Bounded Combinatory Logic

CLS-SMT: Bringing Together Combinatory Logic Synthesis and Satisfiability Modulo Theories

“…There are four rules that control the type inhabitation process. According to these rules, types are assigned to combinatory terms [14]. The first rule (var) allows the usage of any combinator c from the typed repository Γ that has type τ using substitutions.…”

“…The inhabitation problem in general is undecidable. A restriction on variable substitution is needed to ensure decidability [14].…”

“…Similar to dependent types [12], specifications can include arbitrary constants and types can encode precomputed function tables. This specification mechanism is Turing complete in general [14], but in practice we use some restrictions, rendering the existence of terms for the type inhabitation problem decidable. In the current version, (CL)S accepts specifications in almost mathematical notation, allowing to state the example for Γ above as: v a l Gamma = Map( " s t a r t " −> ' Pos ( ' 3 , ' 4 ) , "up" −> ( ' Pos ( ' 3 , ' 4 ) =>: ' Pos ( ' 3 , ' 3 ) ) : & : ( ' Pos ( ' 3 , ' 3 ) =>: ' Pos ( ' 3 , ' 2 ) ) )…”

User Support for the Combinator Logic Synthesizer Framework

“…It could especially help with Petri net and hypergraph based synthesis [21]. To our knowledge, there is no other tool to debug intersection type specifications [14], which are an important area of research [36,27] with multiple recent applications to synthesis [18,12,22] beyond Combinatory Logic [15].…”

“…The Combinatory Logic Synthesizer (CL)S Framework provides a publicly available [8] development tool, which is fully integrated into the Scala programming language and can automatically compose software based on types. Type specifications for (CL)S are based on Combinatory Logic with intersection types [15] and automatic software composition is performed by answering the type inhabitation question: Γ ? : τ .…”