Synthesizing geometry constructions

Abstract

In this paper, we study the problem of automatically solving ruler/compass based geometry construction problems. We first introduce a logic and a programming language for describing such constructions and then phrase the automation problem as a program synthesis problem. We then describe a new program synthesis technique based on three key insights: (i) reduction of symbolic reasoning to concrete reasoning (based on a deep theoretical result that reduces verification to random testing), (ii) extending the instruction set of the programming language with higher level primitives (representing basic constructions found in textbook chapters, inspired by how humans use their experience and knowledge gained from chapters to perform complicated constructions), and (iii) pruning the forward exhaustive search using a goal-directed heuristic (simulating backward reasoning performed by humans). Our tool can successfully synthesize constructions for various geometry problems picked up from high-school textbooks and examination papers in a reasonable amount of time. This opens up an amazing set of possibilities in the context of making classroom teaching interactive.

ArgoTriCS – automated triangle construction solver

“…There are quite few papers about automation of solving construction problems in geometry using geometric approach (Gulwani et al, 2011;Marinković & Janičić, 2012;Schreck, 1994) and few papers on algebraic solving of the same problem (Gao & Chou, 1998b;Guoting, 1992).…”

“…For example, a method developed by Gulwani et al (2011) is derived from the general method for testing and synthesis of pieces of software. Method comes to a construction using probabilistic approach for finding a particular solution which is then used for guiding a search through a huge space of formal function terms.…”

“…There are several approaches for automated solving of construction problems (Gao & Chou, 1998a;Grima & Pace, 2007;Gulwani, Korthikanti, & Tiwari, 2011;Schreck, 1993). However, most, if not all of them, focus on search procedures and do not focus on finding a small portion of geometry knowledge that is underlying the constructions.…”

Code search with input/output queries: Generalizing, ranking, and assessment

“…Programs that under-match a specification provide an opportunity for program synthesis approaches (e.g., Gulwani et al, 2011;Harris and Gulwani, 2011;Solar-Lezama et al, 2006) to bring programs closer to the specification. Additionally, test generation techniques that use the control flow graph could be used to create test case that cover parts of the graph, or cover the graph in particular ways, such as using edge-pair or prime-path coverage (Ammann and Offutt, 2008).…”

“…Also close to our work is program synthesis that uses solvers to derive a function that maps an input to an output (e.g., Gulwani, 2011;Gulwani et al, 2011;Harris and Gulwani, 2011;Jha et al, 2010;Singh and Solar-Lezama, 2011). The domains of applicability include string processing (Gulwani, 2011), spreadsheet table transformations (Harris and Gulwani, 2011), a domain-specific language for geometric constructions , data structure manipulations (Singh and Solar-Lezama, 2011), and loop-free bit manipulation programs (Jha et al, 2010).…”