# Questions tagged [unification]

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### Why is unification so important to inference engines?

I am learning Automated Theorem Proving / SMT solvers / Proof Assistants by myself and post a series of questions about the process, starting here. I keep reading about the Unification Algorithm. ...
674 views

### Unification vs. SAT solver

I read on Wikipedia that unification is a process of solving the satisfability problem. At the same time, I know that such solvers are called "SAT solvers" or "SMT solvers". So, are they different ...
177 views

### Is there a general algorithm to fill holes in terms of the Calculus of Constructions?

Suppose that you extend the Calculus of Constructions with "holes" - i.e., incomplete pieces of code that you didn't fill yet. I wonder if there is an algorithm to fill those roles automatically. For ...
379 views

### Generating constraints to solve dependently-typed metavariables?

In dependent-types, Miller pattern unification is used to solve a decidable fragment of higher-order unification. This allows dependently-typed languages to contain metavariables or implicit arguments....
7k views

### Can someone clarify this unification algorithm?

I've been having trouble understanding a unification algorithm for first order logic, as I don't know what a compound expression is. I googled it, but found nothing relevant. I also don't know what a ...
237 views

### why nominal unification is a first-order unification?

In my understanding, if a unification solves equations of terms that are not higher-order terms, then it is a first-order unification. If a unification solves equations of terms that are higher-order ...
1k views

### What is a unifier?

Currently I read up on unifiers, however have some problem understanding its concept. Thus far I found an example of an equation: add(suc(x); y) $\stackrel{.}{=}$ add(y; suc(z)) and unifiers to it:...
314 views

### Unification — most specific unifier

In unification, given a set of equations, a standard problem is to compute a most general unifier (mgu). I am interested in a somewhat reversed problem. Imagine having a set of equations that do not ...
68 views

### E-Unification: āGoal seekingā pattern matching between directed trees

Preamble Suppose we have some symbols x, y, ... and wildcards (Ī¾), ...
2k views

### How can unifying 2 sentences in first-order logic result in a variable becoming 2 different things?

I'm working on a program which must use inference in first-order logic, and everything is working great except for 1 thing which I don't understand. The book I'm using, "Artificial Intelligence A ...
46 views

### Is unification over regular expression equations doable?

By way of example, suppose I know that $X + a = b + Y$ where $X$ and $Y$ are variables standing for regular expressions, then $(X, Y) = (b, a)$ is a solution to this set of equations. Generalizing ...
488 views

### Term Rewriting vs Unification

How is term rewriting different from unification, and what's the difference between term rewriting languages and logic programming, like Prolog?
923 views

### Skolemization with multiple arguments — how to unify

Edit: answerers keep finding (valid!) problems with my example. I'll try again. The older version is below the horizontal line. Thanks to Klaus below for pointing out the last problem. My ...
176 views

### Type inference of pair (product) types

I am looking into Hindler-Milney type system and I am trying to add support for the pair type. In Pierces book, he introduces special language constructs for creation of pairs and getting their ...
52 views

### “not provable”, what does this to do with unification?

I found one interesting point in nominal unification. Just after proposition 2.16 of Nominal Unification by Urban, Pitts, and Gabbay, it said the following, which I found confusing: For non-ground ...
200 views

### Why this pattern matching fails in Agda?

Consider function wa'' (need natural number definition, either in stdlib or Agda.Builtin.Nat): ...
121 views

### Are these examples of unification problems?

I have been studying unification, especially nominal unification (paper) gets my attention. I read the theory and examples. But I am wondering that what kind of problems occur in unifications. For ...
35 views

### Can high-order unification be applied to programming by example?

In 2007, it has been proven that high-order unification is decidable on the pattern matching case. If that is true, what is stopping someone to write an equation like: ...
58 views

### Is Unification “an Implementation of Existential Quantification”?

I read a comment (I've forgotten the source), "Unification is an implementation of existential quantification." (Emphasis mine.) If true, this point of view clears up many things. For instance why ...
37 views

### Unification Algorithm without Occur Check

I have been reading about Unification algorithm here https://en.wikipedia.org/wiki/Unification_(computer_science)#A_unification_algorithm . And I wonder about the importance of occur check. I know ...
50 views

### 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 ...
178 views

### Most efficient unification algorithm?

I've been trying to find the most efficient unification algorithm by following reference chains on the Wikipedia article on unification. As far as I can tell, the fastest one is the one presented in "...
109 views

### Unification algorithm - need clarification

I have these two terms: {P(a,x,x),P(a,b,c)} I'm supposed to find if the terms and unifiable using the unification algorithm. I'd do the following substitutions: b/x, resulting in : {P(a,b,b),P(a,b,...
20 views

### Need clarification in theorem about syntactic equations and substitutions

In the book "Constraint Handling Rules", Thom FrĆ¼hwirth explains (on p. 50) "Substitutions, variants and equality" over a usual first-order logic language composed of variables, function symbols and ...
51 views

### is this an example of nominal unification?

I am thinking that is the following a problem solved by nominal unification. $\lambda a.X = \lambda b.\lambda c.c$ where we find $X$. The answer is obvious. The reason is that it seems in nominal ...
When learning about a general unification algorithm, we learned the rule decompose, which states unifying $$G \cup \{f(a_0,...a_k)=f(b_0,...,b_k)\} \Rightarrow G \cup \{a_0=b_0,...a_k=b_k\}.$$ The ...