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[[Documentation]] //[[Documentation]] *Syntax **Processes An LMNtal process is a multiset (unordered sequence) of the following: An LMNtal program is written as an LMNtal process, and an LMNtal process is a multiset (bag) of the following: +''atoms'' --- &math(p(X_{1}, \ldots , X_{n})); --- a graph node with a symbolic ''name'' &math(p); and an ordered sequence of ''links'' &math(X_i);, +''cells'' --- { '''Process''' } --- a process recursively enclosed with a ''membrane'' (curly braces,) +''rules'' --- ( '''Head''' :- '''Guard''' | '''Body''' ) --- a rewrite rule for processes, explained below. |BGCOLOR(white):syntactic&br;category|BGCOLOR(white):form|BGCOLOR(white):description| |BGCOLOR(white):''atoms'' | '''p'''('''X'''&size(10){1};, ..., '''X'''&size(10){'''n'''};) | a graph node with a symbolic ''name'' '''p''' and an ordered sequence of ''links'' '''X'''&size(10){'''i'''}; | |BGCOLOR(white):''cells'' | { '''Process''' }&br;'''m'''{ '''Process''' } | a process enclosed with a ''membrane'' (curly braces),&br;optionally with a membrane name '''m''' | |BGCOLOR(white):''rules'' | ( '''Head''' :- '''Body''' ) | a rewrite rule for processes, explained below | Both periods and commas can be used to sequence the elements of a multiset. An LMNtal program is written as an LMNtal process. The elements of a multiset are either separated by commas (e.g., a(X), b(X)) or terminated by periods (e.g., a(X). b(X).) . Links are written using alpha-numeric tokens starting with capital letters (e.g. X, Res.) The other alpha-numeric tokens are treated as names (e.g. foo, 123.) Additionally, quoted symbols can also be used for atom names (e.g. "foo", 'bar'.) Links names are written using alphanumeric tokens starting with capital letters (e.g., X, Res). The other alpha-numeric tokens are treated as atom or membrane names (e.g., foo, 123). Quoted symbols can also be used for atom or membrane names (e.g., "foo", 'bar', [:baz:]). **Term Abbreviation Scheme Since link names stand for endpoints of one-to-one links, each link name in a well-formed process must occur at most twice, and each link name in a well-formed rule must occur exactly twice. All links in LMNtal have at most two occurrences. This enables us to abbreviate ***Term Abbreviation Scheme &math(p(s_1,\ldots,s_m), q(t_1,\ldots,t_n)); to &math(p(s_1,\ldots,s_{k-1},q(t_1,\ldots,t_{n-1}), s_{k+1},\ldots,s_m)); if &math(t_{n}); and &math(s_{k}); are the same '''link'''. Since each link name occurs at most twice, we can abbreviate For example, c(1,c(2,n),L0) is an abbreviated form of c(A,L1,L0),c(B,L2,L1),n(L2),1(A),2(B). >'''p'''('''s'''&size(10){1};, ..., '''s'''&size(10){'''m'''};), '''q'''('''t'''&size(10){1};, ..., '''t'''&size(10){'''n'''};) ***Connector Atoms to Binary atoms with the name = of the form X=Y are called ''connector atoms''. Connector atoms state that the two links in the arguments are identical (in the sense of structural equivalence. Another instance of structural equivalence is the reordering of multiset elements.) For example, ( p(A,X,C),X=B ) is always equivalent to p(A,B,C), as well as to ( p(A,B,X),C=X ), and, finally, to C=p(A,B). >'''p'''('''s'''&size(10){1};, ..., '''s'''&size(10){'''k'''-1};, '''q'''('''t'''&size(10){1};, ..., '''t'''&size(10){'''n'''-1};), '''s'''&size(10){'''k'''+1};, ..., '''s'''&size(10){'''m'''};) // &math(p(s_1,\ldots,s_m), q(t_1,\ldots,t_n)); to // &math(p(s_1,\ldots,s_{k-1},q(t_1,\ldots,t_{n-1}), s_{k+1},\ldots,s_m)); The typical usage of connector atoms can be found in the following example: if '''t'''&size(10){'''n'''}; and '''s'''&size(10){'''k'''}; are the same link name. For example, c(1,c(2,n),L0) is an abbreviation of c(A,L1,L0),c(B,L2,L1),n(L2),1(A),2(B). This can be written also as L0=c(1,c(2,n)) because the semantics of LMNtal regards c(1,c(2,n),L0) and L0=L1, c(1,c(2,n),L1) as identical (see Connectors below) and we can apply term abbreviation to the latter to obtain L0=c(1,c(2,n)) . For an atom name '''p''' and a membrane, '''p'''(..., {...}, ...) stands for a process '''p'''(..., X, ...), {+X, ...} . ***List Notation The Prolog list syntax can be used in LMNtal. List constructor atoms have three arguments and the name '.' . X=[A,B|Rest], Rest=[] and X=[A,B] are abbreviated forms of '.'(A,Tmp,X), '.'(B,Rest,Tmp), '[]'(Rest). //For some reason, the process [A,B|Rest] is parsed as //'.'(A,Tmp), '.'(B,Rest,Tmp). //These two atoms have different arities. ***Connectors Binary atoms of the form X=Y are called ''connector atoms'' or ''connectors''. A connector states that the two link names are considered identical (i.e., interconnected in zero steps). // (in the sense of structural equivalence. Another instance of structural // equivalence is the reordering of multiset elements.) For example, ( p(A,X,C), X=B ) is equivalent to p(A,B,C), as well as to ( p(A,B,X), C=X ) and C=p(A,B). Connectors are typically used in the base case of a recursive definition: ( append([],Y,Res) :- Res=Y ), ( Res=append([A|X],Y) :- Res=[A|append(X,Y)] ) ( append([A|X],Y,Res) :- Res=[A|R], append(X,Y,R)] ) **Rules The basic syntax of a rule is: ( '''Head''' :- '''Body''' ). The basic syntax of a rule is >( '''Head''' :- '''Body''' ). The enclosing parentheses can be omitted if periods are used to delimit the rule. Both of '''Head''' and '''Body''' are ''process templates''. '''Head''' specifies processes to be rewritten and '''Body''' specifies the result of rewriting. Rules work only for the processes residing in the same membrane. The full syntax of a rule that contains '''Guard''' part The full syntax of a rule that contains '''Guards''' will be explained later in [[a separate section>Guards]]. ***Process Templates A process template is a multiset of the following: +''atoms'' --- &math(p(X_{1}, \ldots ,X_{n})); --- same as in a process, +''cells'' --- { '''Template''' } or { '''Template''' }/ --- a process template enclosed with a membrane, +''rules'' --- ( '''Head''' :- '''Guard''' | '''Body''' ) --- allowed only in a Body, +''process contexts'' --- $p or &math(\$p[X_1,\ldots,X_n]); or &math(\$p[X_1,\ldots,X_n|\mathrm{*}X]); --- matches with a multiset of atoms and cells, +''rule contexts'' --- &math(@p); --- matches with a multiset of rules. |BGCOLOR(white):syntactic&br;category|BGCOLOR(white):form|BGCOLOR(white):description| |BGCOLOR(white): ''atoms'' | '''p'''('''X'''&size(10){1};, ..., '''X'''&size(10){'''n'''};) | same as in a process | |BGCOLOR(white): ''cells'' | { '''Template''' }&br;{ '''Template''' }/&br;'''m'''{ '''Template''' }&br;'''m'''{ '''Template''' }/ | a process template enclosed with a membrane | |BGCOLOR(white): ''rules'' | ( '''Head''' :- '''Body''' ) | allowed only in a Body | |BGCOLOR(white): ''process contexts'' | $'''p'''&br;$'''p'''['''X'''&size(10){1};, ..., '''X'''&size(10){'''n'''};]&br;$'''p'''['''X'''&size(10){1};, ..., '''X'''&size(10){'''n'''};|*'''X'''] | matches a multiset of atoms and cells (see below) | |BGCOLOR(white): ''rule contexts'' | @'''p''' | matches a multiset of rules | A membrane template with / (the stable flag) can only match with a ''stable'' cell (i.e., a cell containing no applicable rules). A cell template with a membrane name only matches a cell with the same membrane name. A cell template with '/' (the stable flag) only matches a ''stable'' cell, i.e., a cell containing no applicable rules inside it. The link arguments of a process context specify ***Process Contexts A process context represents a multiset of atoms and cells. The arguments '''X'''&size(10){1};, ..., '''X'''&size(10){'''n'''}; specify the set of free links that must exist in the matched process. The &math(\mathrm{*}X); represents an arbitrary number of extra free links. We will abbreviate &math(p(s_1,\ldots,s_{k-1},X, s_{k+1},\ldots,s_m),\$q[X]); to &math(p(s_1,\ldots,s_{k-1},\$q,s_{k+1},\ldots,s_m));. The optional '''*X''' represents an arbitrary number of extra free links. Note that the current implementation does not fully support process contexts with explicit arguments. The form $'''p''' is an abbreviation of $'''p'''[|*'''X'''], i.e., a multiset of atoms and cells with no constraints on the occurrences of free links. * Any Comments? -the expressions need to be fixed. -- [[nakajima]] &new{2004-02-01 (Sun) 21:43:32}; -fixed the expression using math.inc.php -- [[nakajima]] &new{2004-02-02 (Mon) 21:33:38}; -fixed math.inc.php. no warnings any more. -- [[nakajima]] &new{2004-02-10 (Tue) 12:35:06}; A process context must occur within a membrane in a head. Alternatively, it may either occur - in a head and an input position of a guard, or - in an output position of a guard. #comment Process contexts of the latter kinds are called ''typed process contexts''; see [[Guards]] for details. You can abbreviate >'''p'''('''s'''&size(10){1};, ..., '''s'''&size(10){'''k'''-1};, '''X''', '''s'''&size(10){'''k'''+1};, ..., '''s'''&size(10){'''m'''};), $'''q'''['''X'''] to >'''p'''('''s'''&size(10){1};, ..., '''s'''&size(10){'''k'''-1};, $'''q''', '''s'''&size(10){'''k'''+1};, ..., '''s'''&size(10){'''m'''};). //&math(p(s_1,\ldots,s_{k-1},X, s_{k+1},\ldots,s_m),\$q[X]); to //&math(p(s_1,\ldots,s_{k-1},\$q,s_{k+1},\ldots,s_m));. //Note that the current implementation does not fully // support process contexts with explicit arguments. //* Any Comments? //-the expressions need to be fixed. -- [[nakajima]] &new{2004-02-01 // (Sun) 21:43:32}; //-fixed the expression using math.inc.php -- [[nakajima]] &new{2004-02-02 // (Mon) 21:33:38}; //-fixed math.inc.php. no warnings any more. -- [[nakajima]] &new{2004-02-10 // (Tue) 12:35:06}; //#comment