By D. A. Wolfram

This ebook offers the theoretical beginning of a higher-order common sense programming language with equality, in response to the clausal thought of sorts. A long-sought aim of common sense programming, the clausal idea of varieties is a common sense programming language that enables practical computation as a primitive operation whereas having rigorous, sound, and entire declarative and operational semantics. The language is particularly strong, helping higher-order equational deduction and sensible computation. Its larger order syntax makes it concise and expressive, summary info forms might be expressed in it, and looking for a number of strategies is a uncomplicated operation. the writer proves a few vital and marvelous effects: a Skolem-Herbrand-Gödel theorem for higher-order common sense; a Higher-Order answer Theorem, which include as precise circumstances a few formerly unproven conjectures approximately equational matching and higher-order matching.

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**Extra resources for The Clausal Theory of Types**

**Sample text**

35, for all a the union of all partitions in £a is fixed. {£a}a therefore could only be non-unique if the partitions in a domain £a are not uniquely defined. The partitions to which two closed CTT terms t\ and t2 over S — {II} belong is unique where r(ti) and r(t2) are in To. 35 except the second, they are in the same partition if and only if M \= t\ = t2. 22 of CTT term, it occurs in a prefix and the term is a formula. This formula has the form (Us) where s is a term of type (a —> o). By induction on the number of occurrences of II in a formula, the partition to which the /^-normal form of (st) belongs is unique for every closed CTT term t over S of type a.

The term t/i is a fi-redex of t. r for s. The term t[i <— p{(u, s)}] follows from t by the rule of /3-reduction. 25 The relations p>1/? and t>p are defined as follows: • t $>ips if s follows from t by the rule of /^-reduction. tn if t0 \>xp - • • [>1p tn, where n > 0. 17. 22 Chapter 2. ), and y £ ^(s). The term t/i is a 77-redex of t. The term t[i «— s] follows from t by the rule of 77-reduction. 27 The relations > lr? and [>v are defined as follows: • t t>ivs if s follows from t by the rule of 77-reduction.

Finding a fi-normal form of a term cannot be done in elementary time [183]. A consequence of the next theorem is that normalized terms are unique up to a-conversion. The original version of the theorem is for the untyped A-calculus [29]. The version of this theorem for the /^-calculus is discussed by Huet [98]. -, and Px-calculi have the Church-Rosser property: for all terms t0 G T , if t0 t>ti and t0 t>t2, then there exists a terms £3 and t4 such that £1 >£3, t2 t>t4, and t3 \>at4. 29 also applies to them.