Maximally Parallel Contextual String Rewriting a 1 Traian Florin S , erb˘ anut , ˘ University of Bucharest Diaspora s , tiint , ific˘ a 2016 1 joint work with Liviu P . Dinu Traian Florin S , erb˘ anut , ˘ a (UNIBUC) Maximally Parallel Contextual String Rewriting Diaspora s , tiint , ific˘ a 2016 1 / 26
Motivation Contents Motivation 1 Technical background 2 Contextual String Rewriting 3 Defining maximal parallelism Computing maximal parallel instances Traian Florin S , erb˘ anut , ˘ a (UNIBUC) Maximally Parallel Contextual String Rewriting Diaspora s , tiint , ific˘ a 2016 2 / 26
Motivation Hyphenation rules! (for Romanian) Hyphenation for Romanian seems simple 8 insertion rules for regular words [Dinu, 2003] Regular: no consecutive vowels (to exclude diphthongs, triphthongs). In the rules below v s stand for vowels, c s for consonants ( v 1 , - , cv 2 ) 1 ( v 1 , - , c 1 c 2 v 2 ) if c 1 c 2 ∈ { ch, gh } or ( c 1 , c 2 ) ∈ { b, c, d, f, g, h, p, t } × { l, r } 2 ( v 1 c 1 , - , c 2 v 2 ) if c 1 c 2 � { ch, gh } and ( c 1 , c 2 ) � { b, c, d, f, g, h, p, t } × { l, r } 3 ( v 1 c 1 c 2 , - , c 3 v 2 ) if c 1 c 2 c 3 ∈ { lpt, mpt, mpt , , ncs , , nct, nct , , ndv, rct, rtf, stm } 4 ( v 1 c 1 , - , c 2 c 3 v 2 ) if c 1 c 2 c 3 � { lpt, mpt, mpt , , ncs , , nct, nct , , ndv, rct, rtf, stm } 5 ( v 1 c 1 c 2 , - , c 3 c 4 v 2 ) if c 2 c 3 c 4 ∈ { gst, nbl } 6 ( v 1 c 1 , - , c 2 c 3 c 4 v 2 ) if c 2 c 3 c 4 � { gst, nbl } 7 ( v 1 c 1 c 2 , - , c 3 c 4 c 5 v 2 ) if c 1 c 2 c 3 c 4 c 5 ∈ { ptspr, stscr } 8 Traian Florin S , erb˘ anut , ˘ a (UNIBUC) Maximally Parallel Contextual String Rewriting Diaspora s , tiint , ific˘ a 2016 3 / 26
Motivation Hyphenation example lingvistic˘ a ( v 1 , - , cv 2 ) 1 ( v 1 , - , c 1 c 2 v 2 ) if c 1 c 2 ∈ { ch, gh } or ( c 1 , c 2 ) ∈ { b, c, d, f, g, h, p, t } × { l, r } 2 ( v 1 c 1 , - , c 2 v 2 ) if c 1 c 2 � { ch, gh } and ( c 1 , c 2 ) � { b, c, d, f, g, h, p, t } × { l, r } 3 ( v 1 c 1 c 2 , - , c 3 v 2 ) if c 1 c 2 c 3 ∈ { lpt, mpt, mpt , , ncs , , nct, nct , , ndv, rct, rtf, stm } 4 ( v 1 c 1 , - , c 2 c 3 v 2 ) if c 1 c 2 c 3 � { lpt, mpt, mpt , , ncs , , nct, nct , , ndv, rct, rtf, stm } 5 ( v 1 c 1 c 2 , - , c 3 c 4 v 2 ) if c 2 c 3 c 4 ∈ { gst, nbl } 6 ( v 1 c 1 , - , c 2 c 3 c 4 v 2 ) if c 2 c 3 c 4 � { gst, nbl } 7 ( v 1 c 1 c 2 , - , c 3 c 4 c 5 v 2 ) if c 1 c 2 c 3 c 4 c 5 ∈ { ptspr, stscr } 8 Traian Florin S , erb˘ anut , ˘ a (UNIBUC) Maximally Parallel Contextual String Rewriting Diaspora s , tiint , ific˘ a 2016 4 / 26
Motivation Hyphenation example lingvistic˘ a lin-gvistic˘ a by rule (5) ( v 1 , - , cv 2 ) 1 ( v 1 , - , c 1 c 2 v 2 ) if c 1 c 2 ∈ { ch, gh } or ( c 1 , c 2 ) ∈ { b, c, d, f, g, h, p, t } × { l, r } 2 ( v 1 c 1 , - , c 2 v 2 ) if c 1 c 2 � { ch, gh } and ( c 1 , c 2 ) � { b, c, d, f, g, h, p, t } × { l, r } 3 ( v 1 c 1 c 2 , - , c 3 v 2 ) if c 1 c 2 c 3 ∈ { lpt, mpt, mpt , , ncs , , nct, nct , , ndv, rct, rtf, stm } 4 ( v 1 c 1 , - , c 2 c 3 v 2 ) if c 1 c 2 c 3 � { lpt, mpt, mpt , , ncs , , nct, nct , , ndv, rct, rtf, stm } 5 ( v 1 c 1 c 2 , - , c 3 c 4 v 2 ) if c 2 c 3 c 4 ∈ { gst, nbl } 6 ( v 1 c 1 , - , c 2 c 3 c 4 v 2 ) if c 2 c 3 c 4 � { gst, nbl } 7 ( v 1 c 1 c 2 , - , c 3 c 4 c 5 v 2 ) if c 1 c 2 c 3 c 4 c 5 ∈ { ptspr, stscr } 8 Traian Florin S , erb˘ anut , ˘ a (UNIBUC) Maximally Parallel Contextual String Rewriting Diaspora s , tiint , ific˘ a 2016 4 / 26
Motivation Hyphenation example lingvistic˘ a lin-gvistic˘ a by rule (5) lin-gvis-tic˘ a by rule (3) lin-gvis-ti-c˘ a by rule (1) ( v 1 , - , cv 2 ) 1 ( v 1 , - , c 1 c 2 v 2 ) if c 1 c 2 ∈ { ch, gh } or ( c 1 , c 2 ) ∈ { b, c, d, f, g, h, p, t } × { l, r } 2 ( v 1 c 1 , - , c 2 v 2 ) if c 1 c 2 � { ch, gh } and ( c 1 , c 2 ) � { b, c, d, f, g, h, p, t } × { l, r } 3 ( v 1 c 1 c 2 , - , c 3 v 2 ) if c 1 c 2 c 3 ∈ { lpt, mpt, mpt , , ncs , , nct, nct , , ndv, rct, rtf, stm } 4 ( v 1 c 1 , - , c 2 c 3 v 2 ) if c 1 c 2 c 3 � { lpt, mpt, mpt , , ncs , , nct, nct , , ndv, rct, rtf, stm } 5 ( v 1 c 1 c 2 , - , c 3 c 4 v 2 ) if c 2 c 3 c 4 ∈ { gst, nbl } 6 ( v 1 c 1 , - , c 2 c 3 c 4 v 2 ) if c 2 c 3 c 4 � { gst, nbl } 7 ( v 1 c 1 c 2 , - , c 3 c 4 c 5 v 2 ) if c 1 c 2 c 3 c 4 c 5 ∈ { ptspr, stscr } 8 Traian Florin S , erb˘ anut , ˘ a (UNIBUC) Maximally Parallel Contextual String Rewriting Diaspora s , tiint , ific˘ a 2016 4 / 26
Motivation Parallel hyphenation Approach [Dinu, 2009] Insertion systems can model syllabification Insertion can be done in parallel by sharing the context Maximally parallel application of rules yields one-step hyphenation Why? Suggested as possible at cognitive level by some linguists lingvistic˘ a = ⇒ lingvistic˘ a = ⇒ lin-gvis-ti-c˘ a ( v 1 , - , cv 2 ) ( v 1 c 1 , - , c 2 v 2 ) if c 1 c 2 � { ch, gh } and ( c 1 , c 2 ) � { b, c, d, f, g, h, p, t } × { l, r } ( v 1 c 1 , - , c 2 c 3 v 2 ) if c 1 c 2 c 3 � { lpt, mpt, mpt , , ncs , , nct, nct , , ndv, rct, rtf, stm } Traian Florin S , erb˘ anut , ˘ a (UNIBUC) Maximally Parallel Contextual String Rewriting Diaspora s , tiint , ific˘ a 2016 5 / 26
Motivation Connections with other formalisms Contextual description of rules Insertion/Deletion Systems [Kari, 1991] String Rewriting + Insertion Rules + Deletion Rules + Parallel application K concurrent rewriting [Ros , u, S , erb˘ anut , ˘ a, 2010] Term rewriting + contextual K rules + Parallel application Maximally parallel application of rules L-Systems [Lindenmayer, 1968] Context free grammars with maximally parallel derivation Context Sensitive L-Systems have some notion of sharing However, the context is usually altered during the rewrite P Systems [P˘ aun, 1998] Multiset rewriting with local rules Contextual sharing in the promoter/inhibitor variant Traian Florin S , erb˘ anut , ˘ a (UNIBUC) Maximally Parallel Contextual String Rewriting Diaspora s , tiint , ific˘ a 2016 6 / 26
Motivation Contributions Contextual string rewriting rules as a (slight) variation of rules from Insertion/Deletion Systems as a specialization of K rules to string rewriting Formal definition for (maximal) parallel matching and derivation Transformations to TRSs for simulating one-step maximal parallel derivation Traian Florin S , erb˘ anut , ˘ a (UNIBUC) Maximally Parallel Contextual String Rewriting Diaspora s , tiint , ific˘ a 2016 7 / 26
Technical background Contents Motivation 1 Technical background 2 Contextual String Rewriting 3 Defining maximal parallelism Computing maximal parallel instances Traian Florin S , erb˘ anut , ˘ a (UNIBUC) Maximally Parallel Contextual String Rewriting Diaspora s , tiint , ific˘ a 2016 8 / 26
Technical background Formal languages Alphabets: Σ , . . . are sets of letters: a , b , c , . . . Words: u , v , w , x , y ,. . . are (finite) sequences of letters The set of all words over alphabet Σ is denoted Σ ∗ The empty word is denoted by λ Concatenation of two words is obtained by juxtaposition _ _ and is associative, with unit λ (Σ ∗ , _ _ , λ ) is the free monoid generated by Σ A formal language over Σ is a subset of Σ ∗ . Traian Florin S , erb˘ anut , ˘ a (UNIBUC) Maximally Parallel Contextual String Rewriting Diaspora s , tiint , ific˘ a 2016 9 / 26
Technical background Insertion/Deletion Systems [Kari, 1991] Syntax An insertion-deletion system is a tuple Γ = (Σ , T , A , I , D ) , where Σ is an alphabet T ⊆ Σ is the terminal alphabet A ⊆ Σ ∗ is the set of axioms (starting symbols) I (the insertion rules) and D (the deletion rules) are finite sets of triples of the form ( u , x , v ) , where u , x , v ∈ Σ ∗ with x � λ (Sequential) Semantics Insertion rule ( u , x , v ) specifies insertion of x in context ( u , v ) u v = > u x v Deletion rule ( u , x , v ) specifies deletion of x in context ( u , v ) u x v = > u v The language generated by Γ is L (Γ) = { w ∈ T ∗ | ∃ u ∈ A . u ⇒ ∗ w } Traian Florin S , erb˘ anut , ˘ a (UNIBUC) Maximally Parallel Contextual String Rewriting Diaspora s , tiint , ific˘ a 2016 10 / 26
Technical background Insertion grammars with (maximum) parallel derivation [Dinu, 2009] Syntax An insertion grammar Γ = (Σ , A , I ) is a special case of insertion/deletion systems, where There are no non-terminals ( T = Σ ) There are no deletion rules ( D = ∅ ) Parallel Semantics w ⇛ Γ z iff w = w 0 w 0 w 1 w 1 . . . w r − 1 w r − 1 w r w r , where r ≥ 1 ( r maximal for maximum derivation) for all 1 ≤ i ≤ r , there exist ( u i , x i , v i ) ∈ I such that w i − 1 w i − 1 ends with u i w i w i starts with v i z = w 0 w 0 x 1 w 1 w 1 x 2 . . . w r − 1 w r − 1 x r w r w r Traian Florin S , erb˘ anut , ˘ a (UNIBUC) Maximally Parallel Contextual String Rewriting Diaspora s , tiint , ific˘ a 2016 11 / 26
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