[Paper Review] Decision Problems for Additive Regular Functions

Rajeev Alur, Mukund Raghothaman|arXiv (Cornell University)|Apr 25, 2013

semigroups and automata theory2 references4 citations

TL;DR

This paper solves two fundamental decision problems for Additive Cost Register Automata (ACRA): it characterizes register complexity via a novel notion of register separability and proves it is PSPACE-complete to compute; it further shows that reachability in alternating ACRA games is EXPTIME-complete for non-negative integer costs but undecidable for integer costs, establishing tight complexity bounds for both problems.

ABSTRACT

Additive Cost Register Automata (ACRA) map strings to integers using a finite set of registers that are updated using assignments of the form "x := y + c" at every step. The corresponding class of additive regular functions has multiple equivalent characterizations, appealing closure properties, and a decidable equivalence problem. In this paper, we solve two decision problems for this model. First, we define the register complexity of an additive regular function to be the minimum number of registers that an ACRA needs to compute it. We characterize the register complexity by a necessary and sufficient condition regarding the largest subset of registers whose values can be made far apart from one another. We then use this condition to design a PSPACE algorithm to compute the register complexity of a given ACRA, and establish a matching lower bound. Our results also lead to a machine-independent characterization of the register complexity of additive regular functions. Second, we consider two-player games over ACRAs, where the objective of one of the players is to reach a target set while minimizing the cost. We show the corresponding decision problem to be EXPTIME-complete when costs are non-negative integers, but undecidable when costs are integers.

Motivation & Objective

• To determine the minimum number of registers required to implement a given additive regular function, analogous to state minimization in DFAs.
• To characterize the register complexity of ACRA in a machine-independent way using a new structural property called register separability.
• To analyze the complexity of reachability problems in two-player games over ACRA, where costs are minimized subject to reaching a target.
• To establish tight complexity bounds for the reachability problem in alternating ACRA games under different cost models (non-negative vs. integer costs).

Proposed method

• Introduces the concept of k-separability of registers, where values in k registers can be made arbitrarily far apart in value through string inputs.
• Proves that an ACRA requires at least k registers to compute a function if and only if its registers are k-separable.
• Develops a PSPACE algorithm to decide k-separability by analyzing value differences across string prefixes and suffixes.
• Reduces the register complexity problem to a quantifier-free integer linear programming problem over a bounded domain.
• Models alternating ACRA games as two-player reachability games with cost minimization, where player choices affect register updates.
• Reduces the halting problem for two-counter machines to the reachability problem in ACRA games to prove undecidability over integers.

Experimental results

Research questions

• RQ1 What is the minimum number of registers needed by any ACRA to compute a given additive regular function?
• RQ2 Can register complexity be characterized in a machine-independent way using structural properties of the function?
• RQ3 What is the computational complexity of determining register complexity for a given ACRA?
• RQ4 Is the reachability problem in alternating ACRA games decidable when costs are integers?
• RQ5 How does the complexity of the reachability problem change when costs are restricted to non-negative integers?

Key findings

• Register complexity of an ACRA is characterized by k-separability: a function requires at least k registers if and only if its registers are k-separable.
• Computing the register complexity of a given ACRA is PSPACE-complete, with a matching lower bound established via reduction from quantified Boolean formula evaluation.
• A machine-independent characterization of register complexity is provided using a pumping-style lemma involving strings, cycles, and coefficient vectors.
• The reachability problem in alternating ACRA games is EXPTIME-complete when costs are non-negative integers.
• The reachability problem becomes undecidable when costs are integers, proven via reduction from the halting problem of two-counter machines.
• The undecidability result holds even for budget-0 strategies, showing that no finite strategy tree can encode a winning path in the general integer-cost case.