Mathematical Modeling and Formal Specification Languages CIS 376 Bruce R. Maxim UM-Dearborn

Mathematical Models Abstract representations of a system using mathematical entities and concepts Model should captures the essential characteristics of the system while ignoring irrelevant details If model is to be used for deductive reasoning about the system it needs to provide sufficient analytic power

Mathematical Model Benefits Mathematical model is more concise and more precise than natural language, pseudo-code, or diagrams Mathematical model can be used to calculate and predict system behavior Model can be analyzed using mathematical reasoning to prove system properties or derive new behaviors

Types of Mathematical Models Continuous Models –uses calculus and continuous function theory (e.g. derivatives, integrals, differential equations) –a function f would define the state of the system with an infinite number of states and smooth transitions Discrete Models –based on logic and set theory –state transition functions are used to model transitions among a finite number of states

Discrete Modeling Techniques Informal models –natural language –charts –tables Structural Models (employ formalisms) –data flow diagrams –ER diagrams –object models Formal Models (use formal semantics) –function models –state machine models –formal specification

Object Models Object models represent systems as structured collections of classes and object relations Object models provide a static view of a system Some object models rely on a combination of DFD, ER diagrams, and STD to yield a composite static/dynamic system model Object models are structural in nature and can be useful for creating initial system models that can be mapped to a formal model if needed

Function Models Similar to circuit design work A logic table for a full adder might be captured and represented as Adder(x, y, c input ) = (s, c output ) = ((x + y + c input ) mod 2, (x + y + c input ) div 2) Also similar to the algebraic specification we have seen in data type semantic specification

Function Modeling Example Abstract Stack - 1 Being by defining stack and element to be unspecified types stack: TYPE element: TYPE Use a subtype to define a non-empty stack type nonempty_stack a : TYPE = {s : stack | s /= empty}

Function Modeling Example Abstract Stack - 2 Basic operations are defined as functions push : [element, stack  nonempty_stack] pop: [nonempty_stack  stack] top : [nonempty_stack  element] Behavior is described by axioms Pop_Push : AXIOM pop(push(e, s)) = s Top_Push : AXIOM top(push(e, s)) = e

Function Modeling Example Abstract Stack - 3 Type checking assures that the expression pop(empty) is never allowed The theorem below follows from the type declarations Push_Empty: THEOREM push(e, s) /= empty The use of subtypes and theorems provides a powerful tool for capturing requirements in your type definitions

Formal Specification Language Based on formal mathematical logic with some programming language support (e.g. type system and parameterization) Generally non-executable models Designed to specify what is to be computed and not how the computation should be accomplished Most formal languages are based on axiomatic set theory or higher-order logic

Specification Language Features - part 1 Explicit semantics –language must have a mathematical basis Expressiveness –flexibility –convenience –economy of expression Programming language data types –arrays, structs, strings, sets, lists, etc.

Specification Language Features - part 2 Convenient syntax Diagramatic notation Strong typing –can be richer than ordinary programming languages –provides economy and clarity of expression –type checking provides consistency checks Total functions –most logics assume total functions –subtypes help make total functions more flexible

Specification Language Features - part 3 Axioms and definitions –axioms should be used carefully to avoid introducing inconsistencies –definitional principle ensures well-formed definitions –in some languages type checking assertions will be generated to ensure valid definitions Modularization –ability to break specifications into independent modules –parameterized modules allow for easier reusability

Specification Language Features - part 4 Built-in model of computation –handles simple type checking constraints –enhance proof-checking Maturity –documentation –tool support –theoretical basis –specification library availability –standardization

Importance of Type Checking Specification languages can have much richer type systems than programming languages since type do not have to be implemented directly Type checking can be used to detect faults and inconsistencies Essential model features can be embedded in types and subtypes

System Operations as Functions Basic system operations are often modeled as functions Functions can modify the system state so invariant conditions are often imposed on appropriate combinations of functions (e.g. similar to algebraic specification) Theorems and axioms can be used to model other system invariants employing functions

Logical Errors in Formal Specifications Logical inconsistency –Easiest logic errors to detect Accuracy –Does specification match intended meaning? –System invariants can help in detecting these errors. Completeness –Does specification identify all contingencies? –Are appropriate behaviors defined for all cases? –Peer review can aid in detection.

Techniques for Detecting Specification Errors Manual –Formal specification inspection –Theorem proving, proof checking, and model checking for logic defects Automated –Parsing specification for syntactic correctness –Type checking for semantic consistency –Simulation or animation based on the specification

Formal Specification Process Model Clarify requirements and high level design Articulate implicit assumptions Identify undocumented or unexpected assumptions Expose defects Identify exceptions Evaluate test coverage