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Journal ArticleDOI

Paramodulation with Non-Monotonic Orderings and Simplification

01 Jan 2013-Journal of Automated Reasoning (Springer Netherlands)-Vol. 50, Iss: 1, pp 51-98
TL;DR: A complete ordered paramodulation calculus for non-monotonic orderings which is compatible with powerful redundancy notions including demodulation is presented, hence strictly improving the previous results and making the calculus more likely to be used in practice.
Abstract: Ordered paramodulation and Knuth-Bendix completion are known to remain complete when using non-monotonic orderings. However, these results do not imply the compatibility of the calculus with essential redundancy elimination techniques such as demodulation, i.e., simplification by rewriting, which constitute the primary mode of computation in most successful automated theorem provers. In this paper we present a complete ordered paramodulation calculus for non-monotonic orderings which is compatible with powerful redundancy notions including demodulation, hence strictly improving the previous results and making the calculus more likely to be used in practice. As a side effect, we obtain a Knuth-Bendix completion procedure compatible with simplification techniques, which can be used for finding, whenever it exists, a convergent term rewrite system for a given set of equations and a (possibly non-totalizable) reduction ordering.

Summary (3 min read)

1 Introduction

  • Another source of the interest in dropping ordering requirements is that, in many cases, it is not clear if a particular ordering will be good (e.g. for reducing the search space) in some given problem.
  • The authors show that some redundancy notions w.r.t. (in particular, w.r.t. the reduction ordering r included in ) can be applied in this framework while keeping refutation completeness.
  • Now, if the authors want to add redundancy notions, the first natural choice is to define them w.r.t. the ordering that they use in the ordered paramodulation inference rules.

2 Preliminaries

  • There are some (quasi-)orderings that play a central role in their results (in particular, in the redundancy notions).
  • In order to ease the reading, sometimes the authors will denote marked terms by superindexing their marked subterms with variables.
  • The authors also define the following equivalence relation.
  • Lemma 4 ·>m is stable under substitutions.

4 Paramodulation with Equations

  • In the following the authors will assume that the marked terms of every equation do share the same substitution (if necessary, the substitutions can be extended).
  • Moreover, variables (both marking and non-marking) of each pair of equations will be considered to be disjoint (if necessary, the variables can be renamed).
  • The inference system E for E consists of the following single inference rule:.

Paramodulation:

  • Note that since the marking variables introduced by the previous inference rule are fresh, the authors are mantaining the invariant on disjointness of marking and non-marking variables.
  • Observe that, as a particular case, if both premises have no marks and the leftmost premise can be oriented with respect to the reduction ordering at hand, then condition 4a is always fulfilled, and hence no marks are introduced in the conclusion, making their inference rule coincide with the usual ordered paramodulation inference rule.
  • Here the authors introduce redundancy notions from a static point of view, that is, by first defining the notion of saturated set, regardless of how this saturated set can be obtained.
  • R. Notice that universally quantifying the TRS R allows us to capture the particular TRS RE (see Definition 11) defining the model, which cannot be known in advance.
  • The authors consider the case where sγθ is reducible (the other one is analogous).

5 General Clauses

  • Here the authors extend the presented calculus to general first order clauses, and prove it complete.
  • The authors consider that in each clause with a non-empty antecedent one of its negative equations, the one that is written underlined, has been selected.
  • The authors use the orderings defined in Section 3.2, and assume that all marked terms in the premises of an inference share a unique substitution.
  • The authors inference system G consists of the following four inference rules:.

Equality factoring:

  • The following example illustrates how the previous inference system works.
  • The authors will show that for every terminating ground TRS R and every ground substitution ρ such that Cρ and Dρ are non-topmost variable irreducible w.r.t.
  • In what follows the authors will precisely describe how they can apply forward simplification using marked rewriting.
  • First of all, the authors will show how the empty clause can be derived under their inference system (which introduces the convenient marks).

6 Knuth-Bendix Completion

  • Finding practically useful procedures remained as an open problem for a long time.
  • Also, Devie showed that for left- and right linear E (i.e., no variable occurs more than once in a side of an equation) standard Knuth-Bendix completion finds R (Devie (1990)).
  • But it was not 36 until in Bofill et al (1999, 2003) that a procedure for the general case, not relying on the enumeration of all equational consequences, was presented.
  • From their current result, it directly follows that the method of Bofill et al (2003) can be made compatible (to some amount) with simplification by rewriting.
  • Now the ordered paramodulation rule is also applied on top of the small sides, and hence the authors obtain an interreduced ground TRS for the model.

7 Experiments

  • In order to check if their ideas are feasible at least for small examples the authors have developed a prototype written in Prolog that implements for the equational case the inference system given in Section 4 and the practical notions of redundancy described in Section 5.5.
  • The second one defines a non-strict comparison.
  • The final west-ordering is obtained as the union of the reduction ordering, the subterm relation and all instances of the given definitions.
  • When saturating a set of equations, first of all, the system checks whether some of the equations can be oriented with respect to the reduction ordering, which allows us to make 38 some simplification initially.
  • As explained throughout the paper, these marks have the effect of diminishing the amount of redundancy, as marked subterms are treated as variables for redundancy purposes.

8 Building West Orderings

  • There are some known ways to build west orderings that are not monotonic.
  • Still, the authors have monotonicity for some comparisons, and a monotonic subrelation of a given SPO can be extracted.
  • This is exploited in the method called the Monotonic Semantic Path Ordering, (Borralleras et al (2000)), MSPO for short, which can be used to define the monotonic part of the SPO, and hence a reduction ordering inside the west ordering.
  • Note that case 2 can as well be written the other way round, i.e. with f ∈FP, g /∈FP, preserving well-foundedness.

9 Conclusion

  • By adapting those refutation complete inference systems so that they are compatible with powerful redundancy elimination techniques such as demodulation and, hence, making them more likely to be used in practice.the authors.
  • The authors have proposed some inference systems (for equations and general first order clauses) that work with a pair of orderings: a west ordering , which is used when performing inferences, and a reduction ordering r, included in , which is used for applying simplification by rewriting.
  • As stated in Section 5.5, in their notions of redundancy it is necessary that non-marking variables occurring in the skeleton of a marked term are only instantiated with variables when performing redundancy.
  • The authors results on Knuth-Bendix completion could also be extended in the line of the results on completion with termination tools in Wehrman et al (2006); Winkler and Middeldorp (2010).
  • Note that, in their case fixing in advance the non-monotonic ordering and the reduction ordering included in it (using techniques like the ones given in Section 8) is even harder than in the standard case, where only a reduction ordering is needed.

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jar manuscript No.
(will be inserted by the editor)
Paramodulation with Non-Monotonic Orderings and
Simplification
Miquel Bofill · Albert Rubio
Received: date / Accepted: date
Abstract Ordered paramodulation and Knuth-Bendix completion are known to remain com-
plete when using non-monotonic orderings. However, these results do not imply the com-
patibility of the calculus with essential redundancy elimination techniques such as demod-
ulation, i.e., simplification by rewriting, which constitute the primary mode of computation
in most successful automated theorem provers.
In this paper we present a complete ordered paramodulation calculus for non-monotonic
orderings which is compatible with powerful redundancy notions including demodulation,
hence strictly improving the previous results and making the calculus more likely to be used
in practice.
As a side effect, we obtain a Knuth-Bendix completion procedure compatible with sim-
plification techniques, which can be used for finding, whenever it exists, a convergent term
rewrite system for a given set of equations and a (possibly non-totalizable) reduction order-
ing.
Keywords automated theorem proving · equational reasoning · ordered paramodulation ·
Knuth-Bendix completion
This work has been partially supported by the Spanish MEC/MICINN under grants TIN 2008-04547 and TIN
2010-68093-C02-01. A preliminary version of this work was presented at the IJCAR 2004 conference (Bofill
and Rubio (2004)).
Miquel Bofill
Universitat de Girona
Dept. Inform
`
atica i Matem
`
atica Aplicada
Campus Montilivi, Ed. P4
E-17071 Girona, Spain
Tel.: +34-972418838
Fax: +34-972418792
E-mail: mbofill@ima.udg.edu
Albert Rubio
Universitat Polit
`
ecnica de Catalunya
Dept. Llenguatges i Sistemes Inform
`
atics
C/ Jordi Girona, 1-3
E-08034 Barcelona, Spain
E-mail: rubio@lsi.upc.edu

2
1 Introduction
Knuth-Bendix-like completion techniques and their extensions to ordered paramodulation
for first-order clauses are among the most successful methods for automated deduction with
equality (Bachmair and Ganzinger (1998); Nieuwenhuis and Rubio (2001)). For many years,
all known completeness results for Knuth-Bendix completion and ordered paramodulation
required the term ordering to be well-founded, monotonic and total (or extendable to a to-
tal ordering) on ground terms (Hsiang and Rusinowitch (1991); Bachmair et al (1986); Bach-
mair and Dershowitz (1994); Bachmair and Ganzinger (1994)). In Bofill et al (1999, 2003),
the monotonicity requirement was dropped and well-foundedness and the subterm property
were shown to be sufficient for ensuring refutation completeness of ordered paramodula-
tion (notice that any such ordering can be totalized without losing these two properties).
And in Bofill and Rubio (2002, 2009) it was shown that well-foundedness of the order-
ing suffices for completeness of ordered paramodulation for Horn clauses, i.e., the subterm
property can be dropped as well.
Apart from its theoretical value, these results have several potential applications in con-
texts where the usual requirements are too strong. For example, in deduction modulo built-in
equational theories E, where E-compatibility of the ordering (i.e., s =
E
s
0
t
0
=
E
t im-
plies s t) is needed, finding E-compatible orderings fulfilling the required properties is
extremely complex or even impossible. For instance, when E contains an idempotency ax-
iom f (x,x) ' x, no total E-compatible reduction ordering exists: if s t, then by mono-
tonicity one should have f (s,s) f (s,t) which, by E-compatibility, implies s f (s,t) and
hence non-well-foundedness. Therefore, the techniques for dropping ordering requirements,
among other applications, open the door to deduction modulo many more classes of equa-
tional theories.
Another source of the interest in dropping ordering requirements is that, in many cases,
it is not clear if a particular ordering will be good (e.g. for reducing the search space) in
some given problem. Hence, broadening the range of usable orderings can be helpful in
practice. Indeed, there exist examples of problems for which (unfailing) Knuth-Bendix style
procedures only terminate if we choose a reduction ordering which is not extendable to a
total one.
Example 1
1
Consider the closure under standard Knuth-Bendix completion of the following two
rules:
h(x) f g(x)
h f g(x) f f g(x)
Between these two rules there is a single critical pair since h f g(z) can be rewritten into both
f f g(z) and f g f g(z). Therefore in a Knuth-Bendix completion the equation
f f g(z) ' f g f g(z)
should be added. If we work with a well-founded and monotonic total ordering then the
previous equation is necessarily oriented into
f g f g(x) f f g(x).
1
We thank Christopher Lynch for providing us with this example.

3
(Notice that, otherwise, we will contradict either well-foundedness or monotonicity
or totality on ground terms, because if f f g(x) f g f g(x) then f g(x) and g f g(x)
must be incomparable in any well-founded and monotonic extension of :
If f g(x) g f g(x) then it contradicts the subterm relation, and then by mono-
tonicity we can get an infinite decreasing sequence f g(x) g f g(x) gg f g(x)
·· ·
If g f g(x) f g(x) then by monotonicity we have f g f g(x) f f g(x), leading to
reflexivity and hence to the existence of an infinite decreasing sequence.)
Unfortunately, with the “right” orientation f g f g(x) f f g(x), standard Knuth-Bendix com-
pletion would generate an infinite set of rules of the form
f g f
n
g(x) f
n+1
g(x) with n 1.
However, if we take the unusual orientation, i.e.,
f f g(x) f g f g(x),
the system is already closed, i.e., standard Knuth-Bendix completion would generate only
the additional rule
f f g(x) f g f g(x).
Recall that, as shown above, with this orientation the ordering cannot be extended to
a total one. Notice also that, even if we compute inferences with left-hand sides into right-
hand sides (in the sense of ordered paramodulation) the system would be closed after adding
f f g(x) f g f g(x) to the set. ut
By now we have motivated the interest of dropping ordering requirements. However,
ordered strategies are useful in practice only if compatibility with redundancy is shown.
Simplification of formulae and elimination of redundant formulae are essential components
of automated theorem provers. In fact, in most successful automated theorem provers, sim-
plification is the primary mode of computation, whereas prolific deduction rules are used
only sparingly.
In this direction, here we present a paramodulation based calculus which strictly im-
proves the one in Bofill et al (2003), for which refutation completeness was shown for
non-totalizable reduction orderings, but compatibility with simplification techniques was
left open. On the one hand, regarding the amount of inferences needed to be performed,
the inference system presented here is essentially the same as the one in Bofill et al (2003),
but, on the other hand, this calculus is compatible with powerful redundancy notions which
include demodulation, i.e., simplification by rewriting. Also, as in Bofill et al (2003), we
can apply our results to obtain a Knuth-Bendix style completion procedure, but in this case
compatible with simplification techniques. This procedure can be used for finding, when-
ever it exists, a convergent TRS for a given set of equations and a (possibly non-totalizable)
reduction ordering.
In our calculus, it is assumed that equations are oriented w.r.t. a possibly non-monotonic
ordering which is an extension of a known reduction ordering
r
. As in Bofill et al (2003),
the ordering is required to (i) be well-founded, (ii) fulfill the subterm property and (iii) be
total on ground terms. In the case that is defined on first order terms, we require it to be
stable under substitutions. As shown in Bofill et al (2003), every reduction ordering can be
extended to a total ordering fulfilling these properties (at the expense, however, of possibly
losing monotonicity).

4
We show that some redundancy notions w.r.t. (in particular, w.r.t. the reduction or-
dering
r
included in ) can be applied in this framework while keeping refutation com-
pleteness. In the ground case, if all equations involved in the saturation
2
process turn out
to be orientable with
r
, then demodulation can be fully applied. But, in general, in order
to preserve refutation completeness we must impose some limitations to the terms that can
be simplified. During our saturation process, we mark out some subterms of the clauses,
which become blocked for demodulation (technically, the marked subterms are interpreted
as variables for redundancy purposes). Roughly, the idea is to mark out the terms that are
introduced in the conclusion of an inference that are not smaller w.r.t.
r
than some term in
the maximal premise. Also, some variables need to be marked when performing redundancy
steps (as explained in Section 5.5). In fact, as shown in Example 6 of Section 5.5, refutation
completeness can be lost when applying paramodulation w.r.t. a non-monotonic ordering
extending a reduction ordering
r
, together with demodulation w.r.t.
r
, if no blockings are
introduced at all.
The reason for adding the blockings also has technical roots, coming from the technique
used in the completeness proof, which is based on the model generation technique of Bach-
mair and Ganzinger (1994) and its variant used in Bofill et al (2003). In the latter, in contrast
to the former, the ordering used for orienting the equations and the ordering used for induc-
tion in the completeness proof do not need to coincide. Concretely, equations are oriented
w.r.t. a (possibly) non-monotonic ordering , whereas completeness is proved by induction
w.r.t. a rewrite (and hence monotonic) relation
+
R
, where R is a limit ground rewrite sys-
tem, built up from a subset of equations in the closure. The fact that these two orderings do
not need to coincide is the key for the completeness proof of ordered paramodulation with
non-monotonic orderings given in Bofill et al (2003). Now, if we want to add redundancy
notions, the first natural choice is to define them w.r.t. the ordering that we use in the
ordered paramodulation inference rules. However, since completeness is proved by induc-
tion w.r.t.
+
R
, redundancy notions should be defined w.r.t.
+
R
as well. Unfortunately,
+
R
is unknown during the saturation process, and moreover it is not clear at all how it can be
approximated sufficiently. This is why in Bofill et al (2003) it was left open to what extent
demodulation could be applied in this setting (although some practical redundancy notions
such as tautology deletion and subsumption were shown to preserve completeness).
As said, here our aim is mainly to add demodulation w.r.t. the reduction ordering
r
,
which is included in the (possibly non-monotonic) extension used to orient the equations.
The idea for doing that is to find a well-founded ordering
R
which, roughly, combines
with
+
R
, and then prove completeness by induction w.r.t.
R
. The problem is that, although
r
is a well-founded and monotonic ordering and R is a terminating TRS whose rules are
included in a well-founded extension of
r
, the relation
R
is not well-founded in
general and, in fact,
+
R
can even contradict . Take, e.g., f (b)
r
f (a) and a well-founded
extension of
r
such that f (b) f (a) a b. Then possibly a b R (since the rules
in R are oriented w.r.t. ) and hence the infinite sequence f (a)
R
f (b) f (a)· ·· can be
built.
The idea to circumvent this non-well-foundedness problem is to block the terms that are
introduced by a rewriting step with R that is not included in . In the previous example,
since f (b) f (a), then we rewrite f (a) with a b into f (b) with b blocked, and we
consider that f (b) with b blocked is no longer greater than f (a) w.r.t. . Then, roughly,
if the comparisons with do not take into account the blocked terms, we can combine
2
The saturation of a set of clauses S amounts to the closure of S under the inference system up to redun-
dancy.

5
with
R
in a well-founded way. Altogether, it gives us some amount of redundancy w.r.t.
, while preserving refutation completeness.
A convenient way to represent terms with blocked positions is by means of superindexed
subterms, also called marked terms. For example f (b
x
), where x is a variable, denotes the
term f (b) where the subterm b is blocked. Then, although for performing inferences f (b
x
)
still corresponds to the term f (b), for the redundancy notions it will be seen as f (x), with a
blocked term b in x. In this context f (b
x
) cannot be simplified with f (b) f (a), since f (b)
does not match f (x).
Marked terms resemble the term closures of Bachmair et al (1995). However, their se-
mantics is fairly different: here the blockings only have effect on the redundancy notions
since, although marked terms are seen as variables for redundancy purposes, the ordered
paramodulation inferences are applied at both blocked
3
and unblocked positions. The terms
to be marked will come mainly from non-reductive inferences. Therefore, the more equa-
tions can be handled by the reduction ordering, the less terms will be blocked in the conclu-
sions of the inferences, and the more redundancy will be possible.
The rest of the paper is structured as follows. Preliminaries are presented in Section 2.
Marked terms and orderings on marked terms are defined in Section 3. In Section 4, our
calculus, including the notion of redundancy of inferences, is presented for deduction with
sets of equations. It is extended to general first order clauses in Section 5 where, moreover,
saturation derivations including redundancy of clauses are considered. In Section 6 we show
how, from our results, a Knuth-Bendix completion procedure for finding convergent TRSs
can be obtained. In Section 7 we comment on some experiments. In Section 8 we describe
a way of defining the kind of non-monotonic orderings that we need in this paper, which
could be easily automated. Finally in Section 9 we conclude.
2 Preliminaries
2.1 Terms, Equations and (Equality) Clauses
We use the standard definitions of Nieuwenhuis and Rubio (2001). T (F ,X ) (T (F )) is
the set of (ground) terms over a set of symbols F and a denumerable set of variables X
(over F ). By Var(t) we denote the set of variables occurring in a term t. The subterm of
t at position p is denoted by t|
p
, the result of replacing t|
p
by s in t is denoted by t[s]
p
,
and syntactic equality of terms is denoted by . A context t[ ]
p
is a term t with a hole
at a distinguished position p. A substitution is a partial mapping from variables to terms.
The application of a substitution σ to a term t is denoted by tσ. The composition of two
substitutions σ
1
and σ
2
, denoted by juxtaposition, is defined as the composition of two
functions, that is, tσ
1
σ
2
= (tσ
1
)σ
2
. The substitution σ|
A
is the substitution σ restricted to
the variables of A.
An equation is a multiset of terms {s,t}, denoted s ' t or, equivalently, t ' s. A first-
order clause is a pair of finite multisets of equations Γ (the antecedent) and (the succe-
dent), denoted by Γ . Equations in Γ are called negative literals and equations in
are called positive literals. A Horn clause is a clause with at most one positive literal. The
empty clause is a clause where both Γ and are empty.
3
All the positions under a marked term are considered as blocked for redundancy.

Citations
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TL;DR: Refutationally complete superposition calculi for intentional and extensional \(\lambda \)-free higher-order logic, two formalisms that allow partial application and applied variables, appear promising as a stepping stone towards complete, efficient automatic theorem provers for full higher- order logic.
Abstract: We introduce refutationally complete superposition calculi for intentional and extensional \(\lambda \)-free higher-order logic, two formalisms that allow partial application and applied variables. The calculi are parameterized by a term order that need not be fully monotonic, making it possible to employ the \(\lambda \)-free higher-order lexicographic path and Knuth–Bendix orders. We implemented the calculi in the Zipperposition prover and evaluated them on TPTP benchmarks. They appear promising as a stepping stone towards complete, efficient automatic theorem provers for full higher-order logic.

34 citations


Cites background from "Paramodulation with Non-Monotonic O..."

  • ...Also of interest is Bofill and Rubio’s [22] integration of nonmonotonic orders in ordered paramodulation, a precursor of superposition....

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Abstract: We generalize the Knuth–Bendix order (KBO) to higher-order terms without \(\lambda \)-abstraction. The restriction of this new order to first-order terms coincides with the traditional KBO. The order has many useful properties, including transitivity, the subterm property, compatibility with contexts (monotonicity), stability under substitution, and well-foundedness. Transfinite weights and argument coefficients can also be supported. The order appears promising as the basis of a higher-order superposition calculus.

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  • ...Even without this property, we expect the orders to be usable in a λ-free higher-order generalization of superposition, possibly at the cost of some complications [19]....

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Posted ContentDOI
TL;DR: The authors introduce refutationally complete superposition calculi for intentional and extensional clausal higher-order logic, two formalisms that allow partial application and applied variables, and implement them in the Zipperposition prover and evaluated them on Isabelle/HOL and TPTP benchmarks.
Abstract: We introduce refutationally complete superposition calculi for intentional and extensional clausal $\lambda$-free higher-order logic, two formalisms that allow partial application and applied variables. The calculi are parameterized by a term order that need not be fully monotonic, making it possible to employ the $\lambda$-free higher-order lexicographic path and Knuth-Bendix orders. We implemented the calculi in the Zipperposition prover and evaluated them on Isabelle/HOL and TPTP benchmarks. They appear promising as a stepping stone towards complete, highly efficient automatic theorem provers for full higher-order logic.

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Book ChapterDOI
22 Apr 2017
TL;DR: This new order fully coincides with the standard RPO on first-order terms also in the presence of currying, distinguishing it from previous work and appears promising as the basis of a higher-order superposition calculus.
Abstract: We generalize the recursive path order RPO to higher-order terms without $$\lambda $$-abstraction. This new order fully coincides with the standard RPO on first-order terms also in the presence of currying, distinguishing it from previous work. It has many useful properties, including well-foundedness, transitivity, stability under substitution, and the subterm property. It appears promising as the basis of a higher-order superposition calculus.

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  • ...Nonetheless, we expect the order to be usable for λ-free higher-order superposition, at the cost of some complications [13]....

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TL;DR: In this paper, a paramodulation calculus for equational theorem proving modulo E with constrained clauses is presented, and an inference rule called generalized E-parallel for constrained clauses, which makes their inference system completely basic, meaning that they do not need to allow any paramodulations in the constraint part of a constrained clause for refutational completeness.
Abstract: Unlike other methods for theorem proving modulo with constrained clauses [12, 13], equational theorem proving modulo with constrained clauses along with its simplification techniques has not been well studied. We introduce a basic paramodulation calculus modulo equational theories E satisfying certain properties of E and present a new framework for equational theorem proving modulo E with constrained clauses. We propose an inference rule called Generalized E-Parallel for constrained clauses, which makes our inference system completely basic, meaning that we do not need to allow any paramodulation in the constraint part of a constrained clause for refutational completeness. We present a saturation procedure for constrained clauses based on relative reducibility and show that our inference system including our contraction rules is refutationally complete.

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01 Jan 1990
TL;DR: In this paper, the authors focus on rewrite systems, which are directed equations used to compute by repeatedly replacing sub-terms of a given formula with equal terms until the simplest form possible is obtained.
Abstract: Publisher Summary This chapter focuses on rewrite systems, which are directed equations used to compute by repeatedly replacing sub-terms of a given formula with equal terms until the simplest form possible is obtained. As a formalism, rewrite systems have the full power of Turing machines and may be thought of as nondeterministic Markov algorithms over terms rather than strings. The theory of rewriting is in essence a theory of normal forms. To some extent, it is an outgrowth of the study of A. Church's Lambda Calculus and H. B. Curry's Combinatory Logic. The chapter discusses the syntax and semantics of equations from the algebraic, logical, and operational points of view. To use a rewrite system as a decision procedure, it must be convergent. The chapter describes this fundamental concept as an abstract property of binary relations. To use a rewrite system for computation or as a decision procedure for validity of identities, the termination property is crucial. The chapter presents the basic methods for proving termination. The chapter discusses the question of satisfiability of equations and the convergence property applied to rewriting.

1,551 citations

Book
02 Jan 1991
TL;DR: In this article, the authors focus on rewrite systems, which are directed equations used to compute by repeatedly replacing sub-terms of a given formula with equal terms until the simplest form possible is obtained.
Abstract: Publisher Summary This chapter focuses on rewrite systems, which are directed equations used to compute by repeatedly replacing sub-terms of a given formula with equal terms until the simplest form possible is obtained. As a formalism, rewrite systems have the full power of Turing machines and may be thought of as nondeterministic Markov algorithms over terms rather than strings. The theory of rewriting is in essence a theory of normal forms. To some extent, it is an outgrowth of the study of A. Church's Lambda Calculus and H. B. Curry's Combinatory Logic. The chapter discusses the syntax and semantics of equations from the algebraic, logical, and operational points of view. To use a rewrite system as a decision procedure, it must be convergent. The chapter describes this fundamental concept as an abstract property of binary relations. To use a rewrite system for computation or as a decision procedure for validity of identities, the termination property is crucial. The chapter presents the basic methods for proving termination. The chapter discusses the question of satisfiability of equations and the convergence property applied to rewriting.

1,381 citations

Journal ArticleDOI
TL;DR: Methods of proving that a term-rewriting system terminates are presented, based on the notion of "simplification orderings", orderings in which any term that is homeomorphically embeddable in another is smaller than the other.

630 citations