Computation theory with atoms I. Sets with atoms II. Computation models with atoms S ł awomir Lasota University of Warsaw FoPSS School 2019: Nominal Techniques � 1
II. Computation models with atoms • automata with atoms • Turing machines with atoms • other models of computation � 2
computation theory with atoms orbit-finite automata [Boja ń czyk, Klin, L. 2011, 2014] orbit-finite pushdown automata [Clemente, L. 2015, 2019] orbit-finite Turing machines [Boja ń czyk, Klin, L., Toru ń czyk 2013] [Klin, L., Ochremiak, Toru ń czyk 2014] tractability in orbit-finite computation [Boja ń czyk, Toru ń czyk 2018] programming languages processing orbit-finite objects [Boja ń czyk, Braud, Klin, L. 2012] [Klin, Szynwelski 2016] [Kopczy ń ski, Toru ń czyk 2016, 2017] orbit-finite homomorphism/isomorphism problem [Klin, Kopczy ń ski, Ochremiak, Toru ń czyk 2015] [Klin, L., Ochremiak, Toru ń czyk 2016] [Keshvardoost, Klin, L., Ochremiak, Toru ń czyk 2019] orbit-finite logics [Boja ń czyk, Place 2012] [Klin, Ł e ł yk 2017] [Klin, Eberhart 2019] � 3
In the sequel, atoms are well-behaved : } • have finite vocabulary • are homogeneous hence oligomorphic and • have bounded substructures FO = quantifier free logic • are effective orbits of atoms(n) = substructures generated by n atoms hence quantifier-free logic decidable there is a function b such that substructures generated by n atoms have size bounded by b (n) finitely generated substructures of atoms are computable although may have arbitrarily high complexity � 4
any well-behaved atoms Automata } Nondeterministic automata: • alphabet A = definable sets • states Q orbit-finite sets instead of finite ones • δ ⊆ Q × A × Q • I, F ⊆ Q Deterministic automata: • δ : Q × A → Q • initial state ∊ Q Unambiguous automata, alternating automata: …. � 5
? equality atoms (N, =) Question: Consider an equivariant language accepted by a nondeterministic orbit-finite automaton. Is this language accepted by an equivariant one? What about deterministic automata? Question: Consider an S-supported language accepted by a nondeterministic orbit-finite automaton. Is this language accepted by an S-supported one? What about deterministic automata? � 6
any well-behaved atoms ? • alphabet A • states Q • δ ⊆ Q × ( A ∪ { ε } ) × Q • I, F ⊆ Q Question: do ε -transition increase the power of nondeterministic automata? � 7
any well-behaved atoms input alphabet: atoms "exactly two different atoms appear" language: number of registers may vary from one orbit to another Q = ∪ {reject} atoms ≤ 2 states: transitions: δ : Q × A → Q δ ((), a) = (a) a ∊ atoms δ ((a), b) = (ab) a ≠ b δ ((a), b) = (a) a = b δ ((ab), c) = reject c ≠ a, b () initial state: accepting states: atoms 2 � 8
any well-behaved atoms input alphabet: atoms "exactly two different atoms appear" language: registers are not necessarily ordered Q = ∪ {reject} states: P ≤ 2 (atoms) transitions: δ : Q × A → Q δ ( ∅ , a) = {a} a ∊ atoms δ ({a}, b) = {a, b} a, b ∊ atoms δ ({a, b}, c) = reject c ≠ a, b initial state: ∅ accepting states: P 2 (atoms) � 9
any well-behaved atoms input alphabet: atoms "exactly two different atoms appear" language: 5 {5} 2 5 {2,5} ∅ states have 2 2 → 5 2 → 9 5 four orbits 5 → 7 0 {2} {7,9} 3 2 → 3 2 ... reject {3} ... � 10
any well-behaved atoms input alphabet: atoms language: ’’last letter appears elsewhere and is different than 7” can it be determininized? finitary Q = atoms ∪ {init, accept} states: nondeterminism transitions: δ : Q × A → P fin (Q) δ (init, a) = {init, a} a ∊ atoms, a ≠ 7 δ (a, b) = a a, b ∊ atoms, a ≠ b δ (a, b) = accept a, b ∊ atoms, a = b initial state: init accepting states: accept � 11
any well-behaved atoms input alphabet: atoms language: ’’last letter doesn’t appear elsewhere and is different than 7” Q = atoms ∪ {accept} states: transitions: δ : Q × A → Q δ (a, a) = accept a ∊ atoms δ (a, b) = a a, b ∊ atoms, a ≠ b infinitary nondeterminism initial states: atoms \ {7} accepting states: {accept} � 12
equality atoms (N, =) input alphabet: P 2 (atoms) language: ’’nonempty intersection of all letters, or empty word” can it be determininized? Q = atoms states: transitions: δ : Q × A → Q δ (a, {a,b}) = a a, b ∊ atoms, a ≠ b initial states: atoms accepting states: atoms � 13
equality atoms (N, =) input alphabet: P 2 (atoms) language: ’’nonempty intersection of all letters, or empty word” Q = ∪ {atoms} states: P ≤ 2 (atoms) transitions: δ : Q × A → Q δ (x, y) = x ∩ y initial states: {atoms} all states except ∅ accepting states: � 14
equality atoms (N, =) input alphabet: triples of atoms up to cyclic shift language: sequences like that can be glued into a chain isn’t it states: determininistic? transitions: δ : Q × A → P fin (Q) initial states: {0} accepting states: all states except 0 � 15
total order atoms (Q, <) input alphabet: atoms language: nonempty monotonic words Q = atoms ∪ { - ∞ } states: transitions: δ : Q × A → Q δ ( - ∞ , b) = b b ∊ atoms δ (a, b) = b a, b ∊ atoms, a < b initial state: - ∞ accepting states: atoms � 16
total order atoms (Q, <) input alphabet: atoms language: ’’local minima are monotonic” ? � 17
bit vector atoms (V, +) input alphabet: V language: dependent words = ’’some subsequence of letters sums up to 0’’ can it be Q = atoms ∪ {init} determininized? states: transitions: δ : Q × A → P fin (Q) δ (init, a) = {init, a} a ∊ atoms δ (a, b) = {a, a+b} a, b ∊ atoms initial state: init accepting state: 0 � 18
equality atoms (N, =) Theorem: Every equivariant orbit is isomorphic to atoms(n) modulo G, for some n and G a group of permutations of {1…n}. (Non)deterministic orbit-finite automata slightly generalize register automata: • number of registers (dimension) may vary from one orbit to another • registers are not necessarily ordered • alphabet letters may contain more than one atom not a design decision but a property of orbit-finite sets ordered for total order atoms (Q, <) � 19
equality atoms (N, =) Expressive power nondeterministic nondeterministic = automata with equality atoms register automata with over alphabet atoms × (a finite set) equality tests x = y • likewise for total order atoms (Q, ≤ ) straight set: every orbit isomorphic to atoms(n) for some n straight automata with equality atoms Claim: Every (non)deterministic automaton over a straight alphabet A is equivalent to a straight one � 20
equality atoms (N, =) Straightization (deterministic case) Claim: Every (non)deterministic automaton over a straight alphabet A is equivalent to a straight one straight set: every orbit isomorphic to atoms(n) for some n Think of 1-orbit Q Theorem: Every equivariant orbit is isomorphic to atoms(n)/G, for some n and G a group of permutations of {1…n}. f : atoms(n) ➝ Q support-reflecting f -1 ( δ ) ⊆ atoms (n) × A × atoms (n) • δ ⊆ Q × A × Q ? an orbit of atoms (n) × A atoms (n) • δ : Q × A → Q f f δ Q × A Q � 21
Minimization deterministic deterministic = automata with equality atoms register automata with over alphabet atoms × (a finite set) equality tests x = y do minimize do not minimize � 22
any well-behaved atoms Myhill-Nerode Theorem Theorem: L is recognized by a deterministic automaton iff the set of L-equivalence classes is orbit-finite The equivalence classes are states of the minimal automaton for L Two words are L-equivalent iff they lead the minimal automaton to the same state � 23
Two words are L-equivalent Every equivariant orbit is isomorphic to atoms(n) modulo G, iff for some n and G a group of permutations of {1…n}. they lead the minimal automaton to the same state input alphabet: atoms "exactly two different atoms appear" language: 1 8 18 and 81 are L-equivalent after reading first two different data values, the minimal automaton should not remember their order! this is impossible in register automata! � 24
Two words are L-equivalent Every equivariant orbit is isomorphic to atoms(n) modulo G, iff for some n and G a group of permutations of {1…n}. they lead the minimal automaton to the same state input alphabet: atoms {defdef, defefd, deffde : d, e, f pairwise different} language: 7 579, 795 and 957 are L-equivalent 5 9 after reading first three letters, the minimal automaton should remember their order up to cyclic shift only! again, this is impossible in register automata! � 25
• automata with atoms • Turing machines with atoms • other models of computation � 26
Turing machines } • tape alphabet A • states Q orbit-finite sets instead of finite ones • subset δ ⊆ Q × A × Q × A × { ← , → , ↓ } • subsets I, F ⊆ Q Configurations = A * × Q × A * Deterministic machines: • δ : Q × A → Q × A × { ← , → , ↓ } � 27
Recommend
More recommend
