Complexity of pattern matching for modus ponens logical conclusions

Question

Is a Turing machine with added the following contant-time operation equivalent (in the sense that polynomial time remains polynomial time and exponential time remains exponential time) to a (usual) Turing machine:

By predicates I will mean predicates in first-order predicate calculus. (Note that predicates may have free variables.)

• constant-time modus-ponens resolution (yes or no) and then adding $y$ to the end of this array if yes, for given predicates $x$ and $y$ and an array (or a linked list) of predicates. By definition of modus ponens, it's yes, if and only if some element of the arrays is $X\Rightarrow y$ where $X$ is a pattern matching $x$.

Remark: The above operation is a part of the standard procedure of proof-checking is first-order predicate logic.

If the above hypothesis is false, then what is the running time upped bounds of the above operation in different kinds of Turning machine equivalents (such as Turing machine, Markov algorithms, von Neumann architecture with infinitely many infinitely big words of memory, etc.)?

BTW, is von Neumann architecture with infinitely many infinitely big words of memory a Turning machine equivalent? (I think yes, but not 100% sure.)

Answer 1

This boils down to the question of whether there exists a polynomial-time algorithm for your matching operation.

If there exists such an algorithm, call it $A$, then the answer to your question is yes: given an augmented Turing machine that uses these operations, you can convert it to an ordinary Turing machine by simulating those operations using $A$. This blows up the running time by at most a polynomial. (Invoking a polynomial-time subroutine at most polynomially many times leaves with a running time that remains polynomial.)

If there is no such algorithm, then the answer to your question is no: the matching problem can be solved in polynomial time (indeed, constant time) on an augmented Turing machine, but cannot be solved in polynomial time on an ordinary Turing machine.

I'm not clear on the definition of your matching operation, and I'm not an expert in computational logic, so I don't know what is the computational complexity of the matching operation you have in mind.