Introduction to Formal
Methods
Based on
Jeannette M. Wing. A Specifier's Introduction to Formal
Methods. IEEE Computer, 23(9):8-24, September, 1990.
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Outline
• Definition of formal methods and specification languages
• Pragmatics of formal specifications
• Examples: Z, VDM, Larch, temporal logic, CSP,
transition axioms
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In-class: Read page 8 (10 mins)
• Q: What are formal methods
• At what stages of system development can formal
methods be used?
• Some advantages of formal specifications?
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Formal Methods
• Definition: Formal Methods
Mathematically based techniques that describe system properties,
from which, people can systematically specify, develop, and verify
systems.
• The mathematical foundation allows for a concise and
unambiguous definition of notions such as:
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Consistency
Completeness
Specification
Implementation
Correctness
• Because the semantics are formally defined, they are
amenable to machine analysis and manipulation.
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Formal Methods
• Can be used to specify:
– behavioral properties
– structural properties
– pragmatic considerations, e.g. response time
• Applicable at all phases of the software lifecycle.
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Requirements analysis
Design
V&V
Documentation
Analysis and evaluation
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In-class: Read pages 10-11
• Q1: Three elements of formal specification languages?
• Q2: Example semantic domains?
• Q3: Why an “abstract satisfies relation” on top of the
“satisfies” relation?
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Formal Specification Language
• Triple, <Syn, Sem, Sat>
– Syn: language’s syntactic domain, a set
– Sem: language’s semantic, a set
– Sat: a satisfies relation between Syn and Sem, a subset of Syn X Sem
• Given <Syn, Sem, Sat>
– If Sat(syn, sem), syn is a specification of sem and sem is a specificand of syn.
• Given <Syn, Sem, Sat>
– The specificand set of all specifications syn in Syn is the set of all specificands
sem in Sem such that Sat(syn, sem).
– I.e., Sat doesn’t have to be a function; but why?
– Q: Any other properties of Sat?
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Syntactic Domains
• Defined as a set of symbols and grammatical rules
– Symbols can be constants, variables, and logical connectives
– Grammatical rules define how to combine the symbols into well formed
sentences
– E.g., x.P(x)  Q(x)
• A syntactic domain need not be restricted to text
– Symbols can include boxes, circles, lines, arrows, etc.
– A possible rule could be that “an arrow must be connected at both ends to a box”
• Essentially, the syntactic domain is the set of all possible
well formed specifications that can be expressed using
the symbols, whether textual or graphical.
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Semantic Domains
• Semantic domain
– Set of objects in the universe of what the language can describe;
i.e., meanings or interpretations.
• Examples
– ADTs languages: Algebras, theories, programs
– Concurrent/Distributed: State sequences, event sequences, state and
transition sequences, streams, synchronization trees, partial orders, state
machines
– Programming languages: Functions from input to output, computations,
predicate transformers, machine instructions
• If semantic domain is over programs:
– Implements for satisfies
– implementation for specificand
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Satisfies Relation
• Often need to specify different aspects of a single
specificand (various abstractions), e.g.,
– Functional behavior of a collection of program modules
– Structural relationships between the modules
• Abstraction function for different views
– A semantic abstraction function maps elements of the semantic domain into
equivalence classes
– A partition of the semantic domains
• Abstract satisfies relation between specs and
equivalence classes of semantic domains
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Abstract Satisfies Relation
all methods that
sort arrays
Java
Methods
all methods that
use the set class
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Abstract Satisfies Relation
• Two broad classes of abstraction functions:
– Those that abstract preserving behavior
– Those that abstract preserving structure
• Behavioral specifications
– Constraints on observed behavior
– Functionality such as a mapping from inputs to outputs (Cleanroom)
– Other aspects such as fault tolerance, safety, security, response time, and space
efficiency.
• Structural specifications
– Constraints on the internal composition of specificands
– Capture hierarchical and uses relations
– Denoted by call graphs, data dependency diagrams, etc.
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Properties of Specifications
• Unambiguous
– Given <Syn, Sem, Sat>, a spec. syn is unambiguous if Sat maps syn to exactly
one specificand set.
• Consistent
– Given <Syn, Sem, Sat>, a spec. syn is consistent if Sat maps syn to a nonempty specificand set.
• Complete vs. incomplete (or loose specifications)
– More complete: implementation bias and less freedom
– Less complete: more freedom to programmer and less restrictive
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Proving properties of
Specificands
• Most formal methods include a language that has a welldefined logical inference engine.
• When you prove a specification inferable from the the set
of facts (other specs), you prove a property that a
specificand satisfying the facts (other specs) will have.
– Soundness vs. completeness
• If users are able to prove a surprising result, then
perhaps the base specifications are wrong.
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Pragmatics
• Users
– Writers
– Readers
• Use
• Characteristics
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Use of Formal Methods
• Formal methods can be applied to all phases of system
development, e.g., throughout development lifecycle
• Requirements
– Clarify customer’s stated requirements
– Crystallize vague ideas
– Aid communication between engineer and client, e.g., English to spec / spec to
English tools
• Design
– Aids in decomposition, e.g., by formally specify interfaces between modules
– Aids in refinement, e.g., by ensuring that different levels of abstractions all
satisfy a parent specification
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Use of Formal Methods
• Verification and validation
– Guide the building of test cases, e.g., black-box testing
– Verify the critical sections of implementation
• Documentation
– More precise and concise than natural language
• Analysis and evaluation
– Serve as reference point between what the customer wanted and what was
implemented
– Can be used to find bugs in existing systems that weren’t developed using formal
methods
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Characteristics of FMs
• Model-oriented
– Define a system’s behavior directly by constructing a model
– Model in terms of mathematical structures such as sets, functions, relations, and
sequences
– Use model to show correctness with respect to specifications
– E.g., Sequential and ADT: Z, VDM
Concurrent and distributed: Petri net, CCS, CSP
• Property-oriented
– Define a system behavior indirectly by stating a set of properties that the system
must satisfy
– Properties in the form of axioms
– E.g., Larch, OBJ, Clear, ACT ONE
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Characteristics of FMs
• Visual languages
– HIPO, Structured design, Software Requirements Engineering
Method
• Executable
– OBJ, Prolog
– Q: Should a formal specification language be executable?
• Tool-supported
– Model-checking tools: EMC
– Proof-checking tools: Boyer-Moore Theorem Prover, FDM,HDM,
m-EVES, HOL, LCF, OBJ
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Language Examples
(Symbol Table in Z and VDM)
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Symbol Table – Larch
Larch/CLU
LSL
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Concurrency - Temporal Logic
Based on temporal operators such as:
– P: In all future state
– P: in some future state
– OP: in the next state
Q: Meanings of (1)-(4)?
Notation: <c!m> event of placing message m on channel c.
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Temporal Logic
• (1) Any message transmitted to the right channel must have been
previously placed on the left channel
• (2) Messages are transmitted first in, first out
• (3) All messages are unique
• (4) Each incoming message will eventually be transmitted
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Concurrency – Communicating
Sequential Process (CSP)
Based on a model of traces or event sequences, and
assumes processes communicate by sending messages.
?
prefix
refusal set: refuse to communicate
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Concurrency – Transition Axiom
Axioms for
operations
Temporal logic for
properties
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Summary
• Differences among formal methods
– notation, semantic domain, definition of the
satisfies relation
• But, same purpose
– Let system developers couch their ideas
precisely
– I.e., provide a way to specify and verify
programs in order to provide a deeper
understanding of a system for clients,
designers, implementers, and testers.
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