Binary Decision Diagrams (BDDs) are a data structure for representing and manipulating Boolean functions symbolically. They are particularly effective as the algorithmic basis for symbolic model checking. An Ordered BDD (OBDD) represents a Boolean function as a directed acyclic graph with a common variable ordering, ensuring a unique, reduced representation. OBDDs allow efficient implementation of Boolean operations and are used in symbolic model checking to encode sets and relations as Boolean functions, enabling symbolic operations without explicit state enumeration. OBDDs support variable quantification, which is crucial for model checking. While checking Boolean satisfiability is NP-hard, checking quantified Boolean formulas is PSPACE-complete. OBDDs can efficiently handle variable quantification, providing an advantage over Boolean satisfiability solvers. The relational product operation, used in symbolic model checking, projects system states forward or backward in time. The Boolean function API includes operations for creating, manipulating, and examining Boolean functions. These operations include algebraic operations like AND, OR, and NOT, as well as non-algebraic operations like restriction, composition, quantification, and relational product. The API is implemented using OBDDs, with a shared representation reducing space and enabling strong canonical forms. OBDD implementations use depth-first traversal and memoization to generate reduced graphs. The Apply algorithm recursively computes binary Boolean operations, using a cache to store results and a unique table to avoid duplicate vertices. The NOT operation inverts leaf values, while restriction eliminates variables. Quantification and relational product operations are implemented with specialized routines. Shared OBDD representations allow multiple functions to be represented in a single graph, reducing space and enabling efficient equality checks. Complement edges are used to represent negated functions, enabling efficient NOT operations. The BRB package uses complement edges and ITE (If-Then-Else) operations to implement Boolean functions, improving performance through cache hits. Variable ordering significantly affects OBDD size. Finding an optimal ordering is NP-hard, but heuristics and dynamic reordering (sifting) can improve performance. Zero-suppressed BDDs (ZDDs) are effective for sparse sets, reducing the number of vertices by eliminating redundant paths. Partitioned OBDDs divide variable assignments into subsets, allowing efficient representation of functions over different partitions. Non-Boolean functions are represented using decision diagrams like MTBDDs (multi-terminal BDDs) and DDDs (difference decision diagrams). MTBDDs handle functions with non-Boolean ranges, while DDDs represent constraints on real-valued variables, enabling analysis of timed automata. These representations allow efficient manipulation of complex functions and are crucial for model checking and other applications.Binary Decision Diagrams (BDDs) are a data structure for representing and manipulating Boolean functions symbolically. They are particularly effective as the algorithmic basis for symbolic model checking. An Ordered BDD (OBDD) represents a Boolean function as a directed acyclic graph with a common variable ordering, ensuring a unique, reduced representation. OBDDs allow efficient implementation of Boolean operations and are used in symbolic model checking to encode sets and relations as Boolean functions, enabling symbolic operations without explicit state enumeration. OBDDs support variable quantification, which is crucial for model checking. While checking Boolean satisfiability is NP-hard, checking quantified Boolean formulas is PSPACE-complete. OBDDs can efficiently handle variable quantification, providing an advantage over Boolean satisfiability solvers. The relational product operation, used in symbolic model checking, projects system states forward or backward in time. The Boolean function API includes operations for creating, manipulating, and examining Boolean functions. These operations include algebraic operations like AND, OR, and NOT, as well as non-algebraic operations like restriction, composition, quantification, and relational product. The API is implemented using OBDDs, with a shared representation reducing space and enabling strong canonical forms. OBDD implementations use depth-first traversal and memoization to generate reduced graphs. The Apply algorithm recursively computes binary Boolean operations, using a cache to store results and a unique table to avoid duplicate vertices. The NOT operation inverts leaf values, while restriction eliminates variables. Quantification and relational product operations are implemented with specialized routines. Shared OBDD representations allow multiple functions to be represented in a single graph, reducing space and enabling efficient equality checks. Complement edges are used to represent negated functions, enabling efficient NOT operations. The BRB package uses complement edges and ITE (If-Then-Else) operations to implement Boolean functions, improving performance through cache hits. Variable ordering significantly affects OBDD size. Finding an optimal ordering is NP-hard, but heuristics and dynamic reordering (sifting) can improve performance. Zero-suppressed BDDs (ZDDs) are effective for sparse sets, reducing the number of vertices by eliminating redundant paths. Partitioned OBDDs divide variable assignments into subsets, allowing efficient representation of functions over different partitions. Non-Boolean functions are represented using decision diagrams like MTBDDs (multi-terminal BDDs) and DDDs (difference decision diagrams). MTBDDs handle functions with non-Boolean ranges, while DDDs represent constraints on real-valued variables, enabling analysis of timed automata. These representations allow efficient manipulation of complex functions and are crucial for model checking and other applications.