Determinacy of the query pattern
q()
w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

q :- p(!).
p(X) :- ','(X, failure(a)).
p(X) :- p(X).
failure(b).

Query: q()

(1) PrologDeterminacyProcessorProof (EQUIVALENT transformation)

The root node satisfies the determinacy criterion.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: q\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: q, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: p(!)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: p(!), SCOPE: 2, CLAUSE: 1 | p(!), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: p(!), SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: p(!), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(call(!), failure(a))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(!_3, failure(a))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: failure(a), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: p(!)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 13 [arrowhead = none ];
13 -> 2;

14 [label="ONLY EVAL with clause\nq :- p(!).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 14 [arrowhead = none ];
14 -> 3;

15 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 15 [arrowhead = none ];
15 -> 4;

16 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 16 [arrowhead = none ];
16 -> 5;

17 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 17 [arrowhead = none ];
17 -> 6;

18 [label="ONLY EVAL with clause\np(X5) :- ','(call(X5), failure(a)).\nand substitutionX5 -> !", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 18 [arrowhead = none ];
18 -> 7;

19 [label="ONLY EVAL with clause\np(X9) :- p(X9).\nand substitutionX9 -> !", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 19 [arrowhead = none ];
19 -> 12;

20 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 20 [arrowhead = none ];
20 -> 8;

21 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 21 [arrowhead = none ];
21 -> 9;

22 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 22 [arrowhead = none ];
22 -> 10;

23 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 23 [arrowhead = none ];
23 -> 11;

24 [label="INSTANCE", fontsize=14, style = filled, fillcolor = lightgrey];
12 -> 24 [arrowhead = none , style=dashed];
24 -> 3[style=dashed];

}

(2) YES