# Questions about SMT logic solvers

• I
• olgerm
In summary, according to the undecidability of first order logic, an algorithm that could check consistency of any 1.-order logic formula does not exist. However, Gödels 2. incompleteness theorem does not claim that such an algorithm does not exist. There are computerprograms that could check consistency of any 2.-order logic formula, but no algorithm can check consistency of any 1.- or 2.-order logic formula.
olgerm
Gold Member
Are smt-solvers (like z3) theoretically able to (always correctly) check consistency of any 1.-order logic formula?
Does it follow from The undecidability of first order logic, that an algorithm, that could check consistency of any 1.-order logic formula, does not exist?
How does the algorithm of work in details? (link to explanation)
Are there any algorithms that could check consistency of any 2.-order logic formula?
Does Gödels 2. incompleteness theorem claim, that such algortihm, that could check consistency of any 2.-order logic formula, does not exist.
Are there any computerprograms that could check consistency of any 2.-order logic formula?

To explain what I mean by 1.-order logic formula I will write here a few examples:
• ##\exists_{x_1}(A(x_1))\land \neg \exists_{x_1}(A(x_1))##
• ##\exists_{x_1}(\exists_{x_2}(A(x_1,x_2)\land \exists_{x_3}(A(x_2,x_3)\land A(x_2,x_2))))\land \neg \exists_{x_1}(\exists_{x_2}(\exists_{x_3}( A(x_2,x_1)\land A(x_1,x_3)\land A(x_1,x_1))))##
• ##\exists_{x_1}(m(x_1) \land \neg\exists_{x_2}(M(x_1, x_2) \land \exists_{x_3}(M(x_1, x_3) \land \neg M(x_2, x_3) \land M(x_3, x_1)) \land \exists_{x_3}(M(x_1, x_3) \land M(x_2, x_3)) \land \neg \exists_{x_3}(M(x_2, x_3) \land M(x_3, x_3))))\land \exists_{x_1}(\exists_{x_2}(m(x_2)\land M(x_2,x_1)\land \exists_{x_3}(M(x_1,x_3)\land M(x_2,x_3))\land \exists_{x_3}(M(x_2,x_3)\land M(x_3,x_2)\land \neg M(x_1,x_3)))\land \neg \exists(M(x_2,x_1)\land M(x_2,x_2)))\lor\exists_{x_1}(M(x_1,x_1)\land \neg \exists_{x_2}(M(x_1,x_2)\land M(x_2,x_2)\land M(x_2,x_1)))##

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I have an idea for smt-solver algorithm myself. Some details are still unclear. It is quite long. I can not post it here very well. Were do you think I could find someone who were interested of it? Should I contact makers of z3?

Which mathematical problems z3 can not solve? (I am asking a few examples of simpler mathematical problems/tasks, that z3 can not solve)?

## 1. What is a logic solver in SMT?

A logic solver in SMT (Satisfiability Modulo Theories) is a computer program that is used to determine the satisfiability of a logical formula or set of constraints. It takes in a set of logical statements and checks whether there is a possible assignment of values to the variables that makes the formula true. If such an assignment exists, the formula is considered satisfiable.

## 2. How do SMT logic solvers work?

SMT logic solvers use a combination of decision procedures and heuristics to determine the satisfiability of logical formulas. Decision procedures are algorithms that can efficiently determine the truth value of a logical statement. Heuristics are used to guide the search for a satisfying assignment by making educated guesses about which values to assign to variables.

## 3. What types of problems can SMT logic solvers solve?

SMT logic solvers can be used to solve a wide range of problems, including boolean satisfiability (SAT) problems, constraint satisfaction problems, and problems involving combinations of arithmetic and logical constraints. They are particularly useful in formal verification, software testing, and automated reasoning tasks.

## 4. What are the advantages of using SMT logic solvers?

SMT logic solvers offer several advantages over other methods of solving logical formulas. They are highly efficient and can handle large and complex formulas. They also provide a high level of automation, reducing the need for manual intervention. Additionally, they can handle a variety of logical theories, making them versatile for various applications.

## 5. Are there any limitations to using SMT logic solvers?

While SMT logic solvers are powerful tools, they do have some limitations. They are not capable of solving all types of logical problems, such as undecidable problems or problems with infinite solutions. They also rely on heuristics, which can lead to incorrect solutions in some cases. Additionally, they may not always provide a satisfying assignment even when one exists, making it difficult to find the source of unsatisfiability in a formula.

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