Motivated by the successful application of the theory of regular languages to formal verification of finite-state systems, there is a renewed interest in developing a theory of analyzable functions from strings to numerical values that can provide a foundation for analyzing quantitative properties of finitestate systems. In this paper, we propose a deterministic model for associating costs with strings that is parameterized by operations of interest (such as addition, scaling, and min), a notion of regularity that provides a yardstick to measure expressiveness, and study decision problems and theoretical properties of resulting classes of cost functions. Our definition of regularity relies on the theory of string-to-tree transducers, and allows associating costs with events that are conditional upon regular properties of future events. Our model of cost register automata allows computation of regular functions using multiple “write-only” registers whose values can be combined using the allowed set of operations. We show that classical shortest-path algorithms as well as algorithms designed for computing discounted costs, can be adopted for solving the min-cost problems for the more general classes of functions specified in our model. Cost register automata with min and increment give a deterministic model that is equivalent to weighted automata, an extensively studied nondeterministic model, and this connection results in new insights and new open problems.
A quantitative language @cite_9 @cite_29 @cite_1 over infinite words is a function @math . Such languages are generated by weighted automata, where the value of a word @math is set as the maximal value of all runs over @math . By defining various value functions such as @math , @math , @math , @math , different values can be computed for the run of a weighted automaton on the string @math . Quantitative languages use the fixed syntax of weighted automata, and thereby restricted to having a single weight along each transition in their underlying automata. Moreover, they face similar difficulties in determinization: for interesting models of value functions, the corresponding automata cannot be determinized. An extension of to @math -regular cost functions could prove to be a more expressive and robust model to specify quantitative languages and to analyze their decision problems.
@cite_9 @cite_29 @cite_1
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We develop the theory of regular cost functions over finite trees: aquantitative extension to the notion of regular languages of trees: Cost functions map each input (tree) to a value in @math , and are considered modulo an equivalence relation which forgets about specific values, but preserves boundedness of functions on all subsets of the domain. We introduce nondeterministic and alternating finite tree cost automata for describing cost functions. We show that all these forms of automata are effectively equivalent. We also provide decision procedures for them. Finally, following B "uchi's seminal idea, we use cost automata for providing decision procedures for cost monadic logic, a quantitative extension of monadic second order logic. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . We study efficient translations of Regular Linear Temporal Logic ( ) into automata on infinite words. is a temporal logic that fuses Linear Temporal Logic (LTL) with regular expressions, extending its expressive power to all @math -regular languages. The first contribution of this paper is a novel bottom up translation from into alternating parity automata of linear size that requires only colors @math , @math and @math . Moreover, the resulting automata enjoy the stratified internal structure of hesitant automata. Our translation is defined inductively for every operator, and does not require an upfront transformation of the expression into a normal form. Our construction builds at every step two automata: one equivalent to the formula and another to its complement. Inspired by this construction, our second contribution is to extend with new operators, including universal sequential composition, that enrich the logic with duality laws and negation normal forms. The third contribution is a ranking translation of the resulting alternating automata into non-deterministic automata. To provide this efficient translation we introduce the notion of stratified rankings, and show how the translation is optimal for the LTL fragment of the logic. The 3SUM problem is to decide, given a set of @math real numbers, whether any three sum to zero. It is widely conjectured that a trivial @math -time algorithm is optimal and over the years the consequences of this conjecture have been revealed. This 3SUM conjecture implies @math lower bounds on numerous problems in computational geometry and a variant of the conjecture implies strong lower bounds on triangle enumeration, dynamic graph algorithms, and string matching data structures. In this paper we refute the 3SUM conjecture. We prove that the decision tree complexity of 3SUM is @math and give two subquadratic 3SUM algorithms, a deterministic one running in @math time and a randomized one running in @math time with high probability. Our results lead directly to improved bounds for @math -variate linear degeneracy testing for all odd @math . The problem is to decide, given a linear function @math and a set @math , whether @math . We show the decision tree complexity of this problem is @math . Finally, we give a subcubic algorithm for a generalization of the @math -product over real-valued matrices and apply it to the problem of finding zero-weight triangles in weighted graphs. We give a depth- @math decision tree for this problem, as well as an algorithm running in time @math .