Some Programming Languages for Logspace and Ptime

Abstract

We propose two characterizations of complexity classes by means of programming languages. The first concerns Logspace while the second leads to Ptime. This latter characterization shows that adding a choice command to a Ptime language (the language WHILE of Jones [1]) may not necessarily provide NPtime computations. The result is close to Cook in [2] who used "auxiliary push-down automata". Logspace is obtained through a decidable mechanism of tiering. It is based on an analysis of deforestation due to Wadler in [3]. We get also a characterization of NLogspace. Definition 2. Given a program p, its execution induces a partial function p : T (Cns) n → T (Cns) which maps d 1 , d 2 ,. .. , d n to σ(Y) if the program terminates and where σ is the last store of the computation, otherwise it is undefined. Definition 3. A program is called cons-free, if it does not use an expression of the form c(E 1 ,. .. , E k). We note WHILE cons-free the set of cons-free programs. Theorem 1 (Jones [1]). The set of decision problems computed by cons-free programs is exactely Logspace.

The Power of Non-determinism in Higher-Order Implicit Complexity

“…This also holds true for cons-free programs with non-determinism, as shown in different settings by Bonfante [4], by de Carvalho and Simonsen [7], and by Kop and Simonsen [14], all resulting only in characterisations of deterministic classes such as P. With the exception of [14], all of the above attempts at adding non-determinism consider data order at most 0, and one would expect few changes when passing to higher data orders. This turns out to be patently false as simply increasing to data order 1 already results in an explosion of expressive power.…”

“…Cons-free programs, combined with various limitations on recursion, were introduced by Jones [12], building on ground-breaking work by Goerdt [9,8], and have been studied by a number of authors (see, e.g., [3,4,17,16]). The main difference with our work is that we consider full recursion with full nondeterminism, but impose constraints not present in the previous literature.…”

“…For data order 0, Bonfante [4] shows that adding the choice operator to Jones' language does not increase expressivity. We will recover this result for our generalised language in Section 7.…”

Pure Pointer Programs and Tree Isomorphism

“…This also applies to several alternative characterisations of LOGSPACE by programming language methods, such as [1,11]. As far as we know PURPLE is the first programming formalism for a proper, yet nontrivial subset of LOGSPACE which can therefore be used as a vehicle for relativised separation.…”

Functional Programming in Sublinear Space

“…A number of characterisations of this complexity class have been explored in terms of function algebras [14] and linear logics [15] (there is more work for LOGSPACE-predicates, e.g. [10,5], but we focus on functions here). However, these characterisations are still far away from being real programming languages.…”