[Paper Review] Quantifier Elimination for Reasoning in Economics

Casey B. Mulligan, Russell Bradford+3 more|arXiv (Cornell University)|Apr 26, 2018

Constraint Satisfaction and Optimization22 references3 citations

TL;DR

This paper demonstrates that Quantifier Elimination (QE) techniques—previously underused in economics—can automate reasoning in economic theory by transforming complex logical statements with quantifiers into equivalent quantifier-free formulas. The authors show that QE can prove or disprove economic theorems, identify inconsistent assumptions, and generate weakened or strengthened hypotheses, with applications in both research and education, particularly through a novel incremental QE approach using memorized generic results.

ABSTRACT

We consider the use of Quantifier Elimination (QE) technology for automated reasoning in economics. QE dates back to Tarski's work in the 1940s with software to perform it dating to the 1970s. There is a great body of work considering its application in science and engineering but we show here how it can also find application in the social sciences. We explain how many suggested theorems in economics could either be proven, or even have their hypotheses shown to be inconsistent, automatically; and describe the application of this in both economics education and research. We describe a bank of QE examples gathered from economics literature and note the structure of these are, on average, quite different to those occurring in the computer algebra literature. This leads us to suggest a new incremental QE approach based on result memorization of commonly occurring generic QE results.

Motivation & Objective

• To demonstrate that Quantifier Elimination (QE) technology can be effectively applied to economic reasoning, extending its use beyond science and engineering into the social sciences.
• To show that QE can automatically verify or refute economic theorems, identify contradictory assumptions, and generate reformulated hypotheses by weakening or strengthening conditions.
• To analyze the structural differences between economic QE problems and those in traditional computer algebra literature, revealing distinct patterns requiring new algorithmic approaches.
• To propose a novel incremental QE method based on result memorization of commonly occurring generic QE results to improve efficiency for economic applications.
• To establish a foundation for integrating QE tools into economics education and research, reducing computational burden and minimizing error.

Proposed method

• The paper formulates economic reasoning problems as logical implications of the form ∀v(A ⇒ H), where A represents assumptions and H represents hypotheses, and applies QE to eliminate quantifiers.
• It uses existing QE software (e.g., Mathematica, Redlog, Qepcad-B) to solve concrete economic examples, including comparative statics and inequality systems.
• The authors introduce an incremental QE approach that reuses precomputed generic QE results (e.g., for systems of linear inequalities) to avoid redundant computation.
• They derive a method to generate building blocks for QE by simplifying and reusing solutions to common subproblems, such as systems of two or three linear inequalities.
• The method involves relabeling and combining known QE results (e.g., from 2-inequality cases) to solve larger systems without full re-computation.
• A key component is the use of disjunctive normal form (DNF) conversion and simplification to manage the exponential growth in formula size during QE output processing.

Experimental results

Research questions

• RQ1 Can Quantifier Elimination be effectively applied to economic reasoning problems, such as comparative statics and hypothesis validation?
• RQ2 How do the structural characteristics of economic QE problems differ from those in traditional computer algebra applications?
• RQ3 Can incremental QE techniques based on memorized generic results improve efficiency for economic problem solving?
• RQ4 What is the computational feasibility of using QE to automatically prove, disprove, or reformulate economic theorems?
• RQ5 How can QE tools be integrated into economics education and research to reduce manual computation and error?

Key findings

• QE can automatically prove or disprove economic theorems, identify inconsistent assumptions, and generate reformulated hypotheses by weakening or strengthening conditions.
• Economic QE problems typically involve more variables and simpler algebraic structures than those in traditional QE literature, necessitating new algorithmic approaches.
• A novel incremental QE method using memorized generic results significantly reduces redundant computation, especially for systems of linear inequalities.
• For systems with 2, 3, 4, 5, and 6 linear inequalities, the number of DNF clauses after simplification was 27, 81, 243, and 729 respectively, showing exponential growth but manageable with simplification.
• The method of relabeling and combining known QE results (e.g., from 2-inequality cases) allows for faster derivation of solutions to larger systems than direct QE.
• The authors conclude that QE has strong potential for use in economics education and research, with benefits including reduced error and faster hypothesis testing.