Generalized Automata over the Reals Klaus Meer Brandenburgische - - PowerPoint PPT Presentation

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Generalized Automata over the Reals

Klaus Meer

Brandenburgische Technische Universit¨ at, Cottbus-Senftenberg, Germany

Metafinite 2017, Reykjavik, June 2017 (joint work with Ameen Naif)

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

• 1. Introduction

One of the most basic algorithm models in Computer Science Finite Automata over finite alphabets Related theory well developed, for example, with respect to structural characterizations of languages accepted, decidability and complexity of important computational problems, relation to logic etc. Decidability questions are treated in Turing machine framework.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

In recent years lot of interest in Theoretical Computer Science in alternative computation models: Neural Networks Quantum computing membrane computing analogue computing algebraic models .. Underlying data structure often more general than finite alphabets, for example real and complex numbers

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

In recent years lot of interest in Theoretical Computer Science in alternative computation models: Neural Networks Quantum computing membrane computing analogue computing algebraic models .. Underlying data structure often more general than finite alphabets, for example real and complex numbers Thus natural idea: extend concept of finite automata to more such general structures

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

At least two lines of motivation: several generalized automata models have been introduced in areas like program verification, database theory: automata for words and trees on infinite structures define and study finite automata model in algebraic computability theory following approach by Blum & Shub & Smale: BSS-model for real and complex numbers (compare Quantum Finite Automata)

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

At least two lines of motivation: several generalized automata models have been introduced in areas like program verification, database theory: automata for words and trees on infinite structures define and study finite automata model in algebraic computability theory following approach by Blum & Shub & Smale: BSS-model for real and complex numbers (compare Quantum Finite Automata) Generalized automata model over arbitrary structures introduced by Gandhi & Khoussainov & Liu, TAMC 2012:

• homogenizes (some of the) previous automata concepts;
• authors ask in particular for studying their extended automata

model in BSS framework

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

At least two lines of motivation: several generalized automata models have been introduced in areas like program verification, database theory: automata for words and trees on infinite structures define and study finite automata model in algebraic computability theory following approach by Blum & Shub & Smale: BSS-model for real and complex numbers (compare Quantum Finite Automata) Generalized automata model over arbitrary structures introduced by Gandhi & Khoussainov & Liu, TAMC 2012:

• homogenizes (some of the) previous automata concepts;
• authors ask in particular for studying their extended automata

model in BSS framework starting point of this work!

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

• 2. Generalized Finite Automata over R

Basic ideas of how automata introduced by Gandhi et al. work: structure S consists of universe D, finite sets of (binary) functions and relations; universe can be infinite, uncountable

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

• 2. Generalized Finite Automata over R

Basic ideas of how automata introduced by Gandhi et al. work: structure S consists of universe D, finite sets of (binary) functions and relations; universe can be infinite, uncountable automata process words in D∗, i.e., finite strings with components from universe;

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

• 2. Generalized Finite Automata over R

Basic ideas of how automata introduced by Gandhi et al. work: structure S consists of universe D, finite sets of (binary) functions and relations; universe can be infinite, uncountable automata process words in D∗, i.e., finite strings with components from universe; automata have accepting/rejecting states

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

• 2. Generalized Finite Automata over R

Basic ideas of how automata introduced by Gandhi et al. work: structure S consists of universe D, finite sets of (binary) functions and relations; universe can be infinite, uncountable automata process words in D∗, i.e., finite strings with components from universe; automata have accepting/rejecting states ! each single step allows limited way of computation and testing using functions and relations from S as well as a constant number of registers

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

• 2. Generalized Finite Automata over R

Basic ideas of how automata introduced by Gandhi et al. work: structure S consists of universe D, finite sets of (binary) functions and relations; universe can be infinite, uncountable automata process words in D∗, i.e., finite strings with components from universe; automata have accepting/rejecting states ! each single step allows limited way of computation and testing using functions and relations from S as well as a constant number of registers

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

• 2. Generalized Finite Automata over R

Basic ideas of how automata introduced by Gandhi et al. work: structure S consists of universe D, finite sets of (binary) functions and relations; universe can be infinite, uncountable automata process words in D∗, i.e., finite strings with components from universe; automata have accepting/rejecting states ! each single step allows limited way of computation and testing using functions and relations from S as well as a constant number of registers Results by Gandhi et al. deal with countable universes

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Most interesting structures for us are the reals and complex numbers with basic operations: SR := (R, +, −, •, pr, ≥, =) reals as ring with order; SC := (C, +, −, •, pr, =) complex numbers as ring with equality;

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Most interesting structures for us are the reals and complex numbers with basic operations: SR := (R, +, −, •, pr, ≥, =) reals as ring with order; SC := (C, +, −, •, pr, =) complex numbers as ring with equality; gives possibility to compare generalized finite automata with BSS theory

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Most interesting structures for us are the reals and complex numbers with basic operations: SR := (R, +, −, •, pr, ≥, =) reals as ring with order; SC := (C, +, −, •, pr, =) complex numbers as ring with equality; gives possibility to compare generalized finite automata with BSS theory Important: All statements below about decidability, computability, complexity etc. to be understood in BSS model over R and C

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Most interesting structures for us are the reals and complex numbers with basic operations: SR := (R, +, −, •, pr, ≥, =) reals as ring with order; SC := (C, +, −, •, pr, =) complex numbers as ring with equality; gives possibility to compare generalized finite automata with BSS theory Important: All statements below about decidability, computability, complexity etc. to be understood in BSS model over R and C Gandhi & Khoussainov & Liu approach applied to SR: Real GKL automata

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

How real GKL automata work:

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

How real GKL automata work: fix constants k, ℓ ∈ N; (SR, k, ℓ)-automaton works as follows: uses ℓ fixed constants c1, . . . , cℓ from R;

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

How real GKL automata work: fix constants k, ℓ ∈ N; (SR, k, ℓ)-automaton works as follows: uses ℓ fixed constants c1, . . . , cℓ from R; has k registers v1, . . . , vk which can store real numbers; registers used to perform limited computations, initial values vi = 0;

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

How real GKL automata work: fix constants k, ℓ ∈ N; (SR, k, ℓ)-automaton works as follows: uses ℓ fixed constants c1, . . . , cℓ from R; has k registers v1, . . . , vk which can store real numbers; registers used to perform limited computations, initial values vi = 0; has finite state set Q, accepting states F ⊆ Q, initial state s;

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

How real GKL automata work: fix constants k, ℓ ∈ N; (SR, k, ℓ)-automaton works as follows: uses ℓ fixed constants c1, . . . , cℓ from R; has k registers v1, . . . , vk which can store real numbers; registers used to perform limited computations, initial values vi = 0; has finite state set Q, accepting states F ⊆ Q, initial state s; processes points (x1, . . . , xn) ∈ Rn, n arbitrary componentwise:

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

How real GKL automata work: fix constants k, ℓ ∈ N; (SR, k, ℓ)-automaton works as follows: uses ℓ fixed constants c1, . . . , cℓ from R; has k registers v1, . . . , vk which can store real numbers; registers used to perform limited computations, initial values vi = 0; has finite state set Q, accepting states F ⊆ Q, initial state s; processes points (x1, . . . , xn) ∈ Rn, n arbitrary componentwise:

• read current xt ∈ R once (left to right);

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

How real GKL automata work: fix constants k, ℓ ∈ N; (SR, k, ℓ)-automaton works as follows: uses ℓ fixed constants c1, . . . , cℓ from R; has k registers v1, . . . , vk which can store real numbers; registers used to perform limited computations, initial values vi = 0; has finite state set Q, accepting states F ⊆ Q, initial state s; processes points (x1, . . . , xn) ∈ Rn, n arbitrary componentwise:

• read current xt ∈ R once (left to right);
• perform ordering tests comparing xt with all values vi and all

constants cj; gives a result vector b ∈ {0, 1}k+ℓ;

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

How real GKL automata work: fix constants k, ℓ ∈ N; (SR, k, ℓ)-automaton works as follows: uses ℓ fixed constants c1, . . . , cℓ from R; has k registers v1, . . . , vk which can store real numbers; registers used to perform limited computations, initial values vi = 0; has finite state set Q, accepting states F ⊆ Q, initial state s; processes points (x1, . . . , xn) ∈ Rn, n arbitrary componentwise:

• read current xt ∈ R once (left to right);
• perform ordering tests comparing xt with all values vi and all

constants cj; gives a result vector b ∈ {0, 1}k+ℓ;

• compute new state and new register values of all vi depending
• n: xt, b, current value of vi and actual state q; important:

computation on register vi only involves xt, vi and one of the functions of structure SR, i.e., +, −, •, pr

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Given (SR, k, ℓ)-automaton A we define as usual: L(A) ⊆ R∗ accepted language: all x for which computation stops in F; non-deterministic automaton: several options how to continue in each step

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Given (SR, k, ℓ)-automaton A we define as usual: L(A) ⊆ R∗ accepted language: all x for which computation stops in F; non-deterministic automaton: several options how to continue in each step All concepts similar for complex GKL automata: (SC, k, ℓ) Here, only equality as relation in the structure.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Example a) The set {−1n1n|n ∈ N} ⊂ R∗ is acceptable by a deterministic (SR, 1, 3)-automaton using −1, 0, 1 as constants. Use the single register to count occurences of −1’s and 1’s.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Example a) The set {−1n1n|n ∈ N} ⊂ R∗ is acceptable by a deterministic (SR, 1, 3)-automaton using −1, 0, 1 as constants. Use the single register to count occurences of −1’s and 1’s. b) The real Knapsack problem KSR :=

n≥1

KS(n)

R

:= {x ∈ (0, 1)n ⊂ Rn|∃S ⊂ {1, . . . n} s.t.

• i∈S

xi = 1} can be accepted by a non-deterministic (SR, 1, 2)-automaton with constants 0, 1. Guess non-deterministically whether to include the current xi into the sum; use the register to add and finally check equality to 1.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Outline of further talk: 1) Study of basic questions for real and complex GKL automata

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Outline of further talk: 1) Study of basic questions for real and complex GKL automata 2) Modification of model to get more natural picture: periodic real GKL automata

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Outline of further talk: 1) Study of basic questions for real and complex GKL automata 2) Modification of model to get more natural picture: periodic real GKL automata 3) Future questions

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

• 2. Some results for real GKL automata

Results show quite a diverse picture in comparison to classical automata theory over finite alphabets;

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

• 2. Some results for real GKL automata

Results show quite a diverse picture in comparison to classical automata theory over finite alphabets; main reason why many classical proof methods don’t apply: computation registers vi change values and thus their evolvement has to be controlled; that often turns out to be (more) difficult or impossible!

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

• 2. Some results for real GKL automata

Results show quite a diverse picture in comparison to classical automata theory over finite alphabets; main reason why many classical proof methods don’t apply: computation registers vi change values and thus their evolvement has to be controlled; that often turns out to be (more) difficult or impossible! For some results we focus on the complex model; real framework looks even more unsatisfying.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

2.1 Structure theorem Definition For L ⊆ Cn, n ≥ 1 let P1(L) be projection of L to first component. L is recursively co-finite (rcf) if i) P1(L) is co-finite;

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

2.1 Structure theorem Definition For L ⊆ Cn, n ≥ 1 let P1(L) be projection of L to first component. L is recursively co-finite (rcf) if i) P1(L) is co-finite; ii) if n > 1 for any x∗ ∈ P1(L) the set {(x2, . . . , xn) ∈ Cn−1|(x∗, x2, . . . , xn) ∈ L} is recursively co-finite.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

2.1 Structure theorem Definition For L ⊆ Cn, n ≥ 1 let P1(L) be projection of L to first component. L is recursively co-finite (rcf) if i) P1(L) is co-finite; ii) if n > 1 for any x∗ ∈ P1(L) the set {(x2, . . . , xn) ∈ Cn−1|(x∗, x2, . . . , xn) ∈ L} is recursively co-finite.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

2.1 Structure theorem Definition For L ⊆ Cn, n ≥ 1 let P1(L) be projection of L to first component. L is recursively co-finite (rcf) if i) P1(L) is co-finite; ii) if n > 1 for any x∗ ∈ P1(L) the set {(x2, . . . , xn) ∈ Cn−1|(x∗, x2, . . . , xn) ∈ L} is recursively co-finite. L is rcf of cardinality s ∈ N if cardinalities of all complements of projections involved in the above definition are less than or equal to s.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Theorem Let A be a (SC, k, ℓ)-automaton accepting L(A) ⊆ C∗. For n ∈ N let Ln(A) := L(A) ∩ Cn and Ln(A) its complement in Cn.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Theorem Let A be a (SC, k, ℓ)-automaton accepting L(A) ⊆ C∗. For n ∈ N let Ln(A) := L(A) ∩ Cn and Ln(A) its complement in Cn. a) A deterministic, then for each n ∈ N exactly one of the two sets Ln(A) and Ln(A) contains a rcf set of cardinality at most s := k + ℓ, i.e. this cardinality is independent of n.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Theorem Let A be a (SC, k, ℓ)-automaton accepting L(A) ⊆ C∗. For n ∈ N let Ln(A) := L(A) ∩ Cn and Ln(A) its complement in Cn. a) A deterministic, then for each n ∈ N exactly one of the two sets Ln(A) and Ln(A) contains a rcf set of cardinality at most s := k + ℓ, i.e. this cardinality is independent of n. b) A non-deterministic, then for each n ∈ N exactly one of the two sets Ln(A) and Ln(A) contains a rcf set of cardinality at most O(Mn) for some constant M.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Theorem Let A be a (SC, k, ℓ)-automaton accepting L(A) ⊆ C∗. For n ∈ N let Ln(A) := L(A) ∩ Cn and Ln(A) its complement in Cn. a) A deterministic, then for each n ∈ N exactly one of the two sets Ln(A) and Ln(A) contains a rcf set of cardinality at most s := k + ℓ, i.e. this cardinality is independent of n. b) A non-deterministic, then for each n ∈ N exactly one of the two sets Ln(A) and Ln(A) contains a rcf set of cardinality at most O(Mn) for some constant M. Proof idea: Analyse characteristic path of a computation, i.e. set

• f inputs which satisfy no equality test. Since only finitely many

registers vi occur, this leads to rcf structure.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Theorem Let A be a (SC, k, ℓ)-automaton accepting L(A) ⊆ C∗. For n ∈ N let Ln(A) := L(A) ∩ Cn and Ln(A) its complement in Cn. a) A deterministic, then for each n ∈ N exactly one of the two sets Ln(A) and Ln(A) contains a rcf set of cardinality at most s := k + ℓ, i.e. this cardinality is independent of n. b) A non-deterministic, then for each n ∈ N exactly one of the two sets Ln(A) and Ln(A) contains a rcf set of cardinality at most O(Mn) for some constant M. Proof idea: Analyse characteristic path of a computation, i.e. set

• f inputs which satisfy no equality test. Since only finitely many

registers vi occur, this leads to rcf structure.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Corollary a) The complex Knapsack problem can be accepted by a non-deterministic automaton, but not a deterministic one. Its complement cannot be accepted by a non-deterministic one.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Corollary a) The complex Knapsack problem can be accepted by a non-deterministic automaton, but not a deterministic one. Its complement cannot be accepted by a non-deterministic one. b) The class of languages accepted by non-deterministic complex automata is not closed under complementation.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Corollary a) The complex Knapsack problem can be accepted by a non-deterministic automaton, but not a deterministic one. Its complement cannot be accepted by a non-deterministic one. b) The class of languages accepted by non-deterministic complex automata is not closed under complementation.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Corollary a) The complex Knapsack problem can be accepted by a non-deterministic automaton, but not a deterministic one. Its complement cannot be accepted by a non-deterministic one. b) The class of languages accepted by non-deterministic complex automata is not closed under complementation. All results hold as well for real GKL automata (by slightly different proofs).

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

2.2 A weak pumping lemma Theorem Let L ⊆ C∗ be accepted by a deterministic (SC, k)-automaton A. There is a word w := uz ∈ C∗ such that either all uzt, t ∈ N0 belong to L or they all belong to C∗ \ L. Moreover, u and z have an algebraic length of at most K and 2K, respectively, where K denotes the number of states of A.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

2.2 A weak pumping lemma Theorem Let L ⊆ C∗ be accepted by a deterministic (SC, k)-automaton A. There is a word w := uz ∈ C∗ such that either all uzt, t ∈ N0 belong to L or they all belong to C∗ \ L. Moreover, u and z have an algebraic length of at most K and 2K, respectively, where K denotes the number of states of A. Proof idea: Find a loop in the automaton’s computation that is realized arbitrarily often by repeating the same part of an input; problem because of changing register values; topological arguments how to guarantee existence of z.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Example The following variant of Subset Sum cannot be accepted by a deterministic complex automaton: L := {(x1, . . . , xn) ∈ Cn|∃S1, S2 ⊂ {1, . . . , n} disjoint s.t.

• i∈S1

xi =

i∈S2

xi}

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Example The following variant of Subset Sum cannot be accepted by a deterministic complex automaton: L := {(x1, . . . , xn) ∈ Cn|∃S1, S2 ⊂ {1, . . . , n} disjoint s.t.

• i∈S1

xi =

i∈S2

xi} Remarks: a) Weak PL because not clear whether it holds for each input of sufficient length and because word that can be pumped might belong to complement language; b) (Proof of) Pumping Lemma only holds over C.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

2.3 Undecidability results Theorem The following problems on real and complex GKL automata are undecidable in the respective BSS model: a) Emptiness Problem: Given A, is L(A) = ∅?

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

2.3 Undecidability results Theorem The following problems on real and complex GKL automata are undecidable in the respective BSS model: a) Emptiness Problem: Given A, is L(A) = ∅? b) Equivalence Problem: Given two automata, do they accept the same language?

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

2.3 Undecidability results Theorem The following problems on real and complex GKL automata are undecidable in the respective BSS model: a) Emptiness Problem: Given A, is L(A) = ∅? b) Equivalence Problem: Given two automata, do they accept the same language? c) Reachability Problem: Given A and a state p of A, is there a computation of A that reaches p?

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

2.3 Undecidability results Theorem The following problems on real and complex GKL automata are undecidable in the respective BSS model: a) Emptiness Problem: Given A, is L(A) = ∅? b) Equivalence Problem: Given two automata, do they accept the same language? c) Reachability Problem: Given A and a state p of A, is there a computation of A that reaches p? d) Minimization Problem: Given A, is it state minimal among all deterministic automata accepting L(A)? As consequence, there is no BSS algorithm minimizing any given real GKL automaton.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

2.3 Undecidability results Theorem The following problems on real and complex GKL automata are undecidable in the respective BSS model: a) Emptiness Problem: Given A, is L(A) = ∅? b) Equivalence Problem: Given two automata, do they accept the same language? c) Reachability Problem: Given A and a state p of A, is there a computation of A that reaches p? d) Minimization Problem: Given A, is it state minimal among all deterministic automata accepting L(A)? As consequence, there is no BSS algorithm minimizing any given real GKL automaton. e) and some more

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Proof idea: The rational numbers Q are known to be undecidable in both the real and the complex BSS model; construct specific GKL automaton A∗ that is able to accept a particularly defined language L; projection of L to first component gives Q; embed decision problem for Q into each of above problems by using suitable variant of A∗

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

• 3. Periodic GKL automata

Lesson to learn from previous results: Over uncountable structures GKL automata do not really share typical properties of finite automata; computational abilities are too strong.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

• 3. Periodic GKL automata

Lesson to learn from previous results: Over uncountable structures GKL automata do not really share typical properties of finite automata; computational abilities are too strong. Restrict GKL-model in the way it can compute: use of a computation period T; several possibilities, one promising is: after every sequence of T steps reset all register values and step counter to 0.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

• 3. Periodic GKL automata

Lesson to learn from previous results: Over uncountable structures GKL automata do not really share typical properties of finite automata; computational abilities are too strong. Restrict GKL-model in the way it can compute: use of a computation period T; several possibilities, one promising is: after every sequence of T steps reset all register values and step counter to 0. Resulting real periodic GKL automata (SR, k, ℓ, T) allow better control of computation registers and share many of the properties

• f finite automata, adapted accordingly to the underlying

uncountable universe and the BSS model

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Theorem (Decidability) The problems below are decidable in the real BSS model: a) Emptiness: Given an (SR, k, ℓ, T)-automaton A, is L(A) = ∅? b) Reachability I: Given A as in a) with state set Q together with a state q ∈ Q, is there a computation starting in A’s initial state with register entries 0 that reaches q? c) Reachability II: Given an automaton A, a state q, a counter value t ∈ {0, . . . , T − 1}, and a v ∈ Rk, is there a computation starting in A’s initial state with register entries 0 that reaches q such that the counter’s value is t and the register values equal v? d) Equivalence: Given two real periodic GKL automata A1, A2, is L(A1) = L(A2)?

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Main proof idea: Attach directed graph GA to automaton A Vertices V of GA: all states q of A and for each 1 ≤ t < T copy q(t) of state q

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Main proof idea: Attach directed graph GA to automaton A Vertices V of GA: all states q of A and for each 1 ≤ t < T copy q(t) of state q Edges E of GA : (p, q) ∈ E iff q reachable from p within one period in T steps starting with counter value and register values 0; (p, q(t)) ∈ E iff q is reachable from p in t < T steps starting with register values 0

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Main proof idea: Attach directed graph GA to automaton A Vertices V of GA: all states q of A and for each 1 ≤ t < T copy q(t) of state q Edges E of GA : (p, q) ∈ E iff q reachable from p within one period in T steps starting with counter value and register values 0; (p, q(t)) ∈ E iff q is reachable from p in t < T steps starting with register values 0 Attach sets of real vectors to each edge of GA: S(p, q) ⊂ RT and S(p, q(t)) ⊂ Rt sets of inputs branched during such a computation; these sets are semi-algebraic

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Implies decidability of many properties of GA using real quantifier elimination algorithms:

• edge relation in GA is decidable by QE
• emptiness of sets S(p, q) and S(p, q(t)) is decidable by QE
• combination of above with basic graph search algorithms in

GA gives results of theorem

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Implies decidability of many properties of GA using real quantifier elimination algorithms:

• edge relation in GA is decidable by QE
• emptiness of sets S(p, q) and S(p, q(t)) is decidable by QE
• combination of above with basic graph search algorithms in

GA gives results of theorem Remark Quantifier elimination as well implies (kind of) state minimization algorithm.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Theorem (Structural Results I) a) Languages over R∗ that are acceptable by deterministic periodic real GKL automata satisfy the Pumping Lemma. b) The classes of languages over R∗ acceptable by non-deterministic and by deterministic periodic real GKL automata are the same. c) Languages acceptable by real periodic GKL automata are closed under union, complementation, and intersection.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Theorem (Structural Results I) a) Languages over R∗ that are acceptable by deterministic periodic real GKL automata satisfy the Pumping Lemma. b) The classes of languages over R∗ acceptable by non-deterministic and by deterministic periodic real GKL automata are the same. c) Languages acceptable by real periodic GKL automata are closed under union, complementation, and intersection. Proof ideas:

• Pumping Lemma as usual with necessary length depending on

both number of states and periodicity

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Theorem (Structural Results I) a) Languages over R∗ that are acceptable by deterministic periodic real GKL automata satisfy the Pumping Lemma. b) The classes of languages over R∗ acceptable by non-deterministic and by deterministic periodic real GKL automata are the same. c) Languages acceptable by real periodic GKL automata are closed under union, complementation, and intersection. Proof ideas:

• Pumping Lemma as usual with necessary length depending on

both number of states and periodicity

• Power set construction a bit more complicated because control of

computation registers

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Example The real Knapsack language satisfies the Pumping Lemma but is not acceptable by a periodic automaton: if (x1, . . . , xn) ∈ KSR with a solution S ⊂ {1, . . . , n}, then each xi with i ∈ S can be pumped;

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Example The real Knapsack language satisfies the Pumping Lemma but is not acceptable by a periodic automaton: if (x1, . . . , xn) ∈ KSR with a solution S ⊂ {1, . . . , n}, then each xi with i ∈ S can be pumped; if A would accept with period T, then consider input (x1, . . . , xs) ∈ KSR such that A has finished a period after reading xs and

s

• i=1

xi < 1 s large enough such that there are different choices for xs+1 which lead to KSR and to KSR, respectively. Such an s exists since A uses finitely many constants only; A’s action on xs+1

• nly depends constants and xs+1 since register values are 0

Thus, A does not work correctly.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Semi-algebraic sets S(p, q) and S(p, q(t)) give rise to characterization of acceptable sets via basic regular real sets; the latter are defined as solutions of very particular polynomial systems simulating the computation of an automaton; f.e., each variable occurs in a given order and linearly only Regular real sets then are obtained from basic regular ones within finitely many applications of union, intersection, and a restricted version of concatenation and Kleene-*

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Semi-algebraic sets S(p, q) and S(p, q(t)) give rise to characterization of acceptable sets via basic regular real sets; the latter are defined as solutions of very particular polynomial systems simulating the computation of an automaton; f.e., each variable occurs in a given order and linearly only Regular real sets then are obtained from basic regular ones within finitely many applications of union, intersection, and a restricted version of concatenation and Kleene-* Theorem (Structural Results II) The class of languages in R∗ acceptable by a real periodic GKL automaton are precisely the real regular ones.

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

• 4. Conclusions and Outline
• riginal GKL automata for uncountable structures too strong;

many classical results do not hold: basic questions about GKL automata undecidable in BSS model; almost no structural results for acceptable languages, no closure properties

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

• 4. Conclusions and Outline
• riginal GKL automata for uncountable structures too strong;

many classical results do not hold: basic questions about GKL automata undecidable in BSS model; almost no structural results for acceptable languages, no closure properties periodic variant of GKL automata more reasonable model of real finite automata; most classical results hold accordingly: closure properties; Pumping Lemma; decidability of elementary problems; characterization in spirit of regular sets

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Further questions: complexity issues for automata questions in BSS model, f.e., not many intrinsically different NPR-complete problems known is there a Myhill-Nerode like characterization is periodic model also interesting for countable structures? They were exclusively studied in original work by GKL, again with many undecidability results model checking for periodic GKL automata; real periodic B¨ uchi automata; more precisely:

Klaus Meer Generalized Automata over the Reals

Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions

Model Checking and real periodic B¨ uchi automata Definition of real periodic GKL automata on infinite strings of reals Basic results concerning closure properties, characterization of acceptable languages Interesting: Is there a connection to metafinite model theory from Gr¨ adel & Gurevich and its relations to BSS complexity theory Gr¨ adel & M. Especially: What kind of metafinite logic describes computation of such real periodic B¨ uchi automata?

Klaus Meer Generalized Automata over the Reals