XPost: comp.lang.prolog, comp.theory, sci.math   
   From: polcott333@gmail.com   
      
   On 12/14/2025 4:05 AM, Mikko wrote:   
   > On 13/12/2025 16:43, olcott wrote:   
   >> On 12/13/2025 4:19 AM, Mikko wrote:   
   >>> olcott kirjoitti 12.12.2025 klo 16.19:   
   >>>> On 12/12/2025 2:50 AM, Mikko wrote:   
   >>>>> olcott kirjoitti 11.12.2025 klo 16.17:   
   >>>>>> On 12/11/2025 2:42 AM, Mikko wrote:   
   >>>>>>> olcott kirjoitti 10.12.2025 klo 16.10:   
   >>>>>>>> On 12/10/2025 4:04 AM, Mikko wrote:   
   >>>>>>>>> olcott kirjoitti 8.12.2025 klo 21.09:   
   >>>>>>>>>> On 12/8/2025 3:13 AM, Mikko wrote:   
   >>>>>>>>>>> olcott kirjoitti 5.12.2025 klo 19.43:   
   >>>>>>>>>>>> On 12/5/2025 3:38 AM, Mikko wrote:   
   >>>>>>>>>>>>> olcott kirjoitti 4.12.2025 klo 16.06:   
   >>>>>>>>>>>>>> On 12/4/2025 2:58 AM, Mikko wrote:   
   >>>>>>>>>>>>>>> Tristan Wibberley kirjoitti 4.12.2025 klo 4.32:   
   >>>>>>>>>>>>>>>> On 30/11/2025 09:58, Mikko wrote:   
   >>>>>>>>>>>>>>>>   
   >>>>>>>>>>>>>>>>> Note that the meanings of   
   >>>>>>>>>>>>>>>>>   ?- G = not(provable(F, G)).   
   >>>>>>>>>>>>>>>>> and   
   >>>>>>>>>>>>>>>>>   ?- unify_with_occurs_check(G, not(provable(F, G))).   
   >>>>>>>>>>>>>>>>> are different. The former assigns a value to G, the   
   >>>>>>>>>>>>>>>>> latter does not.   
   >>>>>>>>>>>>>>>   
   >>>>>>>>>>>>>>>> For sufficiently informal definitions of "value".   
   >>>>>>>>>>>>>>>> And for sufficiently wrong ones too!   
   >>>>>>>>>>>>>>>   
   >>>>>>>>>>>>>>> It is sufficiently clear what "value" of a Prolog   
   >>>>>>>>>>>>>>> variable means.   
   >>>>>>>>>>>>>   
   >>>>>>>>>>>>>> % This sentence cannot be proven in F   
   >>>>>>>>>>>>>> ?- G = not(provable(F, G)).   
   >>>>>>>>>>>>>> G = not(provable(F, G)).   
   >>>>>>>>>>>>>> ?- unify_with_occurs_check(G, not(provable(F, G))).   
   >>>>>>>>>>>>>> false.   
   >>>>>>>>>>>>>>   
   >>>>>>>>>>>>>> I would say that the above Prolog is the 100%   
   >>>>>>>>>>>>>> complete formal specification of:   
   >>>>>>>>>>>>>>   
   >>>>>>>>>>>>>> "This sentence cannot be proven in F"   
   >>>>>>>>>>>>>   
   >>>>>>>>>>>>> The first query can be regarded as a question whether "G =   
   >>>>>>>>>>>>> not(provable(   
   >>>>>>>>>>>>> F, G))" can be proven for some F and some G. The answer is   
   >>>>>>>>>>>>> that it can   
   >>>>>>>>>>>>> for every F and for (at least) one G, which is   
   >>>>>>>>>>>>> not(provable(G)).   
   >>>>>>>>>>>>>   
   >>>>>>>>>>>>> The second query can be regarded as a question whether "G =   
   >>>>>>>>>>>>> not(provable   
   >>>>>>>>>>>>> (F, G))" can be proven for some F and some G that do not   
   >>>>>>>>>>>>> contain cycles.   
   >>>>>>>>>>>>> The answer is that in the proof system of Prolog it cannot be.   
   >>>>>>>>>>>>   
   >>>>>>>>>>>> No that it flatly incorrect. The second question is this:   
   >>>>>>>>>>>> Is "G = not(provable(F, G))." semantically sound?   
   >>>>>>>>>>>   
   >>>>>>>>>>> Where is the definition of Prolog semantics is that said?   
   >>>>>>>>>>   
   >>>>>>>>>> Any expression of Prolog that cannot be evaluated to   
   >>>>>>>>>> a truth value because it specifies non-terminating   
   >>>>>>>>>> infinite recursion is "semantically unsound" by the   
   >>>>>>>>>> definition of those terms even if Prolog only specifies   
   >>>>>>>>>> that cannot be evaluated to a truth value because it   
   >>>>>>>>>> specifies non-terminating infinite recursion.   
   >>>>>>>>>   
   >>>>>>>>> Your Prolog implementation has evaluated G = not(provablel(F, G))   
   >>>>>>>>> to a truth value true. When doing so it evaluated each side of =   
   >>>>>>>>> to a value that is not a truth value.   
   >>>>>>>>   
   >>>>>>>> ?- unify_with_occurs_check(G, not(provable(F, G))).   
   >>>>>>>> false.   
   >>>>>>>>   
   >>>>>>>> Proves that   
   >>>>>>>> G = not(provable(F, G)).   
   >>>>>>>> would remain stuck in infinite recursion.   
   >>>>>>>>   
   >>>>>>>> unify_with_occurs_check() examines the directed   
   >>>>>>>> graph of the evaluation sequence of an expression.   
   >>>>>>>> When it detects a cycle that indicates that an   
   >>>>>>>> expression would remain stuck in recursive   
   >>>>>>>> evaluation never to be resolved to a truth value.   
   >>>>>>>>   
   >>>>>>>> BEGIN:(Clocksin & Mellish 2003:254)   
   >>>>>>>> Finally, a note about how Prolog matching sometimes differs   
   >>>>>>>> from the unification used in Resolution. Most Prolog systems   
   >>>>>>>> will allow you to satisfy goals like:   
   >>>>>>>>   
   >>>>>>>> equal(X, X).   
   >>>>>>>> ?- equal(foo(Y), Y).   
   >>>>>>>>   
   >>>>>>>> that is, they will allow you to match a term against an   
   >>>>>>>> uninstantiated subterm of itself. In this example, foo(Y)   
   >>>>>>>> is matched against Y, which appears within it. As a result,   
   >>>>>>>> Y will stand for foo(Y), which is foo(foo(Y)) (because of   
   >>>>>>>> what Y stands for), which is foo(foo(foo(Y))), and so on.   
   >>>>>>>> So Y ends up standing for some kind of infinite structure.   
   >>>>>>>>   
   >>>>>>>> Note that, whereas they may allow you to construct something   
   >>>>>>>> like this, most Prolog systems will not be able to write it   
   >>>>>>>> out at the end. According to the formal definition of   
   >>>>>>>> Unification, this kind of “infinite term” should never come   
   >>>>>>>> to exist. Thus Prolog systems that allow a term to match an   
   >>>>>>>> uninstantiated subterm of itself do not act correctly as   
   >>>>>>>> Resolution theorem provers. In order to make them do so, we   
   >>>>>>>> would have to add a check that a variable cannot be   
   >>>>>>>> instantiated to something containing itself. Such a check,   
   >>>>>>>> an occurs check, would be straightforward to implement, but   
   >>>>>>>> would slow down the execution of Prolog programs considerably.   
   >>>>>>>> Since it would only affect very few programs, most implementors   
   >>>>>>>> have simply left it out 1.   
   >>>>>>>>   
   >>>>>>>> 1 The Prolog standard states that the result is undefined if   
   >>>>>>>> a Prolog system attempts to match a term against an   
   >>>>>>>> uninstantiated subterm of itself, which means that programs   
   >>>>>>>> which cause this to   
   >>>>>>>> happen will not be portable. A portable program should ensure   
   >>>>>>>> that wherever an occurs check might be applicable the built-in   
   >>>>>>>> predicate   
   >>>>>>>> unify_with_occurs_check/2 is used explicitly instead of the normal   
   >>>>>>>> unification operation of the Prolog implementation. As its name   
   >>>>>>>> suggests, this predicate acts like =/2 except that it fails if an   
   >>>>>>>> occurs check detects an illegal attempt to instantiate a variable.   
   >>>>>>>> END:(Clocksin & Mellish 2003:254)   
   >>>>>>>>   
   >>>>>>>> Clocksin, W.F. and Mellish, C.S. 2003. Programming in Prolog   
   >>>>>>>> Using the ISO Standard Fifth Edition, 254. Berlin Heidelberg:   
   >>>>>>>> Springer-Verlag.   
   >>>>>>>   
   >>>>>>> Thank you for the confirmation of my explanation of your error.   
   >>>>>>   
   >>>>>>  >> Y will stand for foo(Y), which is foo(foo(Y)) (because of   
   >>>>>>  >> what Y stands for), which is foo(foo(foo(Y))), and so on.   
      
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