Uncomputability and Undecidability in Economic Theory

Economic theory, game theory and mathematical statistics have all increasingly become algorithmic sciences. Computable Economics, Algorithmic Game Theory ([28]) and Algorithmic Statistics ([13]) are frontier research subjects. All of them, each in its own way, are underpinned by (classical) recursion theory - and its applied branches, say computational complexity theory or algorithmic information theory - and, occasionally, proof theory. These research paradigms have posed new mathematical and metamathematical questions and, inadvertently, undermined the traditional mathematical foundations of economic theory. A concise, but partial, pathway into these new frontiers is the subject matter of this paper. Interpreting the core of mathematical economic theory to be defined by General Equilibrium Theory and Game Theory, a general - but concise - analysis of the computable and decidable content of the implications of these two areas are discussed. Issues at the frontiers of macroeconomics, now dominated by Recursive Macroeconomic Theory, are also tackled, albeit ultra briefly. The point of view adopted is that of classical recursion theory and varieties of constructive mathematics.General Equilibrium Theory, Game Theory, Recursive Macro-economics, (Un)computability, (Un)decidability, Constructivity

Computation in Economics

This is an attempt at a succinct survey, from methodological and epistemological perspectives, of the burgeoning, apparently unstructured, field of what is often – misleadingly – referred to as computational economics. We identify and characterise four frontier research fields, encompassing both micro and macro aspects of economic theory, where machine computation play crucial roles in formal modelling exercises: algorithmic behavioural economics, computable general equilibrium theory, agent based computational economics and computable economics. In some senses these four research frontiers raise, without resolving, many interesting methodological and epistemological issues in economic theorising in (alternative) mathematical modesClassical Behavioural Economics, Computable General Equilibrium theory, Agent Based Economics, Computable Economics, Computability, Constructivity, Numerical Analysis

Towards a Reformulation Based Approach for Efficient Numeric Planning: Numeric Outer Entanglements

Restricting the search space has shown to be an effective approach for improving the performance of automated planning systems. A planner-independent technique for pruning the search space is domain and problem reformulation. Recently, Outer Entanglements, which are relations between planning operators and initial or goal predicates, have been introduced as a reformulation technique for eliminating potential undesirable instances of planning operators, and thus restricting the search space. Reformulation techniques, however, have been mainly applied in classical planning, although many real-world planning applications require to deal with numerical information. In this paper, we investigate the usefulness of reformulation approaches in planning with numerical fluents. In particular, we propose and extension of the notion of outer entanglements for handling numeric fluents. An empirical evaluation, which involves 150 instances from 5 domains, shows promising results

Reasoning About Integrity Constraints for Tree-Structured Data

We study a class of integrity constraints for tree-structured data modelled as data trees, whose nodes have a label from a finite alphabet and store a data value from an infinite data domain. The constraints require each tuple of nodes selected by a conjunctive query (using navigational axes and labels) to satisfy a positive combination of equalities and a positive combination of inequalities over the stored data values. Such constraints are instances of the general framework of XML-to-relational constraints proposed recently by Niewerth and Schwentick. They cover some common classes of constraints, including W3C XML Schema key and unique constraints, as well as domain restrictions and denial constraints, but cannot express inclusion constraints, such as reference keys. Our main result is that consistency of such integrity constraints with respect to a given schema (modelled as a tree automaton) is decidable. An easy extension gives decidability for the entailment problem. Equivalently, we show that validity and containment of unions of conjunctive queries using navigational axes, labels, data equalities and inequalities is decidable, as long as none of the conjunctive queries uses both equalities and inequalities; without this restriction, both problems are known to be undecidable. In the context of XML data exchange, our result can be used to establish decidability for a consistency problem for XML schema mappings. All the decision procedures are doubly exponential, with matching lower bounds. The complexity may be lowered to singly exponential, when conjunctive queries are replaced by tree patterns, and the number of data comparisons is bounded

Action, Time and Space in Description Logics

Description Logics (DLs) are a family of logic-based knowledge representation (KR) formalisms designed to represent and reason about static conceptual knowledge in a semantically well-understood way. On the other hand, standard action formalisms are KR formalisms based on classical logic designed to model and reason about dynamic systems. The largest part of the present work is dedicated to integrating DLs with action formalisms, with the main goal of obtaining decidable action formalisms with an expressiveness significantly beyond propositional. To this end, we offer DL-tailored solutions to the frame and ramification problem. One of the main technical results is that standard reasoning problems about actions (executability and projection), as well as the plan existence problem are decidable if one restricts the logic for describing action pre- and post-conditions and the state of the world to decidable Description Logics. A smaller part of the work is related to decidable extensions of Description Logics with concrete datatypes, most importantly with those allowing to refer to the notions of space and time

IST Austria Technical Report

We consider partially observable Markov decision processes (POMDPs) with ω-regular conditions specified as parity objectives. The class of ω-regular languages extends regular languages to infinite strings and provides a robust specification language to express all properties used in verification, and parity objectives are canonical forms to express ω-regular conditions. The qualitative analysis problem given a POMDP and a parity objective asks whether there is a strategy to ensure that the objective is satis- fied with probability 1 (resp. positive probability). While the qualitative analysis problems are known to be undecidable even for very special cases of parity objectives, we establish decidability (with optimal complexity) of the qualitative analysis problems for POMDPs with all parity objectives under finite- memory strategies. We establish asymptotically optimal (exponential) memory bounds and EXPTIME- completeness of the qualitative analysis problems under finite-memory strategies for POMDPs with parity objectives

Modelling Mixed Discrete-Continuous Domains for Planning

In this paper we present pddl+, a planning domain description language for modelling mixed discrete-continuous planning domains. We describe the syntax and modelling style of pddl+, showing that the language makes convenient the modelling of complex time-dependent effects. We provide a formal semantics for pddl+ by mapping planning instances into constructs of hybrid automata. Using the syntax of HAs as our semantic model we construct a semantic mapping to labelled transition systems to complete the formal interpretation of pddl+ planning instances. An advantage of building a mapping from pddl+ to HA theory is that it forms a bridge between the Planning and Real Time Systems research communities. One consequence is that we can expect to make use of some of the theoretical properties of HAs. For example, for a restricted class of HAs the Reachability problem (which is equivalent to Plan Existence) is decidable. pddl+ provides an alternative to the continuous durative action model of pddl2.1, adding a more flexible and robust model of time-dependent behaviour