Questions tagged [unification]

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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 ...
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0answers
14 views

How can I compute the most general unifier for these two expressions?

I have the following first order logic expressions: $f(g(a, h(b)), g(x, y)),~f(g(z,y), g(y, y))$ and I want to compute the most general unifier for them. If I follow the algorithm found on these ...
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0answers
41 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 ...
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1answer
105 views

Why this pattern matching fails in Agda?

Consider function wa'' (need natural number definition, either in stdlib or Agda.Builtin.Nat): ...
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1answer
61 views

E-Unification: “Goal seeking” pattern matching between directed trees

Preamble Suppose we have some symbols x, y, ... and wildcards (ξ), ...
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0answers
126 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 "...
2
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0answers
87 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,...
2
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1answer
48 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 ...
2
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1answer
102 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 ...
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1answer
50 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 ...
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0answers
15 views

Need clarification in theorem about syntactic equations and substitutions

In the book "Constraint Handling Rules", Thom Frühwirth explains (on p. 50) "Substitutions, variants and equality" over a usual first-order logic language composed of variables, function symbols and ...
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1answer
128 views

Comparison Procedure in Robinson's Unification Algorithm

I'm studying the Principal Type (PT) Algorithm in Basic Simple Type Theory by J. Roger Hindley. One step to find the PT of a term is the Unification of types. The Robinson's Unification Algorithm uses ...
8
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1answer
148 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 ...
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0answers
150 views

what is a disagreement set in nominal unification?

In unification, for example set $s=\{ f(a,Y),f(X,d) \}$ has a disagreement set $\{ (a,X),(Y,d) \}$, since these two elements does not match on their fields. simply speaking that is a set disagreement ...
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1answer
46 views

occur-check, does nominal unification has it?

In unification, there is a "occur-check". Such as $X = a \, X$ fails to find a substitution for $X$ since it appears on right hand side too. The first-order unification, higher-order unification all ...
4
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1answer
1k 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 ...
6
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1answer
203 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 ...
6
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1answer
829 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:...
7
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2answers
338 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....
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1answer
68 views

Unification Functions

I need to apply the unification function to unify the following expression: foo(X,X,Y) and foo(Z,p(Z),w) So far, I've determined that I must substitute 'w's for occurances of 'Y', making foo(X,X,w) ...
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1answer
34 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: ...
3
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1answer
379 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?
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2answers
744 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 ...
1
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1answer
112 views

Unification — removing equations and updating the solution

This question is concerned with the first order unification. Suppose I have a set $D$ of equations and a solution to these equations. Let this solution be a set $S$ of substitutions. Now, suppose I ...
6
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3answers
6k 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 ...
4
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2answers
299 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 ...
3
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1answer
157 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 ...
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1answer
583 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 ...
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3answers
3k views

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. ...