Complexity of the unification algorithm for first-order expressions

CALCOLO ◽  
1975 ◽  
Vol 12 (4) ◽  
pp. 361-371 ◽  
Author(s):  
M. Venturini Zilli
Keyword(s):  
Unification Algorithm ◽  
First Order

scholarly journals A Resolution Calculus for Second-order Logic with Eager Unification

10.29007/zpg2 ◽  
2018 ◽  
Author(s):  
Alexander Leitsch ◽  
Tomer Libal
Keyword(s):  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Unification Algorithm ◽  
Efficient Search ◽  
First Order ◽  
Unification Algorithms ◽  
Second Order Logic ◽  
Resolution Calculus

The efficiency of the first-order resolution calculus is impaired when lifting it to higher-order logic. The main reason for that is the semi-decidability and infinitary natureof higher-order unification algorithms, which requires the integration of unification within the calculus and results in a non-efficient search for refutations.We present a modification of the constrained resolution calculus (Huet'72) which uses an eager unification algorithm while retaining completeness. Thealgorithm is complete with regard to bounded unification only, which for many cases, does not pose a problem in practice.

scholarly journals Unification of Higher-order Patterns modulo Simple Syntactic Equational Theories

2000 ◽  
Vol Vol. 4 no. 1 ◽  
Author(s):  
Alexandre Boudet
Keyword(s):  
Higher Order ◽  
Unification Algorithm ◽  
First Order ◽  
Equational Theories ◽  
Order Case ◽  
International Audience ◽  
Syntactic Theories

International audience We present an algorithm for unification of higher-order patterns modulo simple syntactic equational theories as defined by Kirchner [14]. The algorithm by Miller [17] for pattern unification, refined by Nipkow [18] is first modified in order to behave as a first-order unification algorithm. Then the mutation rule for syntactic theories of Kirchner [13,14] is adapted to pattern E-unification. If the syntactic algorithm for a theory E terminates in the first-order case, then our algorithm will also terminate for pattern E-unification. The result is a DAG-solved form plus some equations of the form λ øverlinex.F(øverlinex) = λ øverlinex. F(øverlinex^π ) where øverlinex^π is a permutation of øverlinex When all function symbols are decomposable these latter equations can be discarded, otherwise the compatibility of such equations with the solved form remains open.

Polytypic unification

1998 ◽  
Vol 8 (5) ◽  
pp. 527-536 ◽  
Author(s):  
PATRIK JANSSON ◽  
JOHAN JEURING
Keyword(s):  
Programming Languages ◽  
Large Class ◽  
Pattern Matching ◽  
Type Inference ◽  
Logic Programs ◽  
Unification Algorithm ◽  
First Order ◽  
Unification Algorithms

Unification, or two-way pattern matching, is the process of solving an equation involving two first-order terms with variables. Unification is used in type inference in many programming languages and in the execution of logic programs. This means that unification algorithms have to be written over and over again for different term types. Many other functions also make sense for a large class of datatypes; examples are pretty printers, equality checks, maps etc. They can be defined by induction on the structure of user-defined datatypes. Implementations of these functions for different datatypes are closely related to the structure of the datatypes. We call such functions polytypic. This paper describes a unification algorithm parametrised on the type of the terms, and shows how to use polytypism to obtain a unification algorithm that works for all regular term types.

scholarly journals Is it possible to unify sequential programs?

10.29007/77z3 ◽  
2018 ◽  
Author(s):  
Tatyana Novikova ◽  
Vladimir Zakharov
Keyword(s):  
Polynomial Time ◽  
Functional Equivalence ◽  
Order Model ◽  
Sequential Programs ◽  
Unification Algorithm ◽  
First Order ◽  
Unification Problem ◽  
Set Up

We introduce a first-order model of imperative sequential programs and set up formally the unification problem in this model: given a pair of programs π<sub>1</sub> and π<sub>2</sub> find a pair of substitutions (θ<sub>1</sub>,θ<sub>2</sub>) such that the instances π<sub>1</sub>θ<sub>1</sub> and π<sub>2</sub>θ<sub>2</sub> of these programs are equivalent, i.e. compute the same function. Since functional equivalence of programs is undecidable, we choose its decidable approximation --- a strong equivalence, --- which is well-known in theory of program schemata. Our main result is a polynomial time unification algorithm for sequential programs w.r.t. strong equivalence of programs.

A UNIFICATION ALGORITHM FOR THE λΠ-CALCULUS

1992 ◽  
Vol 03 (03) ◽  
pp. 333-378 ◽  
Author(s):  
DAVID PYM
Keyword(s):  
Lambda Calculus ◽  
Similar Type ◽  
Unification Algorithm ◽  
First Order ◽  
Typed Lambda Calculus ◽  
Dependent Function ◽  
Suitable Notion ◽  
Taking Care

We present a unification algorithm for the λΠ-calculus, the lambda calculus with first-order dependent function types. Our algorithm is an extension of Huet’s algorithm for the simply-typed lambda calculus to first-order dependent function types. In the simply-typed lambda calculus one attempts to unify a pair of terms of the same type; in the λΠ-calculus types are dependent on terms so one must unify not only terms, but their types as well. Accordingly, we identify a suitable notion of similarity of types, and only attempt to unify a pair of terms of similar type: if the types are not similar then they are not unifiable. Since Huet’s algorithm does not enumerate all of the unifiers of a given pair of terms, the strategy of first unifying pairs of types — by identifying suitable pairs of subterms for unification — is not complete. Accordingly, we must unify terms and their types simultaneously, taking care to maintain all of the conditions that are necessary to ensure the well-formedness of the resulting calculated substitution. Our notion of substitution is an algebraic one: substitutions are morphisms of λΠ-contexts. However, in order to define our algorithm we must work with psubstitutions and pcontexts — substitutions and contexts, respectively, in which variables are replaced by terms of similar, not β(η)-equal, type.

scholarly journals Implementing Connection Calculi for First-order Modal Logics

10.29007/82m9 ◽  
2018 ◽  
Author(s):  
Jens Otten
Keyword(s):  
Modal Logics ◽  
Theorem Prover ◽  
Kripke Semantics ◽  
Unification Algorithm ◽  
First Order ◽  
Automated Theorem Prover ◽  
Problem Library ◽  
And Performance ◽  
Performance Results ◽  
First Order Modal Logic

An implementation of an automated theorem prover for first-order modal logic is presented that works for the constant, cumulative and varying domains of the modal logics D, T, S4 and S5. It is based on the (classical) connection calculus and uses prefixes (or world paths) and a prefix unification algorithm to capture the restrictions given by the Kripke semantics of the standard modal logics. This permits a modular and elegant treatment of the considered modal logics and yields an efficient implementation. Details of the calculus, the implementation and performance results on the QMLTP problem library are presented.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

Stochasting modeling of planetary ring systems

1984 ◽  
Vol 75 ◽  
pp. 461-469 ◽  
Author(s):  
Robert W. Hart
Keyword(s):  
Maximum Entropy ◽  
Ring System ◽  
The Sun ◽  
Ultimate State ◽  
Interparticle Interactions ◽  
Ring Systems ◽  
First Order ◽  
Planetary Ring ◽  
Insight Into

ABSTRACTThis paper models maximum entropy configurations of idealized gravitational ring systems. Such configurations are of interest because systems generally evolve toward an ultimate state of maximum randomness. For simplicity, attention is confined to ultimate states for which interparticle interactions are no longer of first order importance. The planets, in their orbits about the sun, are one example of such a ring system. The extent to which the present approximation yields insight into ring systems such as Saturn's is explored briefly.

Morphological studies of linear (AB)n multiblock copolymers and their blends

1992 ◽  
Vol 50 (1) ◽  
pp. 384-385
Author(s):  
Richard J. Spontak ◽  
Steven D. Smith ◽  
Arman Ashraf
Keyword(s):  
Electron Microscopy ◽  
Microphase Separation ◽  
Multiblock Copolymers ◽  
First Order ◽  
Morphological Studies ◽  
Transmission Electron ◽  
First Order Phase Transition ◽  
Covalently Bonded ◽  
Total Molecular Weight ◽  
First Order Phase

Block copolymers are composed of sequences of dissimilar chemical moieties covalently bonded together. If the block lengths of each component are sufficiently long and the blocks are thermodynamically incompatible, these materials are capable of undergoing microphase separation, a weak first-order phase transition which results in the formation of an ordered microstructural network. Most efforts designed to elucidate the phase and configurational behavior in these copolymers have focused on the simple AB and ABA designs. Few studies have thus far targeted the perfectly-alternating multiblock (AB)n architecture. In this work, two series of neat (AB)n copolymers have been synthesized from styrene and isoprene monomers at a composition of 50 wt% polystyrene (PS). In Set I, the total molecular weight is held constant while the number of AB block pairs (n) is increased from one to four (which results in shorter blocks). Set II consists of materials in which the block lengths are held constant and n is varied again from one to four (which results in longer chains). Transmission electron microscopy (TEM) has been employed here to investigate the morphologies and phase behavior of these materials and their blends.

Exact First Order Finite Element Modeling for Eddy Current NDE

1991 ◽  
Vol 3 (1) ◽  
pp. 235-253 ◽  
Author(s):  
L. D. Philipp ◽  
Q. H. Nguyen ◽  
D. D. Derkacht ◽  
D. J. Lynch ◽  
A. Mahmood
Keyword(s):  
Finite Element ◽  
Finite Element Modeling ◽  
Eddy Current ◽  
Element Modeling ◽  
First Order ◽  
Eddy Current Nde
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