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Logical leaps and quantum connectives: forging paths through predication space

Quantum Informatics for Cognitive, Social, and Semantic Processes: Papers from the AAAI Fall Symposium (FS-10-08) Logical Leaps and Quantum Connectives:
Forging Paths through Predication Space
Trevor Cohen1, Dominic Widdows2, Roger W. Schvaneveldt3, and Thomas C. Rindflesch4
1Center for Cognitive Informatics and Decision Making, University of Texas Health Science Center at Houston 3Department of Applied Psychology, Arizona State University Abstract
These predications are extracted from citations added to MEDLINE, the most comprehensive database of The Predication-based Semantic Indexing (PSI) approach biomedical literature, over the past decade using the encodes both symbolic and distributional information into a SemRep system (Rindflesch and Fiszman, 2003). We semantic space using a permutation-based variant of proceed by presenting the methodological roots and Random Indexing. In this paper, we develop and evaluate a implementation of the PSI model, and follow with a computational model of abductive reasoning based on PSI. discussion of the ways in which abduction can be Using distributional information, we identify pairs of simulated in the PSI space. Finally, we explore the use of concepts that are likely to be predicated about a common quantum-inspired approaches to concept combination to third concept, or middle term. As this occurs without the constrain the process of abduction, with the aim to identify explicit identification of the middle term concerned, we associations between concepts that are of interest for the refer to this process as a “logical leap”. Subsequently, we use further operations in the PSI space to retrieve this purpose of biomedical knowledge discovery. middle term and identify the predicate types involved. On evaluation using a set of 1000 randomly selected cue concepts, the model is shown to retrieve with accuracy Background
concepts that can be connected to a cue concept by a middle term, as well as the middle term concerned, using nearest- Abduction, Similarity and Scientific Discovery
neighbor search in the PSI space. The utility of quantum logical operators as a means to identify alternative paths Abductive reasoning, as defined by the philosopher and logician, C. S. Peirce (1839-1914) is concerned with the generation of new hypotheses given a set of observations. Inductive and deductive reasoning can be applied to Introduction
confirming and disproving hypotheses, but abductive reasoning is concerned with the discovery of hypotheses as The development of alternative approaches to automated candidates for further testing. Abductive reasoning does reasoning has been a concern of the Quantum Interactions not necessarily produce a correct hypothesis, but effective (QI) community since its inception. One line of inquiry has abductive reasoning should lead to plausible hypotheses explored the utility of distributional models of meaning as worthy of further examination and testing. Several factors a means of simulating abduction, the generation of new can be seen to be at work in abductive reasoning hypotheses, in a computationally tractable manner (Bruza, (Schvaneveldt and Cohen, 2010). Among these is Widdows, and Woods, 2006). Another concern has been establishing new connections between concepts. For the combination between symbolic and distributional example, consider information scientist Don Swanson's models, and ways in which mathematical models derived seminal discovery of a therapeutically useful connection from quantum theory might be applied to this end (Clark between Raynaud's disease and fish oil (Swanson, 1986). and Pulman, 2006). This paper describes recent These concepts had not occurred together in the literature, developments along these lines resulting from our work but were connected to one another by Swanson by with Predication-based Semantic Indexing (PSI) (Cohen, identifying potential bridging concepts that did occur with Schvaneveldt, and Rindflesch, 2009), a novel distributional Raynaud's disease (such as blood viscosity). Concepts model that encodes predications, or object-relation-object occurring with such bridging concepts were considered as triplets into a vector space using a variant of the Random candidates for literature-based discovery. Bruza and his Indexing model (Kanerva, Kristofersson, and Holst, 2000). colleagues note that Swanson's discovery is an example of each term is assigned both a sparse elemental vector, and abductive discovery, and argue that, given the constraints a semantic vector of a pre-assigned dimensionality several of the human cognitive system, deductive logic does not orders of magnitude less than the number of terms in the present a plausible model for reasoning of this nature model (usually on the order of 1000). Elemental vectors (Bruza et al., 2006). Rather, associations between terms consist of mostly zero values, but a small number of these derived by a distributional model of meaning, in their case (usually on the order of 10) are randomly assigned as either Hyperspace Analog to Language (Burgess et al., 1998), are +1 or -1, to generate a set of vectors with a high probability presented as an alternative, a line of investigation we have of being close-to-orthogonal to one another on account of also pursued in our recent work on literature-based their sparseness. For each term in the model, the elemental discovery (Cohen, Schvaneveldt, and Widdows, 2009) .
vector for every co-occurring term within a sliding window Specifically, we have been concerned with the ability of moved through the text is added to the term's semantic distributional models to generate indirect inferences, vector. The permutation-based model extends this meaningful estimates of the similarity between terms that approach, using shifting of elements in the elemental do not co-occur with one another in any document in the vector to encode the relative position of terms. Consider database. Such similarities arise because concepts may co- the following approximations of elemental vectors: occur with other terms even though they never co-occur with one another. In the context of Swanson's discovery, this would involve identifying a meaningful association between Raynaud and fish oil. This association would be drawn without the explicit identification of a bridging Vector v2 has been generated from vector v1 by shifting all term. Having identified these associations, it would then be of the elements of this vector one position to the right. possible to employ some more cognitively and These two vectors are orthogonal to one another, and with computationally demanding mechanism such as symbolic high-dimensional vectors it is highly probable that a vector logic to further investigate the nature of the relationship permuted in this manner will be orthogonal, or close-to- between these terms. As proposed by Bruza and his orthogonal, to the vector from which it is derived. It is colleagues, these associations serve as “primordial stimuli possible to reverse this transformation by shifting the for practical inferences drawn at the symbolic level of elements one position to the left to regenerate v1. These cognition” (Bruza, Widdows, and Woods, 2006). The idea properties are harnessed by Sahlgren and his colleagues to that some economical mechanism such as association encode the relative position of terms, providing a might be useful in the identification of fruitful hypotheses computationally convenient alternative to Jones and for further exploration is appealing for both theoretical and Mewhort's Beagle model (Jones and Mewhort, 2007), practical reasons, the latter on account of the explosion in which uses Plate's Holographic Reduced Representation computational complexity that occurs when considering all (Plate, 2003) to achieve similar ends. Both of these possible relations of each potential bridging term in the approaches allow for order-based retrieval. In the case of context of scientific discovery. In addition, there is permutation-based encoding, it is possible, by reversing the empirical evidence that associations drawn subconsciously permutation used to encode position, to extract from the can precede the solution of a problem (Durso, Rea, and resulting vector space a term that occurs frequently in a Dayton, 1994). In the remainder of this paper, we will particular position with respect to another term. For discuss the ways in which similarity/association captured example, in a permutation-based space derived from the by a distributional model of meaning, can support both the Touchstone Applied Sciences corpus, the vector derived by identification and validation of hypotheses drawn from the shifting the elements of the elemental vector for the term biomedical literature. We begin by presenting some recent “president” a position to the left produces a sparse vector technical developments in the field of distributional that is strongly associated with the semantic vectors1 for semantics, to lay the foundation for a discussion of the terms “eisenhower”, “ nixon”, “reagan” and “kennedy”.
Predication-based Semantic Indexing (PSI) (Cohen et al. 2009), a novel distributional model we have developed in Predication-based Semantic Indexing (PSI)
order to simulate aspects of abductive reasoning.
While the incorporation of additional information related to word order facilitates new types of queries, and has been Permutation-based Semantic Indexing
shown to improve performance in certain evaluations In a previous submission to QI (Widdows and Cohen, (Sahlgren et al., 2008), the associations derived between 2009), we discussed a recent variant of the RI model terms are general in nature. However, it has been argued developed by Sahlgren and his colleagues (Sahlgren, Holst, that the fundamental unit of meaning in text and Kanerva, 2008). Based on Pentti Kanerva's work on comprehension is not an individual term, but an object- sparse high-dimensional representations (Kanerva, 2009), relation-object triplet, or proposition. This unit of meaning this model utilizes a permutation operator that shifts the elements of a sparse high-dimensional vector in order to encode the positional relationship between two terms in a sliding window. In sliding-window based variants of RI, is also termed a predication in logic, and is considered to overlap). This constraint is too tight to support scientific be the atomic unit of meaning in memory in cognitive discovery, or to model abduction. Consequently, in the theories of text comprehension (Kintsch, 1998).
current iteration of PSI in addition to adding the predicate- In our recent work (Cohen, Schvaneveldt and appropriate permutation of an elemental vector to the Rindflesch, 2009) we adapt the permutation-based semantic vector of the other concept in a predication, we approach developed by Sahlgren et al to encode object- also add the unpermuted elemental vector for this concept. relation-object triplets, or predications, into a reduced- The procedure to encode the predication “sherry ISA wine” dimensional vector space. These triplets are derived from would then be as follows. First, add the elemental vector all of the titles and abstracts added to MEDLINE, the for sherry to the semantic vector for wine. Next, shift the largest existing repository of biomedical citation data, over elemental vector for sherry right 22 positions and add this the past decade by the SemRep system (see below). To to the semantic vector for wine. The converse would be achieve this end, we assign a sparse elemental vector and a performed as described previously, but both the permuted semantic vector to each unique concept extracted by and unpermuted elemental vectors for wine would be SemRep, and a sequential number to a set of predicate added to the semantic vector for sherry. Encoding of types SemRep recognizes. For example, the predicates predicate-specific and general relatedness in this manner is “TREATS”, “CAUSES” and “ISA” are assigned the analogous to the encoding of “order-based” and “content- numbers 38, 7, and 22 respectively. Rather than use based” relatedness in approaches that capture the relative positional shifting to encode the relative position of terms, position of terms (Sahlgren, Holst and Kanerva 2008). we use positional shifts to encode the type of predicate that links two concepts. Consequently each time the predication “sherry ISA wine” occurs in the set of predications The predications encoded by the PSI model are derived extracted by SemRep, we shift the elemental vector for the from the biomedical literature by the SemRep system. concept “sherry” 22 positions to the right, to signify an SemRep is a symbolic natural language processing system ISA relationship. We then add this permuted elemental that identifies semantic predications in biomedical text. For vector to the semantic vector for “wine”. Conversely, we example, SemRep extracts “Acetylcholine STIMULATES shift the elemental vector for “wine” 22 positions to the Nitric Oxide” from the sentence In humans, ACh evoked a left, and add this permuted elemental vector to the dose-dependent increase of NO levels in exhaled air.
semantic vector for “sherry”. Encoding predicate type in SemRep is linguistically based and intensively depends on this manner facilitates a form of predication-based retrieval structured biomedical domain knowledge in the Unified that is analogous to the order-based retrieval employed by Medical Language System (UMLS SPECIALIST Lexicon, Sahlrgren and his colleagues. For example, permuting the Metathesaurus, Semantic Network (Bodenreider 2004)). At elemental vector for “wine” 22 positions to the left the core of SemRep processing is a partial syntactic produces a sparse vector with the nearest neighboring analysis in which simple noun phrases are enhanced with semantic vectors and association strengths in Table 1 (left).
Metathesaurus concepts. Rules first link syntactic elements (such as verbs and nominalizations) to ontological Table 1. Results of the predication-based queries “?
predicates in the Semantic Network and then find ISA wine” (left) and “? ISA food” (right).
syntactically allowable noun phrases to serve as arguments. A metarule relies on semantic classes associated with Metathesaurus concepts to ensure that constraints enforced by the Semantic Network are satisfied. SemRep provides underspecified interpretation for a range of syntactic structures rather than detailed representation for a limited number of phenomena. Thirty core predications in clinical medicine, genetic etiology of disease, pharmacogenomics, and molecular biology are retrieved. Quantification, tense and modality, and predicates taking predicational arguments are not Further details of the implementation of this model, and addressed. The application has been used to extract examples of the sorts of queries it enables can be found in 23,751,028 predication tokens from 6,964,326 MEDLINE (Cohen, Schvaneveldt and Rindflesch 2009). For the citations (with dates between 01/10/1999 and 03/31/2010). purposes of this paper, we have modified the model in Several evaluations of SemRep are reported in the order to facilitate the recognition of terms that are literature. For example, in Ahlers et al. (2004) .73 meaningfully connected by a bridging term. In PSI, each precision and .55 recall (.63 f-score) resulted from a unique predicate-concept pair is assigned a unique reference standard of 850 predications in 300 sentences (permuted) elemental vector. Consequently, the semantic randomly selected from MEDLINE citations. Kilicoglu et vectors for any two concepts should only be similar to one al. (2010) report .75 precision and .64 recall (.69 f-score) another if they occur in the same predication type with the based on 569 predications annotated in 300 sentences from same bridging concept (discounting unintended random 239 MEDLINE citations. Consequently, the set of predications extracted by SemRep present a considerable that have sufficient data points to generate meaningful resource for biomedical knowledge discovery.
associations and eliminate concepts that carry little information content from the test set. We generate a 500 dimensional PSI space derived from all of the predications Abduction in PSI-space
extracted by SemRep from citations added to MEDLINE over the past decade (n = 22,669.964), excluding negations For the reasons described previously, the stepwise traversal (x does_not_treat y). We also exclude any predication of all concepts in predications with each middle term that involving the predicate “PROCESS_OF”, as these are occurs in a predicate with a cue concept is not plausible as highly prevalent but tend to be uninformative (for example, a computational model of abduction. Consequently, we “tuberculosis PROCESS_OF patients”). For the same have developed a model in which the search for a middle reason, we exclude any concepts that occur more than term is guided by an initial “logical leap” from cue concept We then follow the procedure described previously, taking the nearest neighboring semantic vector of each cue Our model of abduction consists of the following three concept, generating the vector average of these two vectors, searching for the nearest elemental vector and 1. Identification of the nearest neighboring semantic using the decoding process to find the predication that best vector to the semantic vector of a concept of interest links each pair of concepts (cue and middle term, and 2. Identification of a third “middle term” between the target and middle term). We then evaluate these cue concept and the nearest neighbor. This is predications against the original database, to determine accomplished by taking the normalized vector sum (or whether these are accurate. Of the 1000 cue concepts it vector average) of the semantic vectors for these two was possible to evaluate 999, as one concept occurred in concepts, and finding the most similar elemental vector.
predications that were not included in the model (such as 3. Decoding of the predicates that link the three concepts PROCESS_OF) only. Of these 999 concepts, a legitimate identified. For each pair of concepts, this is target concept and middle term were identified for 962 of accomplished by retrieving the elemental vector for one, them, which can be considered as a precision of 0.963 if and the semantic vector for the other, and shifting one of retrieval of a set of accurate relationships from the these by the number corresponding to each encoded predication, to identify the predicate that fits best. For example, the nearest neighboring semantic vector to that of “pastry” represents “rusk”. The nearest neighboring elemental vector to the vector average of these two semantic vectors is the elemental vector for “food”. Decoding these predicates retrieves the predication pair “rusk ISA food; pastry ISA food”.
Such "logical leaps" may correspond to an intuitive sense of association in psychological terms. The underlying mechanism may involve associations arising from related patterns of associated neighbors rather than any direct association. These indirect associations are likely to be weaker than direct associations so detecting and reflecting on them may not occur without some effort directed toward searching for potential hypotheses, Figure 1: Cosine association and accuracy
solutions, or discoveries. Psychological research has provided evidence that such associations occur in learning Accurately retrieved results tended to have a higher cosine and memory experiments (Dougher, et al., 1994, 2007; association between the middle term and the vector Sidman, 2000). Once detected, indirect associations could average constructed from the cue concept and its nearest be pursued in a more conscious/symbolic way to identify neighboring semantic vector, as illustrated in Figure 1, common neighbors or middle terms on the way to which shows the number of accurate and inaccurate results assessing the value of the indirect associations. Our at different association strengths. Table 2 shows the five computational methods can be seen as ways to simulate the most strongly associated middle terms across this test set, generation and evaluation of such potential discoveries.
together with the predicates linking them to the cue and In order to evaluate the extent to which this approach target concepts. In the first example, a can be used to both identify and characterize the nature of meaningful associations, we select at random 1000 UMLS concepts extracted by SemRep from MEDLINE over the past decade. We include only concepts that occur between 10 and 50,000 times in this dataset, to select for concepts this research is to develop computational tools with which scientists can explore the conceptual territory of their domain of interest. Just as users of a vector-based information retrieval system require methods through which to direct their search for documents, there is a need for the development of methods through which a scientist Table 2: “logical leaps”. Cue concepts are in bold, and
might further refine the search for new ideas. nearest neighbors are underlined. cos = cosine.
Quantum Operators in PSI Space
One potential solution to the problem of constraining search is suggested by the analogy drawn between the many senses of a term that may be captured by a term vector in geometric models of meaning, and the many potential states of a particle that are represented by a state vector in quantum mechanics (Widdows and Peters, 2003).     =  With respect to PSI, the semantic vector representing a concept can be viewed as a mixture of elemental vectors representing each predicate-concept pair and concept it   &- > 
occurs with. This analogy supports the application of the operators of quantum logic, as described by Birkhoff and von Neumann (Birkhoff and Von Neumann, 1936), to semantic vectors, resulting in the definition of semantic space operators effecting quantum logical negation and disjunction in semantic space (Widdows and Peters, 2003). Negation
Negation in semantic space involves eliminating an
undesired sense of a term by subtracting that component of a term vector that is shared with a candidate term representing the undesired sense. For example, the term “pop” can be used to eliminate the musical sense of the term “rock” (Widdows, 2004). This is accomplished by projecting the vector for “rock” onto the vector for “pop” (to identify the shared component), and subtracting this projection from the vector for “rock”. The resulting vector will be orthogonal to the vector for “pop”, and as such will not be strongly associated with vectors representing music- related concepts that are similar to the vector for “pop”, but will retain similarity to terms such as “limestone” that represent the geological sense of “rock”. A similar approach can be applied to the semantic vectors generated using PSI, in order to direct the search        #     for related concepts away from a nearest neighbor that has been identified. As is the case with terms, one would anticipate this approach would eliminate not only the specific concept concerned, but also a set of related concepts. Specifically, we anticipate that this approach would identify a new path involving a different middle term (or group of terms), without the explicit identification of the middle term to be avoided beforehand. These examples illustrate the ability of vectors encoded In order to evaluate the extent to which negation can be using PSI to capture similarity between concepts linked by used to identify new pathways in PSI space, we take the a middle term without the need to explicitly retrieve this same set of 1000 randomly selected concepts as cue term. However, at times it may be of greater interest to concepts. For each cue concept, we retrieve the vector for explore some subset of this space, so as to retrieve the concept (cue_concept), and the vector for the nearest concepts linked by specific predicate types. One goal of neighbor previously retrieved (nn_previous). We then use negation to extract the component of cue_concept that is neighboring semantic vector to this combined vector (nn_current). Finally, we take the vector average of cue_concept and nn_current, render this orthogonal to nn_previous using negation, and find the nearest neighboring elemental vector to this combined vector. We then decode the predicates concerned using the permutation operator as described previously. To illustrate the utility of this approach, we present a series of examples in which we attempt logical leaps by applying Table 3: Negation to identify new paths (n=997)
the dissection method to both the cue term and candidate nearest neighbors. For example, consider a logical leap of the form “X ISA Y; Y TREATS Z” where Z is the cue term. In the case of the cue term, we perform the reverse of the “TREATS” permutation prior to dissection. For each target term we perform the “ISA” permutation before dissection. After dissection, we measure the cosine between these transformed vectors to find a best match. The results of this experiment are shown in Table 3. It was possible to obtain results for 997 of the set of 1000. One Table 4: Leaps across specific predicates. * denotes
concept was excluded for the same reason as before, and concepts that do not occur in a predicate with the cue.
another two were excluded as the negation operator produced a zero vector, as these concepts occurred in predications exclusively with a single predicate-concept pair. As anticipated, in every case negation eliminated the concept represented by nn_previous. However, this result could have been obtained using boolean negation, which is the equivalent of simply selecting the next-nearest neighbor, as we have done for comparison purposes. Of greater interest is the extent to which the use of quantum negation eliminates the path across a middle term that was used to identify a previous neighbor. This occurred after quantum negation in 94.1% of cases, as oppose to 27.7% in the case of boolean negation. A concern with the use of this method is that the orthogonalization process may introduce further errors as concept vectors are distorted beyond recognition. However, as shown in Table 3, this process led to only slightly more erroneous predications than were obtained with boolean negation. Interestingly, the set of errors * J "  K; < 46 7(9: - K produced in the original experiment has very few elements in common with the set produced after quantum negation – erroneous predications were produced for only four of the Dual Dissection
We note that it is possible to select for particular predicate types by reversing the permutation operator that corresponds to the predicate of interest. For example, the Table 4 illustrates some examples of dissection-based predication A TREATS B is encoded by shifting the searches. Nearest neighbors for each search are on the left, elemental vector for A, Ae, 38 steps to the right, and and the pattern of the strongest connection through a adding this to the semantic vector for B. The unpermuted middle term in each case is shown on the right. In each vector, Ae, is also added to this vector. Applying the case, the same pattern of strongest connections was shared reverse shift to this semantic vector, to produce B^ should by all five of the nearest neighbors shown, and corresponds produce a vector that retains some remnant of the original to the pattern specified using the dissection-based Ae. As this remnant should be encoded in both the original approach. In all cases, the five nearest neighbors are semantic vector for B, and its permutation, B^, we attempt different than those retrieved by a logical leap search to extract the common components of these vectors using without dissection, and in many cases (denoted by an *), the following procedure, which we will term dissection: the nearest neighbors are concepts that do not occur with the cue concept directly in any predication in the database. These examples illustrate the way in which paired permutations can be used to infer information beyond that which is stated explicitly in the database. The system has inferred plausible treatments for depressive disorders and dysthymic disorder; and gene/protein-disease associations related to prion diseases based on taxonomic relationships extracted from the literature by SemRep. While these examples illustrate only a few possibilities for the application of dual dissection, it was not difficult to generate others. We found that this approach frequently results in logical leaps of the desired form. Pitfalls include a tendency to generalize too generously (for example, therapeutic associations involving high-level middle terms such as “pharmaceutical preparation”), and failure to isolate the desired predicate path. This was encountered with terms that occur in many predication relationships. In these cases, “correct” results would be interspersed amongst results linked to the cue term in other ways.
Dissection and Disjunction
Once vectors representing the desired sense of a concept
have been isolated using this procedure, it is possible to
Conclusion
construct a subspace with these vectors as bases. This In this paper, we develop and evaluate a model of subspace then represents the set {sense1 OR sense2 OR … automated reasoning based on “logical leaps”, in which sense n} and can be modeled using quantum disjunction meaningful associations between concepts derived from (Widdows and Peters 2003), after ensuring the bases of the distributional statistics are used to identify candidates for subspace are orthogonal to one another using the Gram- connection via a third concept, and identify the nature of Schmidt procedure. The association strength between each the relations involved. The chain of predicates constructed semantic vector and this subspace can then be measured by in this manner can subsequently be processed using projecting a semantic vector into the subspace and symbolic methods. Consequently, the vector-based “logical measuring the cosine between the original semantic vector leaps” approach relates to Gardenfors' proposal that conceptual representation at a geometric level might This allows us to broaden the scope of our search. For provide support for symbolic level processes (Gardenfors example, we might expand the query in Leap 1 to <  2000). While this approach is able to infer plausible connections between concepts, this inference occurs at the geometric level, avoiding the computational complexity of extensive symbolic inference. Furthermore, the vector spaces used for these experiments can be retained in RAM to facilitate rapid, dynamic, interactive exploration of biomedical concepts to support discovery. Vector operators derived from quantum logic show promise as a means to direct such searches away from previously trodden paths, and exploratory work suggests there may be ways to adapt "    
these operators to guide search toward conceptual territory #"#    of interest. Of particular interest for future work is the evaluation of the extent to which these operators might be used to model “discovery patterns” (Hristovski, Friedman and Rindflesch 2008), combinations of predications that have been shown useful for literature-based discovery. References
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