtree). Ignore upward context (path). Cannot adapt their behavior to the current location in the tree. Top-down deterministic tree automata Cannot recognize arbitrary regular tree languages. Only consider the upward context. No left-to-right propagation of information. PLAN-X 2004 p.6/22

7 Contexts Path Subtree Left PLAN-X 2004 p.7/22

8 Trees Trees: (ie: ) Regular languages defined by equations: E.g.: PLAN-X 2004 p.8/22

9 Non-uniform automata A specific kind of push-down automata. Combine the advantages of bottom-up and top-down TA. Can totally ignore whole subtrees. Fixed traversal order. Thread a control-state through the traversal. Accumulates knowledge gained from the traversal. Depends on the left context (not only path). PLAN-X 2004 p.9/22

10 Non-uniform automata PLAN-X 2004 p.10/22

11 Non-uniform automata Input: state PLAN-X 2004 p.10/22

12 Non-uniform automata Input: state PLAN-X 2004 p.10/22

13 Non-uniform automata Input: state PLAN-X 2004 p.10/22

14 Non-uniform automata Input: state PLAN-X 2004 p.10/22

15 Non-uniform automata Input: state PLAN-X 2004 p.10/22

16 Non-uniform automata Input: state PLAN-X 2004 p.10/22

17 Compilation Main technical contribution: a compilation algorithm that generates efficient NUA (which ignore many subtrees). Key idea: propagate static information precisely (in the control state) PLAN-X 2004 p.11/22

18 Compilation May ignore right subtree, according to left subtree. PLAN-X 2004 p.12/22

19 Compilation Propagate static information. PLAN-X 2004 p.13/22

20 Compilation Ignore right subtree. PLAN-X 2004 p.14/22

21 Compilation Ignore left subtree (normalization). PLAN-X 2004 p.15/22

22 Compilation Ignore right subtree. PLAN-X 2004 p.16/22

23 Compilation Could also ignore left subtree. PLAN-X 2004 p.17/22

24 Compilation Consider left subtree, but could ignore it. PLAN-X 2004 p.18/22

25 Compilation PLAN-X 2004 p.19/22

26 Compilation PLAN-X 2004 p.20/22

27 Compilation of PM in CDuce Handle capture variables. All CDuce types (open/closed records, integer/characters intervals,... ). Effectively encourages declarative style without degrading performances. PLAN-X 2004 p.21/22

28 Thank you! PLAN-X 2004 p.22/22