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LMNtal
― the language and its implementation Kazunori Ueda, Waseda Univ.
(joint work with Norio Kato, Koji Hara and Ken Mizuno)
November 2005 Copyright (C) 2002-2005 Kazunori Ueda
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Diagrammatic representation of computation and its manipulation hub formed by membrane X
i
X0
Y
o
Y m
Y0
s
A
m g
L1 n
n
i L2
n
n n
cyclic structures
Y
receive
Z
unprotected
m
asynchronous π-calculus
A
L0 b
X
X
X
buffer header
S
m
m
operation right
m
protected by membrane Y
n-to-1 comm.
Y
send X
S1
channel formed by membrane
o
closed unary function
i
f m
o f
c
c
map function
m
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Design goals of LMNtal
A “Turing Machine” for universal computing environments (from wide-area to nanoscale) Serve also as a practical programming language Implementation available! Unifying e.g., processes = messages = data Simple and versatile Computation is manipulation of graphs Membranes express multisets and locality Allows programming by self-organization
Models and languages with membranes + hierarchies
Chemical Abstract Machine Mobile ambients P-systems Bigraphical reactive system LMNtal Seal calculus Kell calculus Brane Calculi (Abstract machines of Systems Biology)
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Syntax of
LMNtal processes Not in Flat LMNtal
P ::= 0 (null) | p(X1, . . . , Xm) (m ≥ 0) (atom) | P, P (molecule) | {P} (cell) | T :- T (rule) Link condition: Each link in P occurs at most twice and each link in a rule occurs exactly twice. Free link of P = link occurring only once P is closed = has no free links
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Syntax of LMNtal process templates
T ::= | | | | | | |
0 (m ≥ 0) p(X1 , . . . , Xm) T, T {T} T :- T @p $p[X1 , . . . , Xm|A] (m ≥ 0) p(*X1 , . . . ,*Xm) (m > 0)
(residual args) A ::= [ ] | *X
Not in Flat LMNtal
(null) (atom) (molecule) (cell) (rule) (rule context) (process context) (aggregate) (empty) (bundle)
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Process contexts $p[X1 ,...,Xm|A]
Express local contexts within cells {p(X)} : matches a cell exactly consisting of p(X) {p(X), $q[|*Y]} : matches a cell containing p(X), X free {p(X), $q[X|*Y]} : matches a cell containing p(X), X local Arguments of a process context $p[X1 , ..., Xm|A] X1 ,...,Xm : must occur as free links A is [] : no other free links may occur A is *X : other free links may occur
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Term notation: Examples A
B
. 2
. 3
ans + * 2
5 a
C
. 5
D
. 7
E
[]
'.' (2,B,A), '.' (3,C,B), '.' (5,D,C), '.' (7,E,D), '[]' (E) or A= '.' (2, '.' (3, '.' (5, '.' (7, '[]' )))) A=[2 | [3 | [5 | [7 | []]]] A=[2,3,5,7]
ans(A), '+' (B,C,A), '*' (D,E,B), 2(D), a(E), 5(C) or ans('+' ('*' (2,a),5)) ans(2*a+5)
ans +
5
2 3
ans(A), {+A, 2,3,5} or ans({2,3,5})
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Term notation: The rule c(A1,X1,X0), c(A2,X2,X1), c(A3,X3,X2), n(X3) ≡ c(A1,c(A2,c(A3,n)),X0) ≡ X0 = Y, c(A1,c(A2,c(A3,n)),Y) ≡ X0 = c(A1,c(A2,c(A3,n))) f(3) ≡ 3(f) ≡ f = 3 ≡ 3 = f ≡ f(X), 3(X)
f 3
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Example: Natural numbers Z= add(0,Y) :- Z= Y. Z= add(s(X),Y) :- Z= s(add(X,Y)). add(0,Y,Z) :- Y=Z. add(s(X),Y,Z0) :- Z0=Z, add(X,Y,Z). Syntax note: processes not in parentheses
can either be comma-separated or periodterminated
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Implementation Available ! Translator to Java running on JDK 1.4 or higher http://www.ueda.info.waseda.ac.jp/lmntal/ Features: (still under development) Arithmetics (int, float) Modules Foreign-language interface to Java Visualizer Interactive mode (Read-Eval-Print Loop) Optimizer Redex/Rule selection strategies Library APIs
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Arithmetics numbers are unary atoms 8 and 3.14 are not numbers, 8(X) and 3.14(X) are. a ground expression evaluates itself to a number
ans(2*a+5), (a(X) :- 3(X)) Î ans(11), (a(X) :- 3(X))
numbers can be compared using guards a(8), (a(X) :- X>5 | b(X), c(X), d(X)), (b(X) :- X>7|)
Î c(8), d(8), (a(X) :- X>5 | b(X), c(X), d(X)), (b(X) :- X>7|) X is not just a link here; it’s a typed process context, representing “a link X and an integer atom (> 5) connected to X”.
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Modules and foreign-language interface { module(io).
}
io.input(Message, X) :- [:/*inline*/ String s = javax.swing.JOptionPane.showInputDialog(null, me.nth(0)); me.setName("done"); me.nthAtom(0).setName(s); :](Message, X). ... I/O, arrays, window toolkit, socket, utilities, . . .
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Structural congruence ( ≡ ) (E1) 0, P (E2) P, Q (E3) P, (Q, R) (E4) P (E5) P ≡ P’ (E6) P ≡ P’ (E7) X=X (E8) X=Y (E9) X = Y, P
multisets ≡ P ≡ Q, P ≡ (P, Q), R ≡ P [Y/X ] if X is a local link of P Ö P, Q ≡ P ’, Q Ö {P } ≡ {P’ } structural connectors ≡ 0 ≡ Y=X ≡ P [Y/X ]
if P is an atom and X is a free link of P
(E10) { X = Y, P } ≡ { P }, X = Y
if X is a free link of P and Y is not a free link of P
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Reduction semantics (R1) (R3) (R4)
P → P’ P, Q → P ’, Q
P → P’ (R2) { P } → { P’ }
Q ≡ P P → P’ P’ ≡ Q’ Q → Q’ { X=Y, P } → X=Y, { P } if X and Y are free links of (X=Y, P)
(R5)
X=Y, { P } → { X=Y, P } if X and Y are free links of P
(R6) Tθ, (T :- U ) → Uθ, (T :- U )
θ is to instantiate process & rule contexts. Links are matched using α-conversion.
Example: Factorization
process context; matches non-rule molecules
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rule context
{ gen(12). gen(N) :- N>=2 | n(N), gen(N-1). gen(N) :- N= PR | X0=[OpL,exp(e(Op,E1,E2)),OpR|X], op(OpL1,PL), op(Op1,P), op(OpR1,PR) .
J Parsing I Unparsing
X0=[N|X] :- int(N) | X0=[exp(N)|X]. ans1([begin, 123, '+', 45, '*', 67, '+', 8, end]) Î ans1([begin, exp(e('+',e('+',123,e('*',45,67)),8)), end]) op('*',4). op('*',4). op('*',4). op('+',3). op('+',3). op('+',3). op(begin,0). op(end,0).
operator precedence table
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Example: Fullerene (C60) /* icosahedron */ dome(L0,L1,L2,L3,L4,L5,L6,L7,L8,L9) :p(T0,T1,T2,T3,T4), p(L0,L1,H0,T0,H4), p(L2,L3,H1,T1,H0), p(L4,L5,H2,T2,H1), p(L6,L7,H3,T3,H2), p(L8,L9,H4,T4,H3). dome(E0,E1,E2,E3,E4,E5,E6,E7,E8,E9), dome(E0,E9,E8,E7,E6,E5,E4,E3,E2,E1). /* icosahedron -> fullerene */ p(L0,L1,L2,L3,L4) :X=c(L0,c(L1,c(L2,c(L3,c(L4,X))))).
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Example: Constraint Handling Rules
CHR ≅ LMNtal − hierarchy + propagation
r1 r2 r3 r4
@ @ @ @
leq(X,X) true. leq(X,Y), leq(Y,X) X=Y. leq(X,Y), leq(Y,Z) ==> leq(X,Z). leq(X,Y), leq(X,Y) leq(X,Y).
SOS of CHR has been encoded in LMNtal Work in progress: built-in support of propagation rules (reaction history management) molecules with infinite multiplicity
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Onoing and Future Work Language Propagation rules and infinite multiplicity Constructs for physical aspects Foundations Type systems Theory of computational resources Reversibility Implementation Optimizing complation of sequential core Integration with static analysis Parallel, distributed, embedded implementation Interoperability Î “Theory meets practice, logic meets physics.”
