Beyond - Reduction in Church ' s !

In this paper, we shall write ! using a notation, item notation, which enables one to make more redexes visible, and shall extend -reduction to all visible redexes. We will prove that ! written in item notation and accommodated with extended reduction, satis es all its original properties (such as Church Rosser, Subject Reduction and Strong Normalisation). The notation itself is very simple: if I translates classical terms to our notation, then I(t1t2) (I(t2) )I(t1) and I( v: :t) ( v)I(t). For example, t (( x7:X4 :( x6:X3 : x5:X1!X2 :x5x4)x3)x2)x1, can be written in our item notation as I(t) (x1 )(x2 )(X4 x7)(x3 )(X3 x6)((X1 ! X2) x5)(x4 )x5 where the visible redexes are based on all the matching -couples. So here, the redexes are based on (x2 )(X4 x7), (x3 )(X3 x6) and (x1 )((X1 ! X2) x5). In classical notation however, only the redexes based on ( x7:X4 : )x2 and ( x6:X3 : )x3 are immediately visible. The third redex, ( x5:(X1!X2): )x2, only becomes visible when the rst two redexes have been contracted. We extend -reduction so that we can contract newly visible redexes even before other redexes have been contracted. So in our example above, (x1 )((X1 ! X2) x5) can be contracted before (x2 )(X4 x7) or (x3 )(X3 x6). This renement (which cannot be done in classical notation) is achieved by generalising the axiom from (t1 )( v)t2 ! t2[v := t1] to (t1 )s( v)t2 ; s(t2[v := t1]) for s consisting of matching -couples only. Hence, as (x2 )(X4 x7)(x3 )(X3 x6) consists of matching couples, we get that I(t); (x2 )(X4 x7)(x3 )(X3 x6)(((x4 )x5)[x5 := x1]). Furthermore, with our item notation, it is possible to re ne reduction by rewriting (or reshu ing) terms so that matching -couples occur adjacent to each other. For example, we can rewrite I(t) above as (x2 )(X4 x7)(x3 )(X3 x6)(x1 )((X1 ! X2) x5)(x4 )x5. We shall formalise term reshu ing and shall show that it is correct and preserves both -reduction and typing.