Logic Programming and Logarithmic Space

Abstract

Abstract. We present an algebraic view on logic programming, related to proof theory and more specifically linear logic and geometry of interaction. Within this construction, a characterization of logspace (deterministic and non-deterministic) computation is given via a synctactic restriction, using an encoding of words that derives from proof theory. We show that the acceptance of a word by an observation (the counterpart of a program in the encoding) can be decided within logarithmic space, by reducing this problem to the acyclicity of a graph. We show moreover that observations are as expressive as two-ways multi-heads finite automata, a kind of pointer machines that is a standard model of logarithmic space computation.

Unary Resolution: Characterizing Ptime

“…Adding a "stack plugin" to observations extends modularly previous works [2,3,5,6] and gives the perfect tool to characterize Ptime. This modularity was inspired by the classical addition of a stack to an automaton, allowing to switch from Logspace to Ptime, and providing a decisive proof technique: memoizationor exponentiation by squaring in our context-implemented as saturation.…”

“…Definition 11 (observation semiring). We define the semirings P ⊥ of flows that do not use the symbols in P; and Σ lr the semiring generated by flows of the form c This simple restriction was shown to characterize (non-deterministic) logarithmic space computation [3,, with a natural subclass of balanced wirings corresponding to the deterministic case. The balanced restriction won't be further considered, even if previous results on the nilpotency problem for balanced wirings [3, p. 54], [8, Theorem IV.12] are required to complete the detailed proof of Theorem 5 [4,8].…”

“…We used in our previous work [3,6] the characterization of Logspace and NLogspace by 2-way k-head finite automata without pushdown stacks [43, pp. 223-225].…”

“…Let us now consider the proof of Ptime-completeness for the set of balanced observations with stacks. It relies on an encoding that is similar to the previously developed encoding of 2-way k-head finite automata (without pushdown stack) by flows [3,Sect. 4.1].…”

“…and offers a straightforward connection with complexity of logic programming [22]. Previous works in this direction led to characterizations of logarithmic space predicates Logspace and co-NLogspace [2,3], by considering restrictions on the height of variables.…”