XPost: comp.theory, sci.math, comp.lang.prolog   
   From: mikko.levanto@iki.fi   
      
   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.   
   >>> As I say non-terminating, thus never resolves to a truth value.   
   >>   
   >> As according to Prolog rules foo(Y) isn't a truth value for any Y   
   >> the above is obviously just an attempt to deive with a distraction.   
   >   
   > That was a quote from the most definitive source   
   > for the Prolog Language.   
      
   As I already said, that source agrees with what I said above.   
      
   > Prolog only has Facts and Rules thus the only   
   > derivation is to a truth value.   
      
   Not true. Prolog also has term expressions that can be used in facts   
   and rules. In particular, in the goal G = not(provable(F, G)) the   
   symbol G is a term expression and provable(F, G) is another term   
      
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