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<Paper uid="P96-1001">
  <Title>Higher-Order Coloured Unification and Natural Language Semantics</Title>
  <Section position="4" start_page="0" end_page="3" type="metho">
    <SectionTitle>
3 Higher-Order Coloured
Unification (HOCU)
</SectionTitle>
    <Paragraph position="0"> There is a restricted form of HOU which allows for a natural modeling of DSP's Primary Occurrence Restriction: Higher-Order Coloured Unification developed independently for theorem proving (Hutter and Kohlhase, 1995). This framework uses a variant of the simply typed A-calculus where symbol occurrences can be annotated with so-called colours and substitutions must obey the following constraint: Given a labeling of occurrences as either primary or secondary, the POR excludes of the set of linguistically valid solutions, any solution which contains a primary occurrence. null For any colour constant c and any c-coloured variable V~, a well-formed coloured substitution must assign to Vc a cmonochrome term i.e., a term whose symbols are c-coloured.</Paragraph>
    <Paragraph position="1"> Here, a primary occurrence is an occurrence that is directly associated with a source parallel element. Neither the notion of direct association, nor that of parallelism is given a formal definition; but given an intuitive understanding of these notions, a source parallel element is an element of the source (i.e. antecedent) clause which has a parallel counterpart in the target (i.e. elliptic or anaphoric) clause. To see how this works, consider example (1) again. In this case, dan is taken to be a primary occurrence because it represents a source parallel element which is neither anaphoric nor controlled i.e. it is directly associated with a source parallel element.</Paragraph>
    <Paragraph position="2"> Given this, equation (lc) becomes (2a) with solutions (2b) and (2c) (primary occurrences are underlined). Since (2c) contains a primary occurrence, it is ruled out by the POR and is thus excluded from the set of linguistically valid solutions.</Paragraph>
    <Paragraph position="3">  (2) a. like(dan, golf)=R(dan) b. R = Ax.like(x, golf) c. R = Ax.like(dan, golf)  Although the intuitions underlying the POR are clear, two main objections can be raised. First, the restriction is informal and as such provides no good basis for a mathematical and computational evaluation. As DSP themselves note, a general theory for the POR is called for. Second, their method is a generate-and-test method: all logically valid solutions are generated before those solutions that violate the POR and are linguistically invalid are eliminated. While this is sufficient for a theoretical analysis, for actual computation it would be preferable never to produce these solutions in the first place.</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
3.1 Modeling the Primary Occurrence
Restriction
</SectionTitle>
      <Paragraph position="0"> Given this coloured framework, the POR is directly modelled as follows: Primary occurrences are pe-coloured whilst free variables are -~pe-coloured. For the moment we will just consider the colours pe (primary for ellipsis) and ~pe (secondary for ellipsis) as distinct basic colours to keep the presentation simple. Only for the analysis of the interaction of e.g.</Paragraph>
      <Paragraph position="1"> ellipsis with focus phenomena (cf. section 4.4) do we need a more elaborate formalization, which we will discuss there.</Paragraph>
      <Paragraph position="2"> Given the above restriction for well-formed coloured substitutions, such a colouring ensures that any solution containing a primary occurrence is ruled out: free variables are -~pe-coloured and must be assigned a -~pe-monochrome term. Hence no substitution will ever contain a primary occurrence (i.e. a pe-coloured symbol). For instance, discourse (la) above is assigned the semantic representation (3a) and the equation (3b) with unique solution (3c). In contrast, (3d) is not a possible solution since it assigns to an -~pe-coloured variable, a term containing a pe-coloured symbol i.e. a term that is not -~pemonochrome. null  (3) a. like(danpe,gol f) A R~pe(peter) b. like(danpe, golf)= R~pe(danpe) c. R~pe = Ax.like(x, golf) d. R~pe = Ax.like(danpe,gOl f)</Paragraph>
    </Section>
    <Section position="2" start_page="0" end_page="3" type="sub_section">
      <SectionTitle>
3.2 HOCU theory
</SectionTitle>
      <Paragraph position="0"> To be more formal, we presuppose a finite set g = {a, b, c, pe, -~pe,...) of colour constants and a  countably infinite supply ~ -- {A, B,...} of colour variables.</Paragraph>
      <Paragraph position="1"> As usual in A-calculus, the set wff of well-formed formulae consists of (coloured 1) constants ca,runs~,runsA,..., (possibly uncoloured) variables x, xa,yb,... (function) applications of the form MN and A-abstractions of the form Ax.M. Note that only variables without colours can be abstracted over. We call a formula M cmonochrome, if all symbols in M are bound or tagged with c.</Paragraph>
      <Paragraph position="2"> We will need the so-called colour erasure IMI of M, i.e. the formula obtained from M by erasing all colour annotations in M. We will also use various elementary concepts of the A-calculus, such as free and bound occurrences of variables or substitutions without defining them explicitly here. In particular we assume that free variables are coloured in all formulae occuring. We will denote the substitution of a term N for all free occurrences of x in M with \[N/x\]M.</Paragraph>
      <Paragraph position="3"> It is crucial for our system that colours annotate symbol occurrences (i.e. colours are not sorts!), in particular, it is intended that different occurrences of symbols carry different colours (e.g. f(xb, Xa)) and that symbols that carry different colours are treated differently. This observation leads to the notion of coloured substitutions, that takes the colour information of formulae into account. In contrast to traditional (uncoloured) substitutions, a coloured substitution a is a pair (at,at), where the term substitution a t maps coloured variables (i.e. the pair xc of a variable x and the colour c) to formulae of the appropriate type and the colour substitution a c maps colour variables to colours. In order to be legal (a g-substitution) such a mapping a must obey the following constraints: * If a and b are different colours, then \[a(xa)\[ = \[a(xb)\[, i.e. the colour erasures have to be equal. * If c E C is a colour constant, then a(xC/) is cmonochrome. null The first condition ensures that the colour erasure of a C-substitution is a well-defined classical substitution of the simply typed A-calculus. The second condition formalizes the fact that free variables with constant colours stand for monochrome subformulae, whereas colour variables do not constrain the substitutions. This is exactly the trait, that we will exploit in our analysis.</Paragraph>
      <Paragraph position="4"> 1Colours axe indicated by subscripts labeling term occurrences; whenever colours axe irrelevant, we simply omit them.</Paragraph>
      <Paragraph position="5"> Note that/37/-reduction in the coloured A-calculus is just the classical notion, since the bound variables do not carry colour information. Thus we have all the known theoretical results, such as the fact that/~/-reduction always terminates producing unique normal forms and that /3T/-equality can be tested by reducing to normal form and comparing for syntactic equality. This gives us a decidable test for validity of an equation.</Paragraph>
      <Paragraph position="6"> In contrast to this, higher-order unification tests for satisfiability by finding a substitution a that makes a given equation M = N valid (a(M) =~ a(N)), even if the original equation is not (M ~Z, N). In the coloured A-calculus the space of (semantic) solutions is further constrained by requiring the solutions to be g-substitutions. Such a substitution is called a C-unifier of M and N. In particular, C-unification will only succeed if comparable formulae have unifiable colours. For instance, introa (Pa, jb, Xa) unifies with introa (Ya, jA, Sa) but not with introa (Pa, ja, sa) because of the colour clash on j.</Paragraph>
      <Paragraph position="7"> It is well-known, that in first-order logic (and in certain related forms of feature structures) there is always a most general unifier for any equation that is solvable at all. This is not the case for higher-order (coloured) unification, where variables can range over functions, instead of only individuals. Fortunately, in our case we are not interested in general unification, but we can use the fact that our formulae belong to very restricted syntactic subclasses, for which much better results are known. In particular, the fact that free variables only occur on the left hand side of our equations reduces the problem of finding solutions to higher-order matching, of which decidability has been proven for the sub-class of third-order formulae (Dowek, 1992) and is conjectured for the general case. This class, (intuitively allowing only nesting functions as arguments up to depth two) covers all of our examples in this paper. For a discussion of other subclasses of formulae, where higher-order unification is computationally feasible see (Prehofer, 1994).</Paragraph>
      <Paragraph position="8">  Some of the equations in the examples have multiple most general solutions, and indeed this multiplicity corresponds to the possibility of multiple different interpretations of the focus constructions. The role of colours in this is to restrict the logically possible solutions to those that are linguistically sound.</Paragraph>
    </Section>
  </Section>
  <Section position="5" start_page="3" end_page="5" type="metho">
    <SectionTitle>
4 Linguistic Applications of the
POR
</SectionTitle>
    <Paragraph position="0"> In section 3.1, we have seen that HOCU allowed for a simple theoretical rendering of DSP's Primary Occurrence Restriction. But isn't this restriction fairly idiosyncratic? In this section, we show that the restriction which was originally proposed by DSP to model VP-ellipsis, is in fact a very general constraint which far from being idiosyncratic, applies to many different phenomena. In particular, we show that it is necessary for an adequate analysis of focus, second occurrence expressions and adverbial quantification.</Paragraph>
    <Paragraph position="1"> Furthermore, we will see that what counts as a primary occurrence differs from one phenomenon to the other (for instance, an occurrence directly associated with focus counts as primary w.r.t focus semantics but not w.r.t to VP-ellipsis interpretation).</Paragraph>
    <Paragraph position="2"> To account for these differences, some machinery is needed which turns DSP's intuitive idea into a fullyblown theory. Fortunately, the HOCU framework is just this: different colours can be used for different types of primary occurrences and likewise for different types of free variables. In what follows, we show how each phenomenon is dealt with. We then illustrate by an example how their interaction can be accounted for.</Paragraph>
    <Section position="1" start_page="3" end_page="4" type="sub_section">
      <SectionTitle>
4.1 Focus
</SectionTitle>
      <Paragraph position="0"> Since (Jackendoff, 1972), it is commonly agreed that focus affects the semantics and pragmatics of utterances. Under this perspective, focus is taken to be the semantic value of a prosodically prominent element. Furthermore, focus is assumed to trigger the formation of an additional semantic value (henceforth, the Focus Semantic Value or FSV) which is in essence the set of propositions obtained by making a substitution in the focus position (cf. e.g. (Kratzer, 1991)). For instance, the FSV of (4a) 2 is (4b), the set of formulae of the form l(j,x) where x is of type e, and the pragmatic effect of focus is to presuppose that the denotation of this set is under consideration. null (4) a. Jon likes SARAH b. {l(j,x) l x e wife} In (Gardent and Kohlhase, 1996), we show that HOU can successfully be used to compute the FSV of an utterance. More specifically, given (part of) an utterance U with semantic representation Sere and foci F1... F n, we require that the following equa-</Paragraph>
      <Paragraph position="2"> On the basis of the Gd value, we then define the FSV, written Gd, as follows: Definition 4.1 (Focus Semantic Value) Let Gd be of type ~ = ~k --~ t and n be the number of loci (n &lt; k), then the Focus Semantic Value derivable from Gd, written G---d, is {Gd(tl... t n) I ti e wife,}.</Paragraph>
      <Paragraph position="3"> This yields a focus semantic value which is in essence Kratzer's presupposition skeleton. For instance, given (4a) above, the required equation will be l(j, s) = Gd(s) with two possible values for Gd: Ax.l(j, x) and Ax.l(j, s). Given definition (4.1), (4a) is then assigned two FSVs namely  (5) a. Gd= {l(j,x) l x e Wife} b. G'--d = {l(j,s) l x ~ Wife}  That is, the HOU treatment of focus overgenerates: (5a) is an appropriate FSV, but not (5b). Clearly though, the POR can be used to rule out (5b) if we assume that occurrences that are directly associated with a focus are primary occurrences. To capture the fact that those primary occurrences are different from DSP's primary occurrences when dealing with ellipsis, we colour occurrences that are directly associated with focus (rather than a source parallel element in the case of ellipsis) pf. Consequently, we require that the variable representing the FSV be -~pf coloured, that is, its value may not contain any pf term. Under these assumptions, the equation for (4a) will be (6a) which has for unique  solution (6b).</Paragraph>
      <Paragraph position="4"> (6) a. l(j, Spf) = FSV~pf(Spf) b. FSV~pf = Ax.l(j, x)</Paragraph>
    </Section>
    <Section position="2" start_page="4" end_page="4" type="sub_section">
      <SectionTitle>
4.2 Second Occurrence Expressions
</SectionTitle>
      <Paragraph position="0"> A second occurrence expression (SOE) is a partial or complete repetition of the preceding utterance and is characterised by a de-accenting of the repeating part (Bartels, 1995). For instance, (Tb) is an SOE  whose repeating part only likes Mary is deaccented. (7) a. Jon only likes MARY.</Paragraph>
      <Paragraph position="1"> b. No, PETER only likes Mary.</Paragraph>
      <Paragraph position="2">  In (Gardent, 1996; Gardent et al., 1996) we show that SOEs are advantageously viewed as involving a deaccented anaphor whose semantic representation must unify with that of its antecedent. Formally, this is captured as follows. Let SSem and TSem be the semantic representation of the source and target clause respectively, and TP 1 ... TP n, SP 1 ... SP n be the target and source parallel elements 3, then the interpretation of an SOE must respect the following equations:</Paragraph>
      <Paragraph position="4"> Given this proposal and some further assumptions about the semantics of only, the analysis of (Tb) involves the following equations:  shared by target and source clause, the second solution is clearly incorrect given that it contains information (j) that is specific to the source clause. Again, the POR will rule out the incorrect solutions, whereby contrary to the VP-ellipsis case, all occurrences that are directly associated with parallel elements (i.e. not just source parallel elements) are taken to be primary occurrences. The distinction is implemented by colouring all occurrences that are directly associated with parallel element ps, whereas the corresponding free variable (An) is coloured as --ps. Given these constraints, the first equation in</Paragraph>
    </Section>
    <Section position="3" start_page="4" end_page="5" type="sub_section">
      <SectionTitle>
4.3 Adverbial quantification
</SectionTitle>
      <Paragraph position="0"> Finally, let us briefly examine some cases of adverbial quantification. Consider the following example from (von Fintel, 1995): Tom always takes SUE to Al's mother.</Paragraph>
      <Paragraph position="1"> Yes, and he always takes Sue to JO's mother.</Paragraph>
      <Paragraph position="2"> In (Gardent and Kohlhase, 1996), we suggest that such cases are SOEs, and thus can be treated as involving a deaccented anaphor (in this case, the anaphor he always takes Sue to _'s mother). Given some standard assumptions about the semantics of 3As in DSP, the identification of parallel elements is taken as given.</Paragraph>
      <Paragraph position="3">  always, the equations constraining the interpretation An of this anaphor are: An(al) = always (Tom take x to al's mother) (Tom take Sue to al's mother) An(jo) = always FSV (Tom take Sue to Jo's mother) Consider the first equation. If An is the semantics shared by target and source clause, then the only possible value for An is )~z.always (Tom take x to z's mother) (Tom take Sue to z's mother) where both occurrences of the parallel element m have been abstracted over. In contrast, the following  take x to z's mother.) (Tom take Sue to al's mother) Once again, we see that the POR is a necessary restriction: by labeling as primary, all occurrences representing a parallel element, it can be ensured that only the first solution is generated.</Paragraph>
    </Section>
    <Section position="4" start_page="5" end_page="5" type="sub_section">
      <SectionTitle>
4.4 Interaction of constraints
</SectionTitle>
      <Paragraph position="0"> Perhaps the most convincing way of showing the need for a theory of colours (rather than just an informal constraint) is by looking at the interaction of constraints between various phenomena. Consider the following discourse  (9) a. Jon likes SARAH b. Peter does too  Such a discourse presents us with a case of interaction between ellipsis and focus thereby raising the question of how DSP' POR for ellipsis should interact with our POR for focus.</Paragraph>
      <Paragraph position="1"> As remarked in section 3.1, we have to interpret the colour -~pe as the concept of being not primary for ellipsis, which includes pf (primary for focus). In order to make this approach work formally, we have to extend the supply of colours by allowing boolean combinations of colour constants. The semantics of these ground colour formula is that of propositional logic, where -~d is taken to be equivalent to the disjunction of all other colour constants.</Paragraph>
      <Paragraph position="2"> Consequently we have to generalize the second condition on C-substitutions * For all colour annotations d of symbols in a(xc) d ~ c in propositional logic.</Paragraph>
      <Paragraph position="3"> Thus X.d can be instantiated with any coloured formula that does not contain the colour d. The HOCU algorithm is augmented with suitable rules for boolean constraint satisfaction for colour equations. null The equations resulting from the interpretation of</Paragraph>
      <Paragraph position="5"> where the first equation determines the interpretation of the ellipsis whereas the second fixes the value of the FSV. Resolution of the first equation yields the value Ax.l(x, Spf) for R~pe. As required, no other solution is possible given the colour constralnts; in particular Ax.l(jpe, Spf) is not a valid solution. The value of R~pe(jpe) is now l(Ppe, 8pf) SO that the second equation is4:</Paragraph>
      <Paragraph position="7"> The first solution yields a narrow focus reading (only SARAH is in focus) whereas the second and the third yield wide focus interpretations corresponding to a VP and an S focus respectively. That is, not only do colours allow us to correctly capture the interaction of the two PORs restricting the interpretation of ellipsis of focus, they also permit a natural modeling of focus projection (cf. (Jackendoff, 1972)).</Paragraph>
    </Section>
  </Section>
  <Section position="6" start_page="5" end_page="6" type="metho">
    <SectionTitle>
5 Another constraint
</SectionTitle>
    <Paragraph position="0"> An additional argument in favour of a general theory of colours lies in the fact that constraints that are distinct from the POR need to be encoded to prevent HOU analyses from over-generating. In this section, we present one such constraint (the so-called weak-crossover constraint) and show how it can be implemented within the HOCU framework.</Paragraph>
    <Paragraph position="1"> In essence, the main function of the POR is to ensure that some occurrence occuring in an equation appears as a bound variable in the term assigned by substitution to the free variable occurring in this equation. However, there are cases where the dual 4Note that this equation falls out of our formal system in that it is untyped and thus cannot be solved by the algorithm described in section 6 (as the solutions will show, we have to allow for FSV and F to have different types). However, it seems to be a routine exercise to augment HOU algorithms that can cope with type variables like (Hustadt, 1991; Dougherty, 1993) with the colour methods from (Hutter and Kohlhase, 1995).</Paragraph>
    <Paragraph position="2">  constraint must be enforced: a term occurrence appearing in an equation must appear unchanged in the term assigned by substitution to the free variable occurring in this equation. The following example illustrates this.</Paragraph>
    <Paragraph position="3"> (Chomsky, 1976) observes that focused NPs pattern with quantified and wh-NPs with respect to pronominal anaphora: when the quantified/wh/focused NP precedes and c-commands the pronoun, this pronoun yields an ambiguity between a co-referential and a bound-variable reading. This is illustrated in example (10) We only expected HIMi to claim that he~ was brilliant where the presence of the pronoun hei gives rise to two possible FSVs s</Paragraph>
    <Paragraph position="5"> thus allowing two different readings: the corefential or strict reading VP\[P E {Ax.ex(x,y,i) I Y E Wife} A P(we) --+ P = Ax.ex(x, i, i)\] and the bound-variable or sloppy reading.</Paragraph>
    <Paragraph position="7"> In contrast, if the quantified/wh/focused NP does not precede and c-command the pronoun, as in (11) We only expected himi to claim that HEi was brilliant there is no ambiguity and the pronoun can only give rise to a co-referential interpretation. For instance, given (11) only one reading arises VP\[P E {Ax.ex(x,i,y) l Y E Wife} A P(we) ~ P = Ax.ex(x, i, i)\] where the FSV is {Ax.ex(x,i,y) l Y E wife}.</Paragraph>
    <Paragraph position="8"> To capture this data, Government and Binding analyses postulate first, that the antecedent is raised by quantifier raising and second, that pronouns that are c-commanded and preceded by their antecedent are represented either as a A-bound variable or as a constant whereas other pronouns can only be represented by a constant (cf. e.g. (Kratzer, 1991)'s binding principle). Using HOCU, we can model this restriction directly. As before, the focus term is pfand the FSV variable -~pf-coloured. Furthermore, we assume that pronouns that are preceded and c-commanded by a quantified/wh/focused antecedent are variable coloured whereas other pronouns are -~pf-coloured. Finally, all other terms are taken to 5We abbreviate exp( x, cl(y, blt( i) ) ) to ex( x, y, i) to increase legibility.</Paragraph>
    <Paragraph position="9"> be --pf-coloured. Given these assumptions, the representation for (10) is ex~o~(we~pf,ipf ,iA) and the corresponding FSV equation R~pf(ipf) -- )~x.eX~pf (x, ipf, in) has two possible solutions</Paragraph>
    <Paragraph position="11"> In contrast, the representation for (11) is ex-.pf(We~of, i~0f, ipf) and the equation is</Paragraph>
    <Paragraph position="13"> Importantly, given the indicated colour constraints, no other solutions are admissible. Intuitively, there are two reasons for this. First, the definition of coloured substitutions ensures that the term assigned to R~0f is -~pf-monochrome. In particular, this forces any occurrences of/of to appear as a bound variable in the value assigned to R~pf whereas in can appear either as i~0f (a colour variable unifies with any colour constant) or as a bound variable - this in effect models the sloppy/strict ambiguity. Second, a colour constant only unifies with itself. This in effect rules out the bound variable reading in (11): if the i~0f occurrence were to become a bound variable, the value of R~of would then Ay.)~x.ex~of(x, y, y) . But then by ~-reduction, R~of(ipf ) would be )~x.ex~of(x, iof,iof ) which does not unify with the right hand side of the original equation i.e ~x.ex.of(x , i-0f, i0f).</Paragraph>
    <Paragraph position="14"> For a more formal account of how the unifiers are calculated see section 6.1.</Paragraph>
  </Section>
  <Section position="7" start_page="6" end_page="8" type="metho">
    <SectionTitle>
6 Calculating Coloured Unifiers
</SectionTitle>
    <Paragraph position="0"> Since the HOCU is the principal computational device of the analysis in this paper, we will now try to give an intuition for the functioning of the algorithm. For a formal account including all details and proofs see (Hutter and Kohlhase, 1995).</Paragraph>
    <Paragraph position="1"> Just as in the case of unification for first-order terms, the algorithm is a process of recursive decomposition and variable elimination that transform sets of equations into solved forms. Since C-substitutions have two parts, a term- and a colour part, we need two kinds (M =t N for term equations and c =c d for colour equations). Sets g of equations in solved form (i.e. where all equations are of the form x = M such that the variable x does not occur anywhere else in M or g) have a unique most general C-unifier a~ that also C-unifies the initial equation.</Paragraph>
    <Paragraph position="2"> There are several rules that decompose the syntactic structure of formulae, we will only present two of them. The rule for abstractions transforms equations of the form )~x.A =t )~y.B to \[c/x\]A =t \[c/y\]B, and Ax.A =t B to \[c/x\]A =t Bc where c is a new constant, which may not appear in any solution. The rule for applications decomposes ha(s1,... ,s n) =t hb(tl,...,t '~) to the set {a =c b, sl =t tl,...,s,~ =t tn}, provided that h is a constant. Furthermore equations are kept in 13~/-normal form.</Paragraph>
    <Paragraph position="3"> The variable elimination process for colour variables is very simple, it allows to transform a set g U {A =c d} of equations to \[d/A\]g U {A =c d}, making the equation {A =c d} solved in the result.</Paragraph>
    <Paragraph position="4"> For the formula case, elimination is not that simple, since we have to ensure that la(XA)l = la(xs)l to obtain a C-substitution a. Thus we cannot simply transform a set gU{Xd =t M} into \[M/Xd\]EU{Xd __t M}, since this would (incorrectly) solve the equations {Xc = fc,Xd = gd}. The correct variable elimination rule transforms $ U {Xd =t M} into a(g) U {Xd =1 M, xc, = M1,...,Xc~ =t Mn}, where ci are all colours of the variable x occurring in M and g, the M i are appropriately coloured variants (same colour erasure) of M, and a is the g-substitution that eliminates all occurrences of x from g.</Paragraph>
    <Paragraph position="5"> Due to the presence of function variables, systematic application of these rules can terminate with equations of the form xc(sl,...,s n) =t hd(tl,...,tm). Such equations can neither be further decomposed, since this would loose unifiers (if G and F are variables, then Ga = Fb as a solution Ax.c for F and G, but {F = G,a = b} is unsolvable), nor can the right hand side be substituted for x as in a variable elimination rule, since the types would clash. Let us consider the uncoloured equation x(a) ~t a which has the solutions (Az.a) and (Az.z) for x.</Paragraph>
    <Paragraph position="6"> The standard solution for finding a complete set of solutions in this so-called flex/rigid situation is to substitute a term for x that will enable decomposition to be applicable afterwards. It turns out that for finding all g-unifiers it is sufficient to bind x to terms of the same type as x (otherwise the unifier would be ill-typed) and compatible colour (otherwise the unifier would not be a C-substitution) that either * have the same head as the right hand side; the so-called imitation solution (.kz.a in our example) or * where the head is a bound variable that enables the head of one of the arguments of x to become head; the so-called projection binding ()~z.z).</Paragraph>
    <Paragraph position="7"> In order to get a better understanding of the situation let us reconsider our example using colours. z(aC/) -- ad. For the imitation solution (~z.ad) we &amp;quot;imitate&amp;quot; the right hand side, so the colour on a must be d. For the projection solution we instantiate ($z.z) for x and obtain ()kz.z)ac, which f~-reduces to ac. We see that this &amp;quot;lifts&amp;quot; the constant ac from the argument position to the top. Incidentally, the projection is only a C-unifier of our coloured example, if c and d axe identical.</Paragraph>
    <Paragraph position="8"> Fortunately, the choice of instantiations can be further restricted to the most general terms in the categories above* If Xc has type f~n --+ c~ and hd has type ~ -~ a, then these so-called general bindings have the following form: G h = ~kzal... z a&amp;quot;.hd(H~l (-5),..., Hem (-5)) where the H i are new variables of type f)-~ ~ Vi and the ei are either distinct colour variables (if c E CI)) or ei = d = c (ifc E C). If his one of the bound variables z ~' , then ~h is called an imitation binding, and else, (h is a constant or a free variable), a projection binding* The general rule for flex/rigid equations transforms {Xc(Sl,...,s n) =t hd(tl,...,tm)} into {Xc(S 1 .... , s n) =t hal(t1,..., tin), Xc =t ~h}, which in essence only fixes a particular binding for the head variable Xc. It turns out (for details and proofs see (Hutter and Kohlhase, 1995)) that these general bindings suffice to solve all flex/rigid situations, possibly at the cost of creating new flex/rigid situations after elimination of the variable Xc and decomposition of the changed equations (the elimination of x changes xc(sl,..., s n) to ~h(sl, ..., s n) which has head h).</Paragraph>
    <Section position="1" start_page="6" end_page="8" type="sub_section">
      <SectionTitle>
6.1 Example
</SectionTitle>
      <Paragraph position="0"> To fortify our intuition on calculating higher-order coloured unifiers let us reconsider examples (10) and (11) with the equations R~pf(ipf) __t ~x.ex~pf(X, ipf, iA) R~pf(ipf) =t Ax.ex~pf(X, i-~pf, ipf) We will develop the derivation of the solutions for the first equations (10) and point out the differences for the second (11). As a first step, the first equation is decomposed to R~pf(ipf, c) :t ex~pf(C, ipf, iA) where c is a new constant* Since R~pf is a variable, we are in a flex/rigid situation and have the possibilities of projection and imitation. The projection bindings Axy.x and )~xy.y for R~pf would lead us to the equations ipf =t eX~pf(C, ipf,iA) and c =t eX~pf (c, ipf, iA), which are obviously unsolvable, since the head constants ipf (and c resp.) and eX~pf  clash 6. So we can only bind R~pf to the imitation binding ~kyx*ex~pf(H~pf(y, x), H~2pf (y, x), H 3 (y, x)). Now, we can directly eliminate the variable R~pf, since there are no other variants. The resulting equation null eX~pf(Hlpf(ipf, c), H2pf (ipf, c), g 3 (ipf, c)) =t eX~pf (c, ipf, iA) can be decomposed to the equations (17) Hlpf(ipf,C) __t c H~pf(ipf, c) =t ipf g3pf(/pf, C) __--t iA Let us first look at the first equation; in this flex/rigid situation, only the projection binding )kzw.w can be applied, since the imitation binding Azw.c contains the forbidden constant c and the other projection leads to a clash. This solves the equation, since (Azw.w)(ipf,c) j3-reduces to c, giving the trivial equation c __t c which can be deleted by the decomposition rules* Similarly, in the second equation, the projection binding Azw.z for H 2 solves the equation, while the second projection clashes and the imitation binding )kzw.ipf is not -~pf-monochrome. Thus we are left with the third equation, where both imitation and projection bindings yield legal solutions: * The imitation binding for H3pf is )kzw.i~pf, and not Azw.iA, as one is tempted to believe, since it has to be -~pf-monochrome. Thus we are left with i~pf =t iA, which can (uniquely) be solved by the colour substitution \[-~pf/A\].</Paragraph>
      <Paragraph position="1"> * If we bind H 3 to Azw.z, then we are left with ~pf Zpf. _-t iA, which can (uniquely) be solved by the colour substitution \[pf/A\].</Paragraph>
      <Paragraph position="2"> If we collect all instantiations, we arrive at exactly the two possible solutions for R~pf in the original equations, which we had claimed in section 5:</Paragraph>
      <Paragraph position="4"> Obviously both of them solve the equation and furthermore, none is more general than the other, since i~pf cannot be inserted for the variable x in the second unifier (which would make it more general than the first), since x is bound* In the case of (11) the equations corresponding 1 __t 2 &amp;quot; __t - and to (17) are H.~pf(e, ipf) - e, H~pf(e, Zpf) - ?,~pf H3pf(ipf) __t ipf. Given the discussion above, it is immediate to see that H 1 has to be instantiated with -~pf the projection binding ~kzw.w, H 2 with the imitation 6For (11) we have the same situation* Here the cor* t responding equation is tpf -- ex~pf(C, i~pf, ipf).</Paragraph>
      <Paragraph position="5"> binding Azw.i~of, since the projection binding leads to a colour clash (i~f =t ipf) and finally H~pf has to be bound to the projection binding Azw.z, since the imitation binding Azw.ipf is not -~pf-monochrome.</Paragraph>
      <Paragraph position="6"> Collecting the bindings, we arrive at the unique solution R~f = Ayx.ex~pf(x, i~pf, x).</Paragraph>
    </Section>
  </Section>
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