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As a counterpoint, Dijkstra [0] makes a distinction between what he calls “postulational” and “operational” (more like what TFA is describing) formal methods. (

by csb6 1y ago

As a counterpoint, Dijkstra [0] makes a distinction between what he calls “postulational” and “operational” (more like what TFA is describing) formal methods. (Sidenote: I think people nowadays would use “denotational” instead of “postulational” (e.g. denotational vs. operational semantics), but Dijkstra wrote this in the 1980s) Instead of thinking of a program as a set of potential execution traces, he advocated thinking of programs as formulas (e.g. pre/postconditions) that can be used to construct proofs without having to simulate any executions. His idea with weakest preconditions was to derive code from the desired logical properties, not to prove that a priori written code satisfies those properties, reckoning that his way would be less work and result in more elegant programs. I am just an amateur but it seems like formal methods as a whole is often conflated with model checking and other operational formal methods. [0] https://www.cs.utexas.edu/~EWD/transcriptions/EWD10xx/EWD1012.html https://www.cs.utexas.edu/~EWD/transcriptions/EWD10xx/EWD101...

mrtesthah 1y ago
>His idea with weakest preconditions was to derive code from the desired logical properties, not to prove that a priori written code satisfies those properties, reckoning that his way would be less work and result in more elegant programs. I love the idea. Are you aware of any such spec-based code generators in widespread use today?
- csb6 1y ago
  The most developed one I know of is Atelier B [0], which uses the B-Method and has a C code generator. I believe this was the tool used to develop several driverless train control systems in the Paris Metro and other subways. There is a free version but unfortunately it is not open source. [0] https://www.atelierb.eu/en/ https://www.atelierb.eu/en/
pron 1y ago
While there is a proof assistant for checking TLA+ proofs and a couple of model checkers for subsets of TLA+, TLA+ itself is no more and no less than a formal language for expressing mathematical formulas that describe dynamical systems. There is no notion of code, program, or execution in TLA+ any more than the formula y(t) = y0 + v0t - 0.5gt^2 has the notion of a ball, a vacuum, or of Earth. When the author talks about "sets of behaviours" (really, classes of behaviours, but that's getting too technical) it is precisely in the same sense that the formula `x > 3` denotes the set of all integers > 3 in a logic over the integers. What's interesting about TLA+ (or, really, about TLA, the temporal logic at the core of TLA+) is how change over time is represented. Rather than with an explicit time variable, TLA intrinsically represents the evolution of the values of variables over time in a way that makes abstraction-refinement relations (the relation between more and less detailed descriptions of a dynamical system) particularly easy to express. Dijkstra's pre/post conditions can be expressed in TLA+ just as anything that could be stated precisely could be expressed in any sufficiently rich mathematical language. We can choose to interpret some TLA+ formulas as "programs" just as we can choose to interpret some formulas as describing the motion of a baseball in a vacuum, but TLA+ allows us to express things that are more abstract than programs, which can be very useful (e.g. no program can express the Quicksort algorithm in its full generality, as that algorithm is too abstract to be a program, but it is nevertheless very useful to show that a particular program is a particular implementation of that algorithm).
- pron 1y ago
  P.S. A model checker doesn't imply anything about the kind of the formal specification used. It is a program that checks whether all assignments of free variables in a formula are satisfying assignments, or "models" for the formula (a model is a satisfying assignement of free variables in a formula, i.e. assignments that make the formula true). It is distinct from an automated proof finder in that it doesn't (necessarily) find a deductive proof in the relevant logic that the forumla is always true, but rather operates in the semantic universe of the logic. For example, the proposition `A ⇒ A` is true for any A in a simple boolean logic. We can determine that either via a deductive proof, which consists of a sequence of application of the deductive rules of the logic, or by examining the relevant truth table; the latter method would be a form of model checking. Neither the use of deductive proofs or of model checking implies something about the nature of the logic. They are just two different ways of determining the truth of a proposition.
  - csb6 1y ago
    > A model checker doesn't imply anything about the kind of the formal specification used. That is true; I was just contrasting the way TFA explained specifications as being definitions of groups of acceptable execution traces (perhaps checked using a model checker) with Dijkstra’s approach of characterizing the acceptable behavior without simulating any executions. Both are of course valid approaches to demonstrate the correctness of a specification.
    - pron 1y ago
      > I was just contrasting the way TFA explained specifications as being definitions of groups of acceptable execution traces (perhaps checked using a model checker) with Dijkstra’s approach of characterizing the acceptable behavior without simulating any executions. I think that's a misunderstanding, as TLA+ doesn't require simulating anything (even the model checker doesn't work like that; it can check a formula that corresponds to an uncountable infinity of behaviours, each of them infinite, in a second [1]), and Dijkstra's approach is separate from the logical semantics. It is true that a TLA formula denotes a set (really, a class) of something called "behaviours" (which could be interpreted as execution traces when a formula is interpreted to describe a program) just as the statement x > 3 denotes the set of all integers greater than 3 (in a logic over the integers), but in neither case is enumerating or "simulating" anything required, either in principle or in practice. It's just simple logical semantics, and Dijkstra's approach is easily expressed in a logic with such simple semantics. In fact, TLA serves as a generalisation of Dijkstra's approach, as that is lacking when dealing with concurrency. While in sequential programs it is easy to specify invariants as pre/post conditions, it is hard to do that in concurrent/distributed programs. [1]: You can ask the TLC model checker to simulate behaviours if you want (it can be useful sometimes), but checking mode, not simulation mode, is the default. Obviously, in simulation mode TLC doesn't work as a model checker. Technically speaking, every TLA formula denotes an uncountable infinity of behaviours of infinite lengths, where between any two states in the behaviour there is an infinite number of steps (it is in this regard that TLA is different from other temporal logics like LTL), and each state in every behaviour is an assignment to an infinite number of variables (in fact, that uncountable infinity is so large that it doesn't even correspond to a set in the set theory sense but to something called a class). The large infinities are what makes TLA so suitable for expressing abstractions and refinement and interactions between components. This may sound strange, but it's actually how most simple logics work. When I said before that `x > 3` in a logic over the integers denotes the set of integers greater than 3, I wasn't entirely precise. What it really denotes is: "x is in the set of integers greater than three and any other imaginable variable could be any integer".

by csb6 1y ago

mrtesthah 1y ago

>His idea with weakest preconditions was to derive code from the desired logical properties, not to prove that a priori written code satisfies those properties, reckoning that his way would be less work and result in more elegant programs. I love the idea. Are you aware of any such spec-based code generators in widespread use today?

csb6 1y ago
The most developed one I know of is Atelier B [0], which uses the B-Method and has a C code generator. I believe this was the tool used to develop several driverless train control systems in the Paris Metro and other subways. There is a free version but unfortunately it is not open source. [0] https://www.atelierb.eu/en/ https://www.atelierb.eu/en/

pron 1y ago

While there is a proof assistant for checking TLA+ proofs and a couple of model checkers for subsets of TLA+, TLA+ itself is no more and no less than a formal language for expressing mathematical formulas that describe dynamical systems. There is no notion of code, program, or execution in TLA+ any more than the formula y(t) = y0 + v0t - 0.5gt^2 has the notion of a ball, a vacuum, or of Earth. When the author talks about "sets of behaviours" (really, classes of behaviours, but that's getting too technical) it is precisely in the same sense that the formula `x > 3` denotes the set of all integers > 3 in a logic over the integers. What's interesting about TLA+ (or, really, about TLA, the temporal logic at the core of TLA+) is how change over time is represented. Rather than with an explicit time variable, TLA intrinsically represents the evolution of the values of variables over time in a way that makes abstraction-refinement relations (the relation between more and less detailed descriptions of a dynamical system) particularly easy to express. Dijkstra's pre/post conditions can be expressed in TLA+ just as anything that could be stated precisely could be expressed in any sufficiently rich mathematical language. We can choose to interpret some TLA+ formulas as "programs" just as we can choose to interpret some formulas as describing the motion of a baseball in a vacuum, but TLA+ allows us to express things that are more abstract than programs, which can be very useful (e.g. no program can express the Quicksort algorithm in its full generality, as that algorithm is too abstract to be a program, but it is nevertheless very useful to show that a particular program is a particular implementation of that algorithm).

pron 1y ago
P.S. A model checker doesn't imply anything about the kind of the formal specification used. It is a program that checks whether all assignments of free variables in a formula are satisfying assignments, or "models" for the formula (a model is a satisfying assignement of free variables in a formula, i.e. assignments that make the formula true). It is distinct from an automated proof finder in that it doesn't (necessarily) find a deductive proof in the relevant logic that the forumla is always true, but rather operates in the semantic universe of the logic. For example, the proposition `A ⇒ A` is true for any A in a simple boolean logic. We can determine that either via a deductive proof, which consists of a sequence of application of the deductive rules of the logic, or by examining the relevant truth table; the latter method would be a form of model checking. Neither the use of deductive proofs or of model checking implies something about the nature of the logic. They are just two different ways of determining the truth of a proposition.
- csb6 1y ago
  > A model checker doesn't imply anything about the kind of the formal specification used. That is true; I was just contrasting the way TFA explained specifications as being definitions of groups of acceptable execution traces (perhaps checked using a model checker) with Dijkstra’s approach of characterizing the acceptable behavior without simulating any executions. Both are of course valid approaches to demonstrate the correctness of a specification.
  - pron 1y ago
    > I was just contrasting the way TFA explained specifications as being definitions of groups of acceptable execution traces (perhaps checked using a model checker) with Dijkstra’s approach of characterizing the acceptable behavior without simulating any executions. I think that's a misunderstanding, as TLA+ doesn't require simulating anything (even the model checker doesn't work like that; it can check a formula that corresponds to an uncountable infinity of behaviours, each of them infinite, in a second [1]), and Dijkstra's approach is separate from the logical semantics. It is true that a TLA formula denotes a set (really, a class) of something called "behaviours" (which could be interpreted as execution traces when a formula is interpreted to describe a program) just as the statement x > 3 denotes the set of all integers greater than 3 (in a logic over the integers), but in neither case is enumerating or "simulating" anything required, either in principle or in practice. It's just simple logical semantics, and Dijkstra's approach is easily expressed in a logic with such simple semantics. In fact, TLA serves as a generalisation of Dijkstra's approach, as that is lacking when dealing with concurrency. While in sequential programs it is easy to specify invariants as pre/post conditions, it is hard to do that in concurrent/distributed programs. [1]: You can ask the TLC model checker to simulate behaviours if you want (it can be useful sometimes), but checking mode, not simulation mode, is the default. Obviously, in simulation mode TLC doesn't work as a model checker. Technically speaking, every TLA formula denotes an uncountable infinity of behaviours of infinite lengths, where between any two states in the behaviour there is an infinite number of steps (it is in this regard that TLA is different from other temporal logics like LTL), and each state in every behaviour is an assignment to an infinite number of variables (in fact, that uncountable infinity is so large that it doesn't even correspond to a set in the set theory sense but to something called a class). The large infinities are what makes TLA so suitable for expressing abstractions and refinement and interactions between components. This may sound strange, but it's actually how most simple logics work. When I said before that `x > 3` in a logic over the integers denotes the set of integers greater than 3, I wasn't entirely precise. What it really denotes is: "x is in the set of integers greater than three and any other imaginable variable could be any integer".