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1 Syntax

lcin supports the syntax of λ-calculus enriched with integers, identifiers and operators.
The supported language is described by the following grammar:

Term → var
     | ( Term Oper Term )
     | ( λ var . Term )
     | num
     | id
Oper → op
     | ε

The symbol λ can be written as '\'.
var and id are arbitary strings of latin characters (uppercase or lowercase), numbers and underscores. However var must start with a lowercase or underscore, while id
must start with an uppercase.
Finally parentheses can be
avoided according to the following rules
• Outmost parentheses can be avoided
• Application is left-associative
• The scope of an abstraction extends as far to the right as possible
• Terms that contain operators are parsed according to the precedence and associativity of these operators (see section 6).




2 Basic function

lcin is an interactive program. When executed it displays the >> prompt and
waits for user input. The most simple usage is to enter a λ-term and press return.
The program performs all β and η reductions and generates the term’s normal
form. The result is printed in a “readable” way, that is only the necessary
parentheses are displayed, church numerals are displayed as integers.
Terms are reduced using the normal order evaluation strategy, that is the
leftmost β or η reduction is performed first. This strategy guarantees that term’s
normal form will be computed in finite time, if it exists. However if a term has no
normal form then execution will not terminate. After the execution the program
displays the number of reductions that were performed and the CPU usage time.




3 Integers

lci supports integers by encoding them as church numerals during parsing.
Integer n will be converted to
c_n ≡ λf.λx.f^n (x)
So, actually, it is just a syntactic sugar. All operations are implemented in
pure λ-calculus and can be used through identifiers and operators. Although the
use of calculus for ordinary operations is sleek, it has has a serious performace
drawback. The complexity of an operation is far from constant, for example the
power-of operator (**) requires an exponential number of reductions.




4 Identifiers

Identifiers are used to represent big λ-terms by defining aliases. For example term
λx.x can be assigned to alias I so that the term I y is equivalent to (λx.x) y.
Aliases must be defined in a file that is read by the program. The file contains 

Command id = Term assings Term to identifier id while command ? Term
causes the evaluation of Term just as if it was entered interactively. File processing
is made using the following command
Consult ’file’
When being started, lcin searches for a file named .lcin_aliases in the src/ place.
This file contains definitions for many basic functions and operators
(integer operations, for expample) and is similar to Haskell’s prelude.hs.
Identifiers are replaced by the corresponding terms during evaluation and
not during parsing. Thus the order of the definitions is not significant. If an
alias is not defined an error message is displayed during evaluation. If no alias
contains itself (directly or indirectly) then aliases are just a syntactic sugar, for
if we replace all of them we get valid λ-terms.