Questions tagged [automated-theorem-proving]
Machine-checked, machine-generated or machine-verified proofs
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how to model time and digital circuits in theorem prover
Most papers that I have seen on proving digital logic focus only on the combinational circuits (ie using boolean logic). But real life circuits also have flip-flops, ie a time element an a clock. How ...
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How to start with verifying a formal proof programmatically?
I heard the terms "theorem provers" and "proof assistants" tossed around before (which I assumed to be the same up until a couple seconds ago), and also of Coq, Idris, Agda, TLA+ (...
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What is the complexity of theorem proving?
I'm learning some computer science and mathematics by myself. I know that proving theorems in ZFC is undecidable in general, but, is there a formal way to express how complex it is? Is it as complex ...
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Automated theorem provers using models
I am wondering if they are automated theorem provers which can take as input known models of the theory, for example to help discard statements which are not correct.
For example, if the statements ...
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Is there an SMT/SAT algorithm for General Predicate Logic (FOL)?
I'm learning how to write my own theorem prover. After skimming Decision Procedures (Kroening & Strichman, 2016), I didn't find any SMT algorithms for solving quantified n-ary predicate formulas. ...
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Can the Calculus of Constructions (without inductives) be used to axiomatize mathematics?
I'm aware that proof assistants like Coq and Agda are based on CIC rather than CoC because there is e.g. no inductive natural number type in CoC. Therefore, for example the proof that addition is ...
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MergeSort's merge function loop invariant
I am reading a proof of correctness for the MergeSort Algorithm. This is the code for the MergeSort and the Merge function:
The correctness of the MergeSort function is easy to prove since the two <...
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DFA and NFA Equivalence Proof
I'm taking a Theory of Computation class and we went over the proof to show that for any NFA there is an equivalent DFA, which I understand the proof fully in this case. But if it were in reverse, for ...
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Building non-classical logics in Agda & Coq
Is it possible to construct different systems of logic in Coq or Agda?
I ask because I'm interested in using a proof assistant to construct (and verify) theorems in things like many-valued logics, ...
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Automated Query Equivalence Solver (MongoDB)
The query-equivalence problem is undecidable. However there are theorem provers that attempt to solve instances of undecidable problems. I am curious how I could go about using an automatic theorem ...
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Can all of math be automated?
Is this correct? Here's my reasoning.
(Definition) Math is anything that can be proved from axioms.
(Assumption) All (finite length) axiom sets can be enumerated.
(Assumption) All (finite length ...
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How to transform an Abstract Syntax Tree (AST) to an Abstract Binding Tree (ABT)? (for machine learning fo theorem proving)
I was reading the HOList paper that applies Graph Neural Networks (GNNs) to the HOL Light (HOList) data set and benchmark for ML for theorem proving. They describe their results etc but there is no ...
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Need help understanding Knuth's proof that: The set of all pure words is well-ordered by the relation >
In the paper linked below by Knuth and Bendix, theorem 1:
The set of all pure words is well-ordered by the relation '$>$'
has a proof that I can't seem to follow despite staring at it all day. I ...
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Do FOL theorem provers accept axiom schemata?
Axiom schemata (such as ZFC) are, in a sense, infinite sets of axioms. Do the ATPs designed to work with FOL (such as Vampire) accept axiom schemata?
I looked in the Vampire "manual" briefly,...
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Automated reasoning with real numbers
I have a large number of equivalences which look like:
$(a \leq 0.54 \wedge b \geq 0.12) \vee (c \gt 0.98)$ $\Leftrightarrow$ $(x \leq 0.25) \vee (x \gt 0.91 \wedge y \geq 0.01)$
This is just an ...
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What were the shortcomings of Robinson's resolution procedure?
Paulson et alii. From LCF to Isabelle/HOL say:
Resolution for first-order logic, complete in principle but frequently disappointing in practice.
I think complete means they can proof any true ...
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Prove simple theorems in Haskell in automated way
I would like to prove in Haskell, whether in vanilla Haskell or using some libraries / tools, some simple theorems such as:
...
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Halting problem vs. automated theorem proving?
In the Theory of Computation tutorial offered by Complexity Tree (I just began the 2nd video), it talks about how the Halting problem was developed to show that mathematics could not be automated. ...
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on coq: Why is the proof complete after proving only for one induction when we have more than one variable?
So I'm learning coq. And I came across the proof for associativity in addition forall (a b c : nat)
Appearntly when we do ...
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How to get an element from an existential proposition in Type theory proof assistant (Lean prover)
I am trying to implement set theory in type theory from scratch, just for self pedagogical purposes. Specifically, I'm using the Lean Prover, and defining the element-of relation from scratch using ...
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How the kernel works in a LCF theorem prover
I heard that there is some mechanism in LCF-based theorem provers that only allows some functions to create values of type theorem. I believe these are based on abstract data types.
Could someone ...
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Do theorem provers demonstrate their own correctness?
I am not very well-versed in the world of theorem proving, much less automated theorem proving, so please correct me if anything I say or assume in my question is wrong.
Basically, my question is: are ...
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Review of Formal Verification and How to Apply it to Greenfield Project [duplicate]
Last year I looked heavily into Formal Verification, such as automated theorem proving, model checking, type systems, symbolic evaluation, and many others. I probably spent a few weeks or maybe months ...
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Is there a correspondence of steps between DPLL and sequent-calculus?
Is there a correspondence between the steps in using DPLL to find out that a formula in propositional logic is unsatisfiable and using sequent calculus to prove that its negation is valid?
And given ...
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Superposition calculus: greater vs greater-or-equal
Bachmair and Ganzinger 1991, 'Rewrite-Based Equational Theorem Proving With Selection and Simplification', specifies the criterion for using an equation as, by some appropriate ordering, ...
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Superposition calculus: Elimination of redundant atoms
Bachmair and Ganzinger (1991), 'Rewrite-Based Equational Theorem Proving With Selection and Simplification', section 5.2, 'Simplification and Deletion Techniques', page 17, 'Elimination of redundant ...
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Bachmair and Ganzinger, ordering of equations
Bachmair and Ganzinger (1991), 'Rewrite-Based Equational Theorem Proving With Selection and Simplification', page 4, defines an order on equations. (This is an arcane piece of machinery but a critical ...
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Roadmap to formal verification
I would like to learn about different approaches to formal verification of software programs that goes beyond what Wikipedia has to offer. Ideally one would not only get an overview but also ...
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Unification algorithm that directly finds multiple substitutions?
Systems of formal logic generally have inference rules that
require certain expressions to be syntactically the same
in multiple steps. Typically two steps are involved, as
for modus ponens, where ...
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Type theory based automated theorem prover?
I know that there exist type theory based proof-checker, and I know that there are logic/sequent-calculus based automated theorem provers.
But I haven’t heard of a type-theory based automated theorem ...
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Why aren’t (type theory based) automated theorem provers efficient?
Note: my main experience with theorem provers is with Type theory based proof-assistant, rather than an actual automated theorem prover.
It is obvious why naive automated theorem proving is ...
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Finding libraries of formalized mathematics
I want to formalize Interval Newton Methods in Isabelle. As a research, I have the curiosity of whether there is anything done in the field. For that, I need to know what mathematical contents have ...
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Are there any interesting terms in pure LF or $\lambda\Pi$?
In my searching, I've seen that if Church numerals are encoded in a dependently typed Lambda calculus, that we can't derive induction or that $0 \neq 1$.
I know that LF and the dependently typed ...
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Has the concept of using a hand checked simple theorem prover to validate more complex theorem provers been explored before?
More specifically, has anyone used a chain of theorem provers to validate a highly evolved theorem prover, starting with a very simple hand checked prover such that each new theorem prover is used to ...
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Differentiate arguments in SPASS prover formula
I have a formula in FOL:
$\forall x \exists y: B(x) \implies C(y)$
and in SPASS: forall([X], exists[Y], implies(B(X),C(Y)))
I want to check the formula: $\exists x:...
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What kind of math problems can be solved by automated theorem provers?
Can I prove following statements with using available automated theorem provers?
$(a+b)^2=a^2+b^2+2ab$.
If $ 11 \mid 2a-3b$, then $ 11 \mid 7a-5b $.
If $ ax^2+bx+c=0$, then $x=\frac{-b\pm\sqrt{b^2-...
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How to define function type in AGDA
The "function" type $\rightarrow$ is predefined in Agda. But how would one define it if it was not predefined? Specifically I am talking about $\rightarrow$ in:
...
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Modern presentation of Ackermann's "Solvable Cases?"
Ackermann's book "Solvable Cases of the Decision Problem" discusses decidable instances of first order logic, particularly monadic logic, and so called "equality formulas". However, the book is from ...
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Theorem Prover for complexity theoretic reductions
Can complexity theoretic reductions that lead to proofs of say NP completeness be formalized using an existing theorem prover such as Coq?
If so, can you provide an outline of how to formalize a ...
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Monadic Second Order Logic for Dummies
I am programmer with a grip on automata, but not on logic.
I read in papers that the two are very tightly related.
Deterministic Finite Automata (DFA), Tree Automata and Visibly Pushdown Automata are ...
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Why do TPTP Performance plots look like this?
CASC is the premier Automated Theorem Prover competition performed annually at the Conference on Automated Deduction (CADE). The 2017 event has finished on the 9th of August this year.
During this ...
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Procedure to automatically solve field theorems in a SMT solver
I'm working with the Welder proof assistant. Basically, this system uses basic inference rules to modify the goal one wants to proof. At a latter step, the modified goals are passed to a SMT solver to ...
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Sorts and constructors for modelling classes in a theorem prover
I have been working in the Welder theorem prover for some time now. But I'm confused in the way they handle data-types. I'm not familiar with the terminology of sorts and constructors. Here is how one ...
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What was the major breakthrough between Hoare-Floyd logic and Scott–Strachey semantics?
I'm reading through a commentary on Milner's "The use of machines to assist in rigorous proof" by Mike Gordon. In this paper, he explains how LCF was born from the ideas of denotational semantics by ...
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Verification condition in case of array theory
As far as i can understand the problem of checking safety property, can be reduce to solving for inductive invariants in VC's and there is following parts to it:
Generate VC's from the program.
Solve ...
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Formal specification; Logical formula
* Initial Question *
I'm trying to write a logical formula consists of three Boolean variable C1, C2, C3.
My program takes graph as input and checks properties about them. C1 represents presence of ...
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Is it possible to build short proofs of arbitrary folds over a huge list?
With the use of Merkle Trees, you can prove the presence of an element of a very big list, with an amount of information close to just logarithm of the size of the whole tree. Merkle proofs, thus, ...
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Why Church-encoded types aren't sufficient to express inductive proofs?
I've heard some claims that the calculus of constructions without inductive types isn't powerful enough to express proofs by induction. Is that correct? If so, why isn't the Church-encoding sufficient ...
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How to self-learn automated theorem proving? [duplicate]
Year 12 high school student here, but my ambition is set on Automated Theorem Proving. I wish to apply it to, of course, mathematics and computer science, and maybe if possible other applied subjects ...
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Framework or tools to generate theorem prover/solver/reasoner for new logic
I have new logic which has syntax and semantics in usual natural languages and I have to create theorem prover/solver/reasoner for this logic. Is there framework or tool set that can generate such ...
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