Undecidability of the Post Correspondence Problem in Coq

Transcription

1 Undecidaility of the Post Correspondence Prolem in Coq Bachelor Talk Edith Heiter Advisors: Prof. Dr. Gert Smolka, Yannick Forster August 23, 2017

2 PCP and Undecidaility SR to MPCP Turing Machines Reach to SR Halt to SR Conclusion What to Expect? Formalized decision prolems: Post correspondence prolem (PCP) modified Post correspondence prolem (MPCP) word prolem in string-rewriting systems halting prolem for Turing machines Formal definition and verification of reductions from the literature proving PCP undecidale: Hopcroft et al. (2006) Davis et al. (1994) Wim H. Hesselink (2015) constructive Coq development 1

3 PCP and Undecidaility SR to MPCP Turing Machines Reach to SR Halt to SR Conclusion The Post Correspondence Prolem print sprint dog doge dog doge eats at dogeatsprint dogeatsprint eats at print sprint Assume a fixed alphaet Σ. strings Σ := L Σ instance P of type pcp := L (Σ Σ ) S is a match if concat (map π 1 S) = concat (map π 2 S), areviated as C 1 S = C 2 S S is a match for P if S [ ], S P, and S is a match Definition (Post correspondence prolem) PCP P := S. S is a match for P 2

4 PCP and Undecidaility SR to MPCP Turing Machines Reach to SR Halt to SR Conclusion The Modified Post Correspondence Prolem print sprint print sprint dog doge dog doge dog doge eats at eats at eats at print sprint Assume a fixed alphaet Σ. strings Σ := L Σ instance (d, P) of type mpcp := (Σ Σ ) pcp S is a match if C 1 S = C 2 S S is a match for P if S [ ], S P, and S is a match Definition (Modified Post correspondence prolem) MPCP (d, P) := S. (d :: S) is a match for (d :: P) 3

5 PCP and Undecidaility SR to MPCP Turing Machines Reach to SR Halt to SR Conclusion Undecidaility in Coq Definition (Undecidaility) A class P : X P is undecidale if the halting prolem (Halt) reduces to P. Definition (Reduction) Let P : X P and Q : Y P e two classes. A reduction of P to Q is a function f : X Y such that x. P x Q (f x). Halt MPCP Hopcroft et al. (2006) Davis et al. (1994) SR Hesselink (2015) PCP 4

6 PCP and Undecidaility SR to MPCP Turing Machines Reach to SR Halt to SR Conclusion String-Rewriting Systems Σ := {a, } R := {a/a, aa/a} aa R aa aa R a finite alphaet of symols finite set of rewrite rules u/v R x R y y R z xuy R xvy z R z x R z Definition: Word prolem in string-rewriting systems SR (R, x, y) := x R y 5

7 PCP and Undecidaility SR to MPCP Turing Machines Reach to SR Halt to SR Conclusion Reducing String Rewriting to MPCP Word prolem aa R a with R = {a/a, aa/a} aa aa aa a $ $aa a a a a a a a a aa a a $ $ $ $aa $aa aa $aa $aa aa $aa aa aa $aa aa aa a $ $aa aa aa a $ $aa aa aa $aa aa aa a copy dominoes transfer unchanged symols to the next string rewrite dominoes simulate a single rewrite consecutive strings are separated y { } { } { $ f (R, x, y) :=, y $, a $x $ a a : Σ u v } u/v R 6

8 PCP and Undecidaility SR to MPCP Turing Machines Reach to SR Halt to SR Conclusion Correctness Proof x R y MPCP (f (R, x, y)) Let x, y and z e strings over Σ and R a set of rewrite rules. Lemma If x R y, then there is a match for the MPCP instance f (R, x, y). Lemma Let A f (R, x, y). If C 1 A = z (C 2 A), then z R y. Proof. Size induction on A with a generalized claim for all z. A more general lemma yields either z R y or z R m and C 1 A = m (C 2 A ) for a smaller list A. The inductive hypothesis yields m R y. Theorem (SR reduces to MPCP) SR (R, x, y) MPCP (f (R, x, y)) 7

9 PCP and Undecidaility SR to MPCP Turing Machines Reach to SR Halt to SR Conclusion Intermediate Result Halt MPCP SR PCP Hopcroft et al. (2006) Davis et al. (1994) RSR 8

10 PCP and Undecidaility SR to MPCP Turing Machines Reach to SR Halt to SR Conclusion Turing Machines 1 and the Halting Prolem aa Ba A Ba tape := leftof a A midtape B a A rightof a B (a : Σ) (A B : Σ ) Turing machine M := (Q, δ, q 0, H) over finite alphaet Σ transition function δ : Q Σ Q Σ {L, N, R} halting function H : Q B configurations conf : Q tape and step function ˆδ : conf conf ˆδ (q, aa) = (q, caa) if δ (q, ) = (q, c, R) ˆδ (q, aa) = (q, aa) if δ (q, ) = (q,, L) 1 Andrea Asperti and Wilmer Ricciotti (2015) 9

11 PCP and Undecidaility SR to MPCP Turing Machines Reach to SR Halt to SR Conclusion Turing Machines 2 and the Halting Prolem final configurations H c := H (π 1 c) = true reachaility predicate: ˆδ c c H c c c c c Definition: Reachaility Reach (M, c 1, c 2 ) := c 1 c 2 Definition: Halting prolem Halt (M, t):= c f. (q 0, t) c f H cf 2 Andrea Asperti and Wilmer Ricciotti (2015) 10

12 PCP and Undecidaility SR to MPCP Turing Machines Reach to SR Halt to SR Conclusion Reducing Reachaility to String Rewriting f (M, c 1, c 2 ) := (R, x, y)f (M, c 1, c 2 ) := (R, c 1, c 2 )f (M, c 1, c 2 ) := (, c 1, c 2 ) a q 0 a a q1 a aa q1 aa q0 aa qf q 0 aa aq 1 a aq 1 a aaq 0 aq f a string encoding : conf Γ with Γ := q : Q a : Σ c (q, ) (q, aa) (q, BaA) (q, Ba ) c q q aa BqaA Baq each rewrite rule realizes one ˆδ-step q 0 a/aq 1 represents δ (q 0, a ) = (q 1,, R) aq 0 /q f a and q 0 /q f represent δ (q 0, ) = (q f,, L) contains rules that simulate the result of δ (q, a ) and δ (q, ) for all non final states q : Q and symols a : Σ 11

13 PCP and Undecidaility SR to MPCP Turing Machines Reach to SR Halt to SR Conclusion Translating the Transition Function into Rewrite Rules δ (q 1, ) = (q 2, write, move) u v u v write move q 1 q 2 c q 1 q 2 c L q 1 q 2 q 1 q 2 N q 1 q 2 q 1 q 2 R q 1 c q 1 c R q 1 q 2 c q 1 q 2 c L q 1 q 2 q 1 q 2 N q 1 q 2 q 1 q 2 R δ (q 1, a ) = (q 2, write, move) u v u v write move q 1 a q 2 a c q 1 a q 2 c a L q 1 a q 2 a N q 1 a a q 2 R q 1 a q 2 c q 1 a q 2 c L q 1 a q 2 N q 1 a q 2 R 12

14 PCP and Undecidaility SR to MPCP Turing Machines Reach to SR Halt to SR Conclusion Correctness Proof Lemmas If c is not a final configuration, then c ˆδ c. If c z, then z = ˆδ c and c is not a final configuration. Proof. Both lemmas require large case analyses on the tape of configuration c and the result of transitions. Theorem (Reach reduces to SR) c 1 c 2 c 1 c 2 13

15 PCP and Undecidaility SR to MPCP Turing Machines Reach to SR Halt to SR Conclusion Reducing the Halting Prolem to String Rewriting f (M, t) := (R, (q 0, t), y)f (M, t) := (R, (q 0, t), ε)f (M, t) := ( D, (q 0, t), ε) (q 0, t) c f if and only if (q 0, t) c f provide rules enaling c f ε for all final configurations c f : D := { (q f s/q f ), (sq f /q f ), (q f /ε) q f Q H, s Σ {, } } q 0 aa aq f a D aq f D aq f D aq f D q f q f D ε Theorem (Halt reduces to SR) ( c f. (q 0, t) c f H cf ) (q 0, t) D ε 14

16 PCP and Undecidaility SR to MPCP Turing Machines Reach to SR Halt to SR Conclusion Undecidaility Result Halt MPCP SR PCP Reach RSR Hopcroft et al. (2006) Davis et al. (1994) Realization of one Turing machine transition reduction via SR: q 0 a/aq 1 direct reduction to MPCP: q 0 a aq 1 a a 15

17 PCP and Undecidaility SR to MPCP Turing Machines Reach to SR Halt to SR Conclusion Future Work Formalize undecidaility proofs ased on reductions of PCP: prolems related to context-free grammars: inclusion and non-emptiness of intersection (Hopcroft et al. 2006, Hesselink 2015) satisfiaility prolem for variants of specification formalisms (Finkeiner and Hahn 2016, Song and Wu 2014) validity of first-oder formulas (Schöning 2009) secrecy prolem for security protocols (Tiplea et al. 2005) Show PCP λ and Turing undecidale: implement the reductions in the weak call-y-value λ-calculus L (Forster and Smolka 2017) formalize the computational equivalence of L and Turing machines (Dal Lago and Martini 2008) 16

18 References Andrea Asperti and Wilmer Ricciotti. A formalization of multi-tape Turing machines. Theoretical Computer Science, 603:23 42, Martin D. Davis, Ron Sigal, and Elaine J. Weyuker. Computaility, Complexity, and Languages: Fundamentals of Theoretical Computer Science. Academic Press, Wim H. Hesselink. Post s correspondence prolem and the undecidaility of context-free intersection. Manuscript, July John E. Hopcroft, Rajeev Motwani, and Jeffrey D. Ullman. Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Emil L Post. A variant of a recursively unsolvale prolem. Bulletin of the American Mathematical Society, 52(4): , Axel Thue. Proleme üer Veränderungen von Zeichenreihen nach gegeenen Regeln. J. Dywad, 1914.

19 Coq Development Spec Proof Σ Definitions MPCP to PCP SR to MPCP Halt to SR Halt to MPCP SR to RSR RSR to PCP PCP undecidaility Halt, SR, MPCP, PCP : 955 Halt, SR, RSR, PCP : 1112 Halt, MPCP, PCP : 1043

20 Proof (SR to MPCP) Lemma If z R y, then there is some A f (R, x, y) such that C 1 A = z (C 2 A). Proof. Induction on. Lemma If x R y, then there is a match for the MPCP instance f (R, x, y). Proof. The list $ $x :: A is a match for the MPCP instance.

21 Proof (SR to MPCP) Lemma Let A f (R, x, y). If C 1 A = z m (C 2 A), then either z R y and m = [ ] or A = B + :: A, C 1 B = z, C 2 B = m, and z R m for some A, B, m. Proof. Induction on A for all strings z and m. Let A = d :: A. z = [ ]: z = az : y $ $, u v with u = [ ], and y $ $, u v, and a a are candidates for d are candidates for d

22 Proof (Reach to SR) Lemma If c z, then z = ˆδ c and c is not a final configuration. Proof. Let c = (q, t). We have (q, t) = xuy and z = xvy with u/v. Case analysis on tape t. Assume t =. (q, ) = q = xuy. If u/v = q 1 / aq 2 simulating δ (q 1, ) = (q 2, a, R), then q = xq 1 y yields q = q 1 and ˆδ c = aq2 = x aq 2 y = z. Remark: It is important that. Assume a configuration (q 1, ) = q 1 and δ (q 1, ) = (q 2, a, R). The only applicale rewrite rule is (q 1 / aq 2 ) and ˆδ(q1, ) = (q 2, a ) = aq 2. If the only one tape delimiter is, the rule (q 1 / aq 2 ) for the right end of the tape is also suitale. But aq 2 (q 2, a ) = aq 2.

23 Proof (Halt to SR) Lemmas 1. If c f is a final configuration, then c f D ε. 2. If c D z for some z, then c is a final configuration. 3. If c D ε, then c c f for some final configuration c f. Proof (3). Induction on the derivation with a generalized claim for all c. c = ε is contradictory. c D z: If the rewrite rule is from, we use the inductive hypothesis and z D ε, otherwise the lemma aove.

24 Reducing Restricted String Rewriting to PCP { $ f (R, x, y):=, y $,, $x $ Example: } { a, ã ã a { } a : Σ u, ũ ṽ v } u/v R R := {aa/a, a/a}, x := aa and y := a. Since aa R a holds, we should e ale to construct a match for the PCP instance { } $, a $,,, a, ã, $aa $ ã a,, aa, a a a aa a, a a, $ $aa aa ã ã a a $ $ $aa ã a $ $aa ã a $

25 Reducing the Halting Prolem to MPCP tape leftof midtape rightof c (q, ) (q, aa) (q, BaA) (q, Ba ) c q q aa BqaA Baq Encoding of configurations using a lank symol. $ $ q s aa q s a q 0 a a a... q 0 q f q f q f... q f q f q f $ $ initial domino transition dominoes for all non final states copy dominoes for all symols and, deletion dominoes for all final states final dominoes for all final states

26 Reducing MPCP to PCP f { 1, 10111, } { = $#1#0#1#1#1 $#1#0#, #1, #1#0#1#1#1 1#1#1# 1#0#, #1#0 0#, } #$ $ Both instances are solvale: $#1#0#1#1#1 $#1#0# #1 1#1#1# #1 1#1#1# #1#0 0# #$ $ interleave the domino components with # symols starting to the left of the first symol in the top string and to the right in the ottom string delete empty dominoes since the interleaving has no effect provide an additional copy of the first MPCP domino starting at the top and the ottom with $# provide an extra domino adding the missing # at the top row