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Equations and Logic on Words Equations and logic on words Sam van Gool Utrecht University TACL, Nice 17 June 2019 Overview Logic on words Duality Equations between words Equations between languages 1 / 26 Overview Logic on words Duality Equations between words Equations between languages 1 / 26 I Solution 1: a (deterministic) automaton A: 0 1 1 0 q0 q1 q2 1 0 Answer yes iff A accepts w. ∗ I Solution 2: a homomorphism ': f0; 1g ! S3 defined by 0 7! (1 2); 1 7! (0 1): Answer yes iff the permutation '(w) sends 0 to 1. Regular languages: example I A programming problem: given a natural number in binary, w 2 f0; 1g∗, determine if w is congruent 1 modulo 3. 2 / 26 ∗ I Solution 2: a homomorphism ': f0; 1g ! S3 defined by 0 7! (1 2); 1 7! (0 1): Answer yes iff the permutation '(w) sends 0 to 1. Regular languages: example I A programming problem: given a natural number in binary, w 2 f0; 1g∗, determine if w is congruent 1 modulo 3. I Solution 1: a (deterministic) automaton A: 0 1 1 0 q0 q1 q2 1 0 Answer yes iff A accepts w. 2 / 26 Regular languages: example I A programming problem: given a natural number in binary, w 2 f0; 1g∗, determine if w is congruent 1 modulo 3. I Solution 1: a (deterministic) automaton A: 0 1 1 0 q0 q1 q2 1 0 Answer yes iff A accepts w. ∗ I Solution 2: a homomorphism ': f0; 1g ! S3 defined by 0 7! (1 2); 1 7! (0 1): Answer yes iff the permutation '(w) sends 0 to 1. 2 / 26 Regular languages: example I A programming problem: given a natural number in binary, w 2 f0; 1g∗, determine if w is congruent 1 modulo 3. I Solution 1: a (deterministic) automaton A: 0 1 1 0 q0 q1 q2 1 0 Answer yes iff A accepts w. I Solution 3: an MSO sentence ': 9Q09Q19Q2(Q0(first) ^ Q1(last)^ 8x[0(x) ^ Q0(x) ! Q0(Sx)] ^ [1(x) ^ Q0(x) ! Q1(Sx)] ^ ::: ): Answer yes iff w satisfies the formula '. 2 / 26 Regular languages Regular languages are subsets L ⊆ Σ∗ which are ... I recognizable by a finite automaton; I invariant under a finite index monoid congruence; I definable by a monadic second order sentence. Myhill-Nerode 1958; Büchi 1960 3 / 26 I Semantics. A word w = a1 ::: an gives a structure W . I The underlying set of W is f1;:::; ng. W I The natural linear order < interprets the binary predicate <. W I For every letter a 2 Σ, a := fi 2 f1;:::; ng: ai = ag. Logic on words I Syntax. Monadic Second Order (MSO) logic over <, Σ. I Basic propositional connectives: ^, :. I Quantification over first-order variables x, y, . and monadic second-order variables P, Q, ::: . I Relational signature: x < y, a(x) for a 2 Σ. 4 / 26 Logic on words I Syntax. Monadic Second Order (MSO) logic over <, Σ. I Basic propositional connectives: ^, :. I Quantification over first-order variables x, y, . and monadic second-order variables P, Q, ::: . I Relational signature: x < y, a(x) for a 2 Σ. I Semantics. A word w = a1 ::: an gives a structure W . I The underlying set of W is f1;:::; ng. W I The natural linear order < interprets the binary predicate <. W I For every letter a 2 Σ, a := fi 2 f1;:::; ng: ai = ag. 4 / 26 Logic on words I Syntax. Monadic Second Order (MSO) logic over <, Σ. I Semantics. A word w = a1 ::: an gives a structure W . ∗ I For a sentence ', L' := fw 2 Σ j w j= 'g. I A language L is regular iff L = L' for some ' in MSO. I Shortcuts such as S(x), first, last, ⊆, ... are MSO-definable. 5 / 26 I aaaa j= ', but aaaaa 6j= '. I W j= ' iff W has even length. : 9P 9xP (x) ^ P ⊆ a ^ 8y (8x[P(x) ! x < y]) ! b(y) . I aacbaccaabbb j= ', but aacbaccaabbc 6j= '. I W j= ' iff W has a non-empty subset of a-positions after which there are only b-positions. 0 : 9x a(x) ^ 8y[x < y ! (:a(y) ^ b(y))] . I “There is a last a-position, with only b-positions after that.” and 0 are equivalent, and 0 is first order. Question. Does such an equivalent first order formula exist for '? Logic on words: examples ': 9P P(first) ^ :P(last) ^ 8x(P(x) $ :P(S(x)) . 6 / 26 j= ', but aaaaa 6j= '. I W j= ' iff W has even length. : 9P 9xP (x) ^ P ⊆ a ^ 8y (8x[P(x) ! x < y]) ! b(y) . I aacbaccaabbb j= ', but aacbaccaabbc 6j= '. I W j= ' iff W has a non-empty subset of a-positions after which there are only b-positions. 0 : 9x a(x) ^ 8y[x < y ! (:a(y) ^ b(y))] . I “There is a last a-position, with only b-positions after that.” and 0 are equivalent, and 0 is first order. Question. Does such an equivalent first order formula exist for '? Logic on words: examples ': 9P P(first) ^ :P(last) ^ 8x(P(x) $ :P(S(x)) . I aaaa 6 / 26 but aaaaa 6j= '. I W j= ' iff W has even length. : 9P 9xP (x) ^ P ⊆ a ^ 8y (8x[P(x) ! x < y]) ! b(y) . I aacbaccaabbb j= ', but aacbaccaabbc 6j= '. I W j= ' iff W has a non-empty subset of a-positions after which there are only b-positions. 0 : 9x a(x) ^ 8y[x < y ! (:a(y) ^ b(y))] . I “There is a last a-position, with only b-positions after that.” and 0 are equivalent, and 0 is first order. Question. Does such an equivalent first order formula exist for '? Logic on words: examples ': 9P P(first) ^ :P(last) ^ 8x(P(x) $ :P(S(x)) . I aaaa j= ', 6 / 26 : 9P 9xP (x) ^ P ⊆ a ^ 8y (8x[P(x) ! x < y]) ! b(y) . I aacbaccaabbb j= ', but aacbaccaabbc 6j= '. I W j= ' iff W has a non-empty subset of a-positions after which there are only b-positions. 0 : 9x a(x) ^ 8y[x < y ! (:a(y) ^ b(y))] . I “There is a last a-position, with only b-positions after that.” and 0 are equivalent, and 0 is first order. Question. Does such an equivalent first order formula exist for '? Logic on words: examples ': 9P P(first) ^ :P(last) ^ 8x(P(x) $ :P(S(x)) . I aaaa j= ', but aaaaa 6j= '. I W j= ' iff W has even length. 6 / 26 I aacbaccaabbb j= ', but aacbaccaabbc 6j= '. I W j= ' iff W has a non-empty subset of a-positions after which there are only b-positions. 0 : 9x a(x) ^ 8y[x < y ! (:a(y) ^ b(y))] . I “There is a last a-position, with only b-positions after that.” and 0 are equivalent, and 0 is first order. Question. Does such an equivalent first order formula exist for '? Logic on words: examples ': 9P P(first) ^ :P(last) ^ 8x(P(x) $ :P(S(x)) . I aaaa j= ', but aaaaa 6j= '. I W j= ' iff W has even length. : 9P 9xP (x) ^ P ⊆ a ^ 8y (8x[P(x) ! x < y]) ! b(y) . 6 / 26 j= ', but aacbaccaabbc 6j= '. I W j= ' iff W has a non-empty subset of a-positions after which there are only b-positions. 0 : 9x a(x) ^ 8y[x < y ! (:a(y) ^ b(y))] . I “There is a last a-position, with only b-positions after that.” and 0 are equivalent, and 0 is first order. Question. Does such an equivalent first order formula exist for '? Logic on words: examples ': 9P P(first) ^ :P(last) ^ 8x(P(x) $ :P(S(x)) . I aaaa j= ', but aaaaa 6j= '. I W j= ' iff W has even length. : 9P 9xP (x) ^ P ⊆ a ^ 8y (8x[P(x) ! x < y]) ! b(y) . I aacbaccaabbb 6 / 26 but aacbaccaabbc 6j= '. I W j= ' iff W has a non-empty subset of a-positions after which there are only b-positions. 0 : 9x a(x) ^ 8y[x < y ! (:a(y) ^ b(y))] . I “There is a last a-position, with only b-positions after that.” and 0 are equivalent, and 0 is first order. Question. Does such an equivalent first order formula exist for '? Logic on words: examples ': 9P P(first) ^ :P(last) ^ 8x(P(x) $ :P(S(x)) . I aaaa j= ', but aaaaa 6j= '. I W j= ' iff W has even length. : 9P 9xP (x) ^ P ⊆ a ^ 8y (8x[P(x) ! x < y]) ! b(y) . I aacbaccaabbb j= ', 6 / 26 0 : 9x a(x) ^ 8y[x < y ! (:a(y) ^ b(y))] . I “There is a last a-position, with only b-positions after that.” and 0 are equivalent, and 0 is first order. Question. Does such an equivalent first order formula exist for '? Logic on words: examples ': 9P P(first) ^ :P(last) ^ 8x(P(x) $ :P(S(x)) . I aaaa j= ', but aaaaa 6j= '. I W j= ' iff W has even length. : 9P 9xP (x) ^ P ⊆ a ^ 8y (8x[P(x) ! x < y]) ! b(y) . I aacbaccaabbb j= ', but aacbaccaabbc 6j= '. I W j= ' iff W has a non-empty subset of a-positions after which there are only b-positions. 6 / 26 I “There is a last a-position, with only b-positions after that.” and 0 are equivalent, and 0 is first order. Question. Does such an equivalent first order formula exist for '? Logic on words: examples ': 9P P(first) ^ :P(last) ^ 8x(P(x) $ :P(S(x)) . I aaaa j= ', but aaaaa 6j= '. I W j= ' iff W has even length. : 9P 9xP (x) ^ P ⊆ a ^ 8y (8x[P(x) ! x < y]) ! b(y) .
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