Index: doc/working/resolver_design.md
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 ## Conversion Costs ##
-Each possible resolution of an expression has a _cost_ consisting of four 
-integer components: _unsafe_ conversion cost, _polymorphic_ specialization 
-cost, _safe_ conversion cost, and a _count_ of conversions. 
-These components form a lexically-ordered tuple which can be summed 
-element-wise; summation starts at `(0, 0, 0, 0)`.
+Each possible resolution of an expression has a _cost_ tuple consisting of 
+the following components: _unsafe_ conversion cost, _polymorphic_ 
+specialization cost, _safe_ conversion cost, a count of _explicit_ 
+conversions, and _qualifier_ conversion cost. 
+These components are lexically-ordered and can be summed element-wise; 
+summation starts at `(0, 0, 0, 0, 0)`.
 
 ### Lvalue and Qualifier Conversions ###
@@ -1224,4 +1225,193 @@
 programmers.
 
+## Resolver Architecture ##
+
+### Function Application Resolution ###
+Our resolution algorithm for function application expressions is based on 
+Baker's[3] single-pass bottom-up algorithm, with Cormack's[4] single-pass 
+top-down algorithm applied where appropriate as an optimization. 
+Broadly speaking, the cost of this resolution per expression will be 
+proportional to `i^d`, where `i` is the number of interpretations of each 
+program symbol, and `d` is the maximum depth of the expression DAG. 
+Since `d` is determined by the user programmer (in general, bounded by a small 
+constant), opportunities for resolver optimization primarily revolve around 
+minimizing `i`, the number of interpretations of each symbol that are 
+considered.
+
+[3] Baker, Theodore P. A one-pass algorithm for overload resolution in Ada. 
+ACM Transactions on Programming Languages and Systems (1982) 4:4 p.601-614
+
+[4] Cormack, Gordon V. An algorithm for the selection of overloaded functions 
+in Ada. SIGPLAN Notices (1981) 16:2 p.48-52
+
+Unlike Baker, our system allows implicit type conversions for function 
+arguments and return types; the problem then becomes to find the valid 
+interpretation for an expression that has the unique minimal conversion cost, 
+if such exists. 
+Interpretations can be produced both by overloaded names and implicit 
+conversions applied to existing interpretations; we have proposals to reduce 
+the number of interpretations considered from both sources. 
+To simplify the problem for this discussion, we will consider application 
+resolution restricted to a domain of functions applied to variables, possibly 
+in a nested manner (e.g. `f( g( x ), y )`, where `x` and `y` are variables and 
+`f` and `g` are functions), and possibly in a typed context such as a variable 
+initialization (e.g. `int i = f( x );`); the other aspects of Cforall type 
+resolution should be able to be straightforwardly mapped into this model. 
+The types of the symbol tables used for variable and function declarations 
+look somewhat like the following:
+
+	variable_table = name_map( variable_name, variable_map )
+	
+	function_table = name_map( function_name, function_map )
+	
+	variable_map = multi_index( by_type( variable_type ), 
+	                            variable_decl_set )
+
+	function_map = multi_index( by_int( n_params ),
+								by_type( return_type ),
+								function_decl_set )
+
+`variable_name` and `function_name` are likely simple strings, with `name_map` 
+a hash table (or perhaps trie) mapping string keys to values. 
+`variable_decl_set` and `function_decl_set` can be thought of for the moment 
+as simple bags of typed declarations, where the declaration types are linked 
+to the graph of available conversions for that type. 
+In a typed context both the `variable_decl_set` and the `function_decl_set` 
+should be able to be selected upon by type; this is accomplished by the 
+`by_type` index of both `variable_map` and `function_map`. 
+The `by_int` index of `function_map` also provides a way to select functions 
+by their number of parameters; this index may be used to swiftly discard any 
+function declaration which does not have the appropriate number of parameters 
+for the argument interpretations being passed to it; given the likely small 
+number of entries in this map, it is possible that a binary search of a sorted 
+vector or even a linear search of an unsorted vector would be more efficient 
+than the usual hash-based index.
+
+Given these data structures, the general outline of our algorithm follows 
+Baker, with Cormack's algorithm used as a heuristic filter in typed contexts. 
+
+In an untyped context, we use a variant of Baker's bottom-up algorithm. 
+The leaves of the interpretation DAG are drawn from the variable symbol table, 
+with entries in the table each producing zero-cost interpretations, and each 
+implicit conversion available to be applied to the type of an existing entry 
+producing a further interpretation with the same cost as the conversion. 
+As in Baker, if two or more interpretations have the same type, only the 
+minimum cost interpretation with that type is produced; if there is no unique 
+minimum cost interpretation than resolution with that type is ambiguous, and 
+not permitted. 
+It should be relatively simple to produce the list of interpretations sorted 
+by cost by producing the interpretations via a breadth-first search of the 
+conversion graph from the initial interpretations provided in the variable 
+symbol table.
+
+To match a function at one of the internal nodes of the DAG, we first look up 
+the function's name in the function symbol table, the appropriate number of 
+parameters for the arguments that are provided through the `by_int` index of 
+the returned `function_map`, then go through the resulting `function_decl_set` 
+searching for functions where the parameter types can unify with the provided 
+argument lists; any such matching function produces an interpretation with a 
+cost that is the sum of its argument costs. 
+Though this is not included in our simplified model, this unification step may 
+include binding of polymorphic variables, which introduces a cost for the 
+function binding itself which must be added to the argument costs. 
+Also, checking of function assertions would likely be done at this step as 
+well, possibly eliminating some possible matching functions (if no suitable 
+assertions can be satisfied), or adding further conversion costs for the 
+assertion satisfaction. 
+Once the set of valid function interpretations is produced, these may also be 
+expanded by the graph of implicit conversions on their return types, as the 
+variable interpretations were. 
+
+This implicit conversion-based expansion of interpretations should be skipped 
+for the top-level expression if used in an untyped (void) context, e.g. for 
+`f` in `f( g ( x ) );` or `x` in `x;`. 
+On the other hand, if the top-level expression specifies a type, e.g. in 
+`int i = f( x );`, only top level expressions that return that type are 
+relevant to the search, so the candidates for `f` can be filtered first by 
+those that return `int` (or a type convertable to it); this can be 
+accomplished by performing a top-down filter of the interpretations of `f` by 
+the `by_type` index of the `function_map` in a manner similar to Cormack's[4] 
+algorithm. 
+
+In a typed context, such as an initialization expression 
+`T x = f( g( y ), z );`, only interpretations of `f( g( y ), z )` which have 
+type `T` are valid; since there are likely to be valid interpretations of 
+`f( g( y ), z )` which cannot be used to initialize a variable of type `T`, we 
+can use this information to reduce the number of interpretations considered. 
+Drawing from Cormack[4], we first search for interpretations of `f` where the 
+return type is `T`; by breadth-first-search of the conversion graph, it should 
+be straightforward to order the interpretations of `f` by the cost to convert 
+their return type to `T`. 
+We can also filter out interpretations of `f` with less than two parameters, 
+since both `g( y )` and `z` must produce at least one parameter; we may not, 
+however, rule out interpretations of `f` with more than two parameters, as 
+there may be a valid interpretation of `g( y )` as a function returning more 
+than one parameter (if the expression was `f( y, z )` instead, we could use an 
+exact parameter count, assuming that variables of tuple type don't exist). 
+For each compatible interpretation of `f`, we can add the type of the first 
+parameter of that interpretation of `f` to a set `S`, and recursively search 
+for interpretations of `g( y )` that return some type `Si` in `S`, and 
+similarly for interpretations of `z` that match the type of any of the second 
+parameters of some `f`. 
+Naturally, if no matching interpretation of `g( y )` can be found for the 
+first parameter of some `f`, the type of the second parameter of that `f` will 
+not be added to the set of valid types for `z`. 
+Each node in this interpretation DAG is given a cost the same way it would be 
+in the bottom-up approach, with the exception that when going top-down there 
+must be a final bottom-up pass to sum the interpretation costs and sort them 
+as appropriate. 
+
+If a parameter type for some `f` is a polymorphic type variable that is left 
+unbound by the return type (e.g. `forall(otype S) int f(S x, int y)`), the 
+matching arguments should be found using the bottom-up algorithm above for 
+untyped contexts, because the polymorphic type variable does not sufficiently 
+constrain the available interpretations of the argument expression. 
+Similarly, it would likely be an advantage to use top-down resolution for 
+cast expressions (e.g. `(int)x`), even when those cast expressions are 
+subexpressions of an otherwise untyped expression. 
+It may also be fruitful to switch between the bottom-up and top-down 
+algorithms if the number of valid interpretations for a subexpression or valid 
+types for an argument exceeds some heuristic threshold, but finding such 
+a threshold (if any exists) will require experimental data. 
+This hybrid top-down/bottom-up search provides more opportunities for pruning 
+interpretations than either a bottom-up or top-down approach alone, and thus 
+may be more efficient than either. 
+A top-down-only approach, however, devolves to linear search through every 
+possible interpretation in the solution space in an untyped context, and is 
+thus likely to be inferior to a strictly bottom-up approach, though this 
+hypothesis needs to be empirically validated.
+
+Both Baker and Cormack explicitly generate all possible interpretations of a 
+given expression; thinking of the set of interpretations of an expression as a 
+list sorted by cost, this is an eager evaluation of the list. 
+However, since we generally expect that user programmers will not often use 
+high-cost implicit conversions, one potentially effective way to prune the 
+search space would be to first find the minimal-cost interpretations of any 
+given subexpression, then to save the resolution progress of the 
+subexpressions and attempt to resolve the superexpression using only those 
+subexpression interpretations.
+If no valid interpretation of the superexpression can be found, the resolver 
+would then repeatedly find the next-most-minimal cost interpretations of the 
+subexpressions and attempt to produce the minimal cost interpretation of the 
+superexpression. 
+This process would proceed until all possible subexpression interpretations 
+have been found and considered. 
+
+A middle ground between the eager and lazy approaches can be taken by 
+considering the lexical order on the cost tuple; essentially, every 
+interpretation in each of the classes below will be strictly cheaper than any 
+interpretation in the class after it, so if a min-cost valid interpretation 
+can be found while only generating interpretations in a given class, that 
+interpretation is guaranteed to be the best possible one:
+
+1. Interpretations without polymorphic functions or implicit conversions
+2. Interpretations without polymorphic functions using only safe conversions
+3. Interpretations using polymorphic functions without unsafe conversions
+4. Interpretations using unsafe conversions
+
+In this lazy-eager approach, all the interpretations in one class would be 
+eagerly generated, while the interpretations in the next class would only be 
+considered if no match was found in the previous class.
+
 ## Appendix A: Partial and Total Orders ##
 The `<=` relation on integers is a commonly known _total order_, and 
