Wide machines with a regular input channel
Lax822549.WideRegChannel · concepts/Lax822549/WideRegChannel.lean · lax-822549
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Definition
The wide machine with a regular channel reads its input from a channel laid out along the addresses: a cell reads an element of the instance when it lies on the segment of the cells carrying an input. Wide acceptance with a regular channel holds of a well-formed instance whose machine, so reading its input, accepts within its steps.
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| 1 | import Mathlib.Algebra.BigOperators.Finprod |
| 2 | import Mathlib.Data.Set.Finite.Lemmas |
| 3 | import Mathlib.Data.Fintype.EquivFin |
| 4 | import Mathlib.Data.Set.Card |
| 5 | import Mathlib.SetTheory.Cardinal.Finite |
| 6 | import Mathlib.Logic.Equiv.Prod |
| 7 | import Mathlib.Order.PiLex |
| 8 | import Mathlib.Data.Prod.Lex |
| 9 | import Mathlib.ModelTheory.Order |
| 10 | import Mathlib.ModelTheory.Semantics |
| 11 | import Mathlib.ModelTheory.Complexity |
| 12 | import Mathlib.Tactic.FinCases |
| 13 | import Mathlib.Logic.Equiv.Fin.Basic |
| 14 | import Mathlib.Data.Finite.Sigma |
| 15 | import Mathlib.Data.Fintype.Lattice |
| 16 | import Mathlib.Order.Lattice.Nat |
| 17 | import Mathlib.Data.Fintype.Pigeonhole |
| 18 | import Mathlib.Dynamics.FixedPoints.Basic |
| 19 | import Lax822549.WideMachines |
| 20 | import Lax904597.Machines |
| 21 | import Lax904597.Problems |
| 22 | import Lax485149.Problems |
| 23 | |
| 24 | /-! |
| 25 | --- |
| 26 | title: Wide machines with a regular input channel |
| 27 | type: definition |
| 28 | --- |
| 29 | The wide machine with a regular channel reads its input from a channel laid |
| 30 | out along the addresses: a cell reads an element of the instance when it |
| 31 | lies on the segment of the cells carrying an input. Wide acceptance with a |
| 32 | regular channel holds of a well-formed instance whose machine, so reading |
| 33 | its input, accepts within its steps. |
| 34 | -/ |
| 35 | |
| 36 | namespace Lax822549.WideRegChannel |
| 37 | |
| 38 | open Lax822549.WideMachines Lax904597.Machines |
| 39 | |
| 40 | open FirstOrder |
| 41 | |
| 42 | open Language Structure |
| 43 | |
| 44 | section Machine |
| 45 | |
| 46 | variable {A : Type} [wide.Structure A] |
| 47 | |
| 48 | /-- The initial tape of the register channel: the cell of `x` holds the input |
| 49 | symbol of `x`. -/ |
| 50 | def wpInpReg : WPoint A → WPoint A → Prop |
| 51 | | Sum.inl s, Sum.inr y => ∃ x, WMRegSeg s x ∧ WMInp x y |
| 52 | | _, _ => False |
| 53 | |
| 54 | variable (A) in |
| 55 | /-- **The wide machine an instance describes, at the register channel**: the |
| 56 | machine of `wideData` with its input written on the file |
| 57 | of the elements that carry input, instead of on the ruler of all the segments. |
| 58 | Every other field is the same one. -/ |
| 59 | def wideRegData : TMData (WPoint A) := |
| 60 | { wideData A with Inp := wpInpReg } |
| 61 | |
| 62 | end Machine |
| 63 | |
| 64 | open Lax904597.Problems Lax485149.Problems |
| 65 | |
| 66 | /-- **Wide acceptance with a regular channel.** -/ |
| 67 | def WideRegAccept : DecisionProblem wide := |
| 68 | DecisionProblem.ofPred fun A _ => TMData.WellFormed (wideRegData A) ∧ TMData.Accepts (wideRegData A) |
| 69 | |
| 70 | end Lax822549.WideRegChannel |
| 71 |
From Mathlib
Mathlib.Algebra.BigOperators.FinprodMathlib.Data.Finite.SigmaMathlib.Data.Fintype.EquivFinMathlib.Data.Fintype.LatticeMathlib.Data.Fintype.PigeonholeMathlib.Data.Prod.LexMathlib.Data.Set.CardMathlib.Data.Set.Finite.LemmasMathlib.Dynamics.FixedPoints.BasicMathlib.Logic.Equiv.Fin.BasicMathlib.Logic.Equiv.ProdMathlib.ModelTheory.ComplexityMathlib.ModelTheory.OrderMathlib.ModelTheory.SemanticsMathlib.Order.Lattice.NatMathlib.Order.PiLexMathlib.SetTheory.Cardinal.FiniteMathlib.Tactic.FinCases
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