About
By regarding matching as an assignment of occurrences of strings to each part of an expression, regular expressions are resource (limited occurrences of strings) consumption (exclusive assignment/matching of them).
Kind
- Decidable (or only complete?)
- Orderd/non-commutative
- Linear/non-deterministic (⊇ regular/deterministic)
- Unitless/promonoidal
TODO
- Is there such as selective/projective mul to concatenate only one side of And?
- Representation of the four units, or do we need them? (Decidability is affecting)
- 1 as Seq::empty(), 𝟏 ≡ νX.X
- 𝟎 ≡ μX.X
- ⊤ as Inf(Interval::full())
- ⊥ as a & b when a ≠ b (?A)
- One as Seq(['a']) vs Interval('a', 'a')
- Relatipnship between nominality, input-ness/string, and function arguments/abstraction, the Integral trait and the De Bruijn index
- TextStart and TextEnd as left / right units
- Interpretation of
match, is it a judgementa : Aor a test that one is divisible by another (quotient is equal to zero)a / b, one (string) is contained by another (regex) a → b? Proof search - Two types of negations for each positive connectives: rev and div for Seq, not and sub for Interval
- a.le(b) → a.and(b) = a, a.add(b) = b
- Equational reasoning, bisimulation
- Induction
- True(Seq<I>, Box<Repr<I>>), assertions, dependent types, backreference
- action a.P, a ⊙ P, guard, scalar product
- Normalisation vs equality rules
- characteristic function/morphism
- weighted limit, weighted colimit, end, coend
- parameterise some laws as features
- Spherical conic (tennis ball)
Xor
- lookahead/behind, multiple futures, communication and discard,
ignorecombinator - Split, subspace
Correspondence
| formal language theory /automata theory /set theory | linear logic | repr | type theory /category theory /coalgebra | len | process calculus | probability theory /learning theory | quantum theory |
|---|---|---|---|---|---|---|---|
| a ∈ L (match) | a : A (judgement) | ||||||
| $∅$ | $0$ (additive falsity) | Zero | 𝟎 (empty type) | nil, STOP | |||
| $⊤$ (additive truth) | True | ||||||
| a | a | One(Seq(a)) | len(a) | (sequential composition, prefix) | |||
| $ε$ (empty) ∈ {ε} | $1$ (multiplicative truth) | Seq([]) | * : 𝟏 (unit type) | 0 | SKIP | ||
| . | Interval(MIN, MAX) | 1 | |||||
| ab / $a · b$ (concatenation) | $a ⊗ b$ (multiplicative conjunction/times) | Mul(a, b) | $a ⊗ b$ (tensor product) | len(a) + len(b) | P ||| Q (interleaving) | ||
| a|b (alternation),$a ∪ b$ (union) | $a ⊕ b$ (additive disjuction/plus) | Or(a, b) | $a + b$ (coproduct) | max(len(a), len(b)) | (deterministic choice) | ||
| a* (kleen star),..|aa|a|ε | $μX.𝟏 + (L(a) ⊗ X)$ | ||||||
| TODO | $!a$ (exponential conjunction/of course) | Inf(a) | $νX.𝟏 & a & (X ⊗ X)$ | (replication) | |||
| a*? (non greedy) | |||||||
| TODO | $?a$ (exponential disjunction/why not) | Sup(a) | $µX.⊥ ⊕ a ⊕ (X ⅋ X)$ | ||||
| a? | a + 1 | Or(Zero, a) | |||||
| a{n,m} (repetition) | a | Or(Mul(a, Mul(a, ..)), Or(..)) | |||||
| [a-z] (class) | Interval(a, z) | ||||||
[^a-z] (negation) |
TODO this is complementary op | Neg(a)/-a |
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| a† (reverse) | right law vs left law | a.rev() | len(a) | ||||
| $a / b$ (right quotient) | $a ⊸ b$ | Div(a, b) | len(a) - len(b) | (hiding) | |||
| a \ b (left quotient) | Div(a, b) |
(hiding) | |||||
| a ⅋ b (multiplicative disjunction/par) | Add(a, b) | $a ⊕ b$ (direct sum) | (nondeterministic choice) | ||||
| $a ∩ b$ (intersection) | a & b (additive conjunction/with) | And(a, b) | $a × b$ (product) | (interface parallel) | |||
a(?=b) (positive lookahead) |
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a(?!b) (negative lookahead) |
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(?<=a)b (positive lookbehind) |
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(?<!a)b (negative lookbehind) |
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| $a ⊆ b, a ≤ b$ (containmemt) | $a ≤ b (≃ a = b ⅋ a < b)$ | a.le(b) | |||||
| a⊥ (dual) | a.dual() | ||||||
| a = b (equality) | a = b (identity type) |
About symbols
Symbols are grouped and assigned primarily by additive/multiplicative distinciton. They are corresponding to whether computation exits or rather continues; though concatenation ⊗/Mul/* has conjunctive meaning, computation doesn't complete/exit at not satisfying one criterion for there are still different ways of partition to try (backtracking). Though RegexSet ⅋/Add/+ has disjunctive meaning, computation doesn't complete/exit at satisfying one criterion to return which regex has match. On the other hand, alternation ⊕/Or/| and intersection &/And/& early-break, hence order-dependent. When I add Map variant to Repr to execute arbitrary functions, this order-dependency suddenly becomes to matter for those arbitrary functions can have effects, for example, modification/replacement of input string. (Effects are additive.)
| additive | multiplicative | exponential | |
|---|---|---|---|
| positive/extensional | $⊕$ $0$ Or | $⊗$ $1$ Mul | ! |
| negative/intensional | & ⊤ And | ⅋ ⊥ Add | ? |
Properties/Coherence
- All connectives are associative
- Additive connectives are commutative and idempotent
TODO:
- Seq::empty(), ε - can be empty because negative
- Interval::full() - can't be empty because positive
| name | regular expressions | linear logic | type theory | repr |
|---|---|---|---|---|
| or-associativity | $a | (b | c) = (a | b) | c$ | $a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ c$ | Or(a, Or(b, c)) = Or(Or(a, b), c) | |
| or-idempotence | $a | a = a$ | $a ⊕ a = a$ | Or(a, a) = a | |
| or-unit | $a ⊕ 0 = 0 ⊕ a = a$ | Or(a, Zero) = Or(Zero, a) = a | ||
| mul-unit | $a · ε = ε · a = a$ | $a ⊗ 1 = 1 ⊗ a = a$ | Mul(a, One('')) = Mul(One(''), a) = a | |
| right-distributivity | $a · (b | c) = (a · b) | (a · c)$ | $a ⊗ (b ⊕ c) = (a ⊗ b) ⊕ (a ⊗ c)$ | Mul(a, Or(b, c)) = Or(Mul(a, b), Mul(a, c)) | |
| left-distributivity | $(a | b) · c = (a · c) | (b · c)$ | $(a ⊕ b) ⊗ c = (a ⊗ c) ⊕ (b ⊗ c)$ | Mul(Or(a, b), c) = Or(Mul(a, c), Mul(b, c)) | |
| $ε^† = ε$ | ||||
| (a & b)† = (b†) & (a†) | Rev(Mul(a, b)) = Mul(Rev(b), Rev(a)) | |||
| Mul(One(a), One(b)) = One(ab) | ||||
| right-distributivity | a ⅋ (b & c) = (a ⅋ b) & (a ⅋ c) | Add(a, And(b, c)) = And(Add(a, b), Add(a, c)) | ||
| left-distributivity | $(a & b) ⅋ c = (a ⅋ c) & (b ⅋ c)$ | Add(And(a, b), c) = And(Add(a, c), Add(b, c)) | ||
| and-idempotence | $a & a = a$ | And(a, a) = a | ||
| exponential | $(a & b)* = a* · b*$ | $!(A & B) ≣ !A ⊗ !B$ | ||
| exponential | $(a | b)? = a? ⅋ b?$ | $?(A ⊕ B) ≣ ?A ⅋ ?B$ |
Relationship among additive, multiplicative and exponential
- exp(a + b) = exp(a) * exp(b)
Linearity (which)
- f(a + b) = f(a) + f(b)
- (a → b) + (b → a) (non-constructive)
- functions take only one argument
- presheaves of modules
𝜕, derivation
| math | repr |
|---|---|
| $𝜕(a ⊗ b) ≃ 𝜕(a) ⊗ b + a ⊗ 𝜕(b)$ | der(Mul(a, b)) = Or(Mul(der(a), b), Mul(a, d(b))) |
- d(Zero) = Zero
- d(Or(a, b)) = Or(d(a), d(b))
- d(Mul(a, b)) = Or(Mul(d(a), b), Mul(a, d(b)) *
- d(Inf(a)) = Mul(d(a), Inf(a))
- a : D(a)
True
- And(True, a) ->
Stream processor
- νX.μY. (A → Y) + B × X + 1
Flags (TODO)
-
i, CaseInsensitive -
m, MultiLine -
s, DotMatchesNewLine -
U, SwapGreed -
u, Unicode -
x, IgnoreWhitespace -
Recouse non-greedy pattern to _
Algorithms (TODO)
- Bit-pararell
- aho-corasick
- Boyer–Moore
- memchr
Semantics (TODO)
- Coherent space
References
- Completeness for Categories of Generalized Automata, Guido Boccali, Andrea Laretto, Fosco Loregian, Stefano Luneia, 2023 [arXiv]
- Beyond Initial Algebras and Final Coalgebras, Ezra Schoen, Jade Master, Clemens Kupke, 2023 [arXiv]
- The Functional Machine Calculus, Willem Heijltjes, 2023 [arXiv]
- The Functional Machine Calculus II: Semantics, Chris Barrett, Willem Heijltjes, Guy McCusker, 2023 [arXiv]
- Coend Calculus, Fosco Loregian, 2020 [arXiv]
- Partial Derivatives for Context-Free Languages: From μ-Regular Expressions to Pushdown Automata, Peter Thiemann, 2017 [arXiv]
- Proof Equivalence in MLL Is PSPACE-Complete, Willem Heijltjes, Robin Houston, 2016 [arXiv]
- Linear Logic Without Units, Robin Houston, 2013 [arXiv]
- Imperative Programs as Proofs via Game Semantics, Martin Churchill, Jim Laird, Guy McCusker, 2013 [arXiv]
- Regular Expression Containment as a Proof Search Problem, Vladimir Komendantsky, 2011 [inria]