Чи існує логіка без індукції, яка захоплює велику частину P?

Іммерман-Варді теорема стверджує , що PTIME (або P) є саме клас мов , які можуть бути описані пропозиції першого порядку логіки разом з оператором фіксованого точкою, по класу впорядкованих структур. Оператор з фіксованою точкою може бути або найменш фіксованою (як вважають Іммерман та Варді), або інфляційною фіксованою. (Стефан Кройцер, Експресивна еквівалентність найменшої та інфляційної логіки з фіксованою точкою , Аннали чистої та прикладної логіки 130 61–78, 2004).

Юрій Гуревич висловив думку про те, що немає логіки, що фіксує PTIME ( Логіка та виклик інформатики , в сучасних тенденціях теоретичної інформатики, ред. Егон Боргер, 1–57, Computer Science Press, 1988), а Мартін Грое заявив, що він є менш впевнений ( Квест про логіку захоплення PTIME , FOCS 2008).

Оператор з фіксованою точкою призначений для збору потужності рекурсії. Фіксовані точки є потужними, але мені очевидно, що вони необхідні.

Чи існує оператор X, який не заснований на фіксованих точках, таким, що FOL + X фіксує (великий) фрагмент PTIME?

Редагувати: Наскільки я розумію, лінійна логіка може виражати лише твердження про структури, які мають досить обмежувальну форму. Я в ідеалі хотів би побачити посилання на або ескіз на логіку, яка може виражати властивості довільних наборів реляційних структур, уникаючи нерухомих точок. Якщо я помиляюсь у виразній силі лінійної логіки, тоді вказівник чи натяк буде вітатися.

You want to have a look at what some people call Grädel's Theorem. You can find it in Papadimitriou's book "Computational Complexity" (it's Theorem 8.4 in page 176) or in Grädel's original paper.

In a nutshell, Grädel's Theorem is to P what Fagin's Theorem is to NP. It states that on the class of finite structures with a successor relation, the collection of polynomial-time decidable properties coincides with those expressible in the Horn-fragment of existential second-order logic. These are the sentences of second-order logic of the form

$$(\exists R)(\forall x)(\phi)$$ where $R$ is a sequence of second-order relation variables, $x$ is a sequence of first-order variables, and $\phi$ is a quantifier-free formula that, when written in CNF form, is a conjunction of $R$-Horn clauses (i.e. clauses that have at most one non-negated atom involving the variables in $R$).

As far as I understand, linear logic can only express statements about structures that have quite restrictive form. I would ideally like to see a reference to, or a sketch of, a logic that can express properties of arbitrary sets of relational structures, while still avoiding fixed points. If I am wrong about the expressive power of linear logic then a pointer or hint would be welcome.

This isn't right: all residuated commutative monoidal lattices are models of linear logic. Here's an easy way to create such a lattice from finite graphs. Start with the set

$$M = \{(g, n) \;|\; g\mbox{ is a finite graph and }n \subseteq \mathrm{nodes}(g)\}$$

So our forcing relation will be $(g,n) \models \phi$, and the intuition is that $n$ is the set of nodes "owned" by the formula $\phi$. There is a partial operation $(\cdot) : M \times M \to M$, defined as:

$$(g,n) \cdot (g',n') = \left\{\begin{array}{ll} (g, n \uplus n') & \mbox{when } g = g' \land n \cap n' = \emptyset \\ \mathrm{undefined} & \mbox{otherwise} \end{array}\right.$$

This combines two elements by mergining their owned sets, if the graphs are equal and the owned sets are disjoint.

Now, we can give a model of linear logic as follows:

$$\begin{array}{lcl} (g,n) \models I & \iff & n = \emptyset \\ (g,n) \models \phi \otimes \psi & \iff & \exists n_1,n_2.\; n = n_1 \uplus n_2 \mbox{ and } (g,n_1) \models \phi \mbox{ and } (g,n_2) \models \psi \\ (g,n) \models \phi \multimap \psi & \iff & \forall n'.\; \mbox{if } n \cap n' = \emptyset \mbox{ and } (g,n') \models \phi \mbox{ then } (g,n \uplus n') \models \psi \\ (g, n) \models \top & \iff & \mbox{always} \\ (g,n) \models \phi \land \psi & \iff & (g,n) \models \phi \mbox{ and } (g,n) \models \psi \end{array}$$

This model is actually a variant of the ones used in separation logic, which is widely used in the verification of heap-manipulating programs. (If you like, think of the graph as the pointer structure of the heap, and the analogy is exact!)

This isn't really the right way to think about linear logic, though: its real intuitions are proof-theoretic, and the connection to complexity comes via the computational complexity of the cut-elimination theorem. The model theory of linear logic is the shadow cast by its proof theory.

There are recent exciting results concerning the search for a logic capturing PTIME. The famous example by Cai, Fürer and Immerman showing that LFP+C does not capture PTIME was based on a seemingly artificial class of graphs, though. Of course, it was constructed for the particular task of demonstrating the restrictions of LFP+C. Only recently it was shown by Dawar that the class is not artificial at all. It can rather be seen as an example for the fact that LFP+C cannot solve linear equation systems!

Hence Dawar, Grohe, Holm and Laubner extended logics by operators from linear algebra, for example by an operator to define the rank of a definable matrix. The resulting logic LFP+rank can express strictly more than LFP+C, in fact, there is no known PTIME property that LFP+rank cannot express.

Even FO+rk is surprisingly powerful, it can express deterministic and symmetric transitive closure. It is still open whether it can express the general transitive closure of a graph.

Depending on what you mean by "capture," Soft Linear Logic and Polynomial Time by Yves Lafont may be of interest. There is a 1-1 correspondence to proofs in this logic and PTIME algorithms that take a string as input and output 0 or 1.

The Wikipedia article on Linear Logic is here. It's not a fixpoint logic. The intuition of "classical logic over $C^*$ algebras instead of boolean algebras" is the easiest for me to grasp.

Some older work on this problem, again in the Linear Logic vein is, Jean-Yves Girard, Andre Scedrov, and Philip Scott. Bounded linear logic: A modular approach to polynomial-time computability. Theoretical Computer Science, 97(1):1–66, 1992.

More recent work includes Bounded Linear Logic, Revisited by Ugo Dal Lago and Martin Hofmann.