Exploiting Graph Structure to Summarize and Compress ......Exploiting Graph Structure to Summarize...
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Exploiting Graph Structure to Summarize and Compress Relational Knowledge Scott E. Friedman SIFT, LLC 319 1st Ave North, Suite 400 Minneapolis, MN 55401 [email protected] Abstract AI systems consume volumes of relational knowledge to ex- tract patterns and model the world. We face the challenges of scaling to growing volumes of knowledge, identifying more sophisticated patterns and concepts, and compressing and indexing relational knowledge for future access. This paper describes the domain-general path-specific isomorphic abstraction (ψ-abstraction) approach for encoding relational knowledge as relational graphs, and exploiting paths and cy- cles in these graphs to guide compression and pattern ex- traction. ψ-abstraction automatically summarizes relational graphs by (1) identifying cyclic and acyclic paths; (2) gen- eralizing the graph along those paths; (3) explicitly anno- tating generalizable sequences; (4) encoding qualitative rela- tions over generalized quantity sequences; and (5) optionally pruning generalized knowledge to compress the graph. We demonstrate ψ-abstraction on twelve examples spanning di- verse domains: 2D spatial; 3D spatial; temporal (calendar) concepts; abstract feedback cycles; narratives; cell signaling; and anatomy. We evaluate ψ-abstracted knowledge for data compression, pattern identification, and structural similarity of examples within and across categories. Introduction Intelligent systems— ranging from higher-order cogni- tive models, scientific collaboration systems, and domain- specific data miners— consume volumes of relational knowledge to learn informative models of the world and ex- tract patterns of interest. The increasing volume of knowl- edge presents the first challenge that motivates this paper: addressing scalability demands of relational knowledge to meet storage and efficiency requirements. A second— and more central— challenge is improving symbolic AI systems’ ability to learn sophisticated patterns and categories from examples represented with relational knowledge. For example, symbolic AI systems consume relational knowledge to induce logic programs, perform comparative analysis (e.g., McLure, Friedman, and For- bus, 2015), induce qualitative and quantitative models (e.g., Friedman, Taylor, and Forbus, 2009), and perform case- based reasoning. These techniques compute mappings, hy- potheses, and abstractions over specific symbols and state- ments within the relational knowledge, but most of these ap- proaches are less effective at automatically learning and ex- pressing categories involving arbitrary many symbols in ag- gregations, sequences, and cycles. This is a representational (or re-representational) challenge for cognitive systems. This paper presents path-specific isomorphic abstraction (ψ-abstraction) designed to address the above scalability and representational challenges. ψ-abstraction traverses relational knowledge as a graph and selectively general- izes isomorphic subgraphs using an enhanced SAGE (Se- quential Analogical Generalization of Exemplars) algorithm (McLure, Friedman, and Forbus, 2015). This compactly represents arbitrarily-long relational paths and cycles (e.g., of temporal, causal, flow, or spatial relations) within a ma- trix of isomorphic entities and values. ψ-abstraction then augments the initial relational knowledge to explicitly de- scribe this matrix as an ordered sequence of entities, and it describes bounded qualitative relationships over general- ized qualitative or numerical values. Finally, ψ-abstraction optionally removes the generalized graph structure, leaving the generalized, compressed sequence description and qual- itative descriptions. Rerepresenting the graph with ψ-abstraction jointly im- proves similarity-based reasoning and compacts the repre- sentation. The according claims of this paper are as follows: 1. ψ-abstraction reduces the relational knowledge required to express graphs with repetitive or sequential structure, with potentially lossless compression. 2. ψ-abstraction increases the structural similarity of cate- gory exemplars that involve sequences and cycles. We support these claims with experimental results of ψ- abstraction on fourteen examples spanning diverse domains: 2D spatial representations; 3D spatial representations; tem- poral calendar concepts; abstract feedback systems; narra- tives; human anatomy; and cell signalling. We measure information theoretic compression, sequence identification, and the increased structural similarity of within-category ex- amples (e.g., two stacks of blocks, with different height) and across-category examples (e.g., a stack and a ladder) after processing them with ψ-abstraction. Since ψ-abstraction in- creases within-category and across-category similarity, it is a promising knowledge representation strategy for learning- by-generalization as well as learning with near-misses. We begin by describing background and related work and then we describe the ψ-abstraction approach and our exper- iments to support the above claims. We close with a discus-
Transcript of Exploiting Graph Structure to Summarize and Compress ......Exploiting Graph Structure to Summarize...
Exploiting Graph Structure to Summarize and Compress Relational Knowledge
Scott E. FriedmanSIFT, LLC
319 1st Ave North, Suite 400Minneapolis, MN 55401
Abstract
AI systems consume volumes of relational knowledge to ex-tract patterns and model the world. We face the challengesof scaling to growing volumes of knowledge, identifyingmore sophisticated patterns and concepts, and compressingand indexing relational knowledge for future access. Thispaper describes the domain-general path-specific isomorphicabstraction (ψ-abstraction) approach for encoding relationalknowledge as relational graphs, and exploiting paths and cy-cles in these graphs to guide compression and pattern ex-traction. ψ-abstraction automatically summarizes relationalgraphs by (1) identifying cyclic and acyclic paths; (2) gen-eralizing the graph along those paths; (3) explicitly anno-tating generalizable sequences; (4) encoding qualitative rela-tions over generalized quantity sequences; and (5) optionallypruning generalized knowledge to compress the graph. Wedemonstrate ψ-abstraction on twelve examples spanning di-verse domains: 2D spatial; 3D spatial; temporal (calendar)concepts; abstract feedback cycles; narratives; cell signaling;and anatomy. We evaluate ψ-abstracted knowledge for datacompression, pattern identification, and structural similarityof examples within and across categories.
IntroductionIntelligent systems— ranging from higher-order cogni-tive models, scientific collaboration systems, and domain-specific data miners— consume volumes of relationalknowledge to learn informative models of the world and ex-tract patterns of interest. The increasing volume of knowl-edge presents the first challenge that motivates this paper:addressing scalability demands of relational knowledge tomeet storage and efficiency requirements.
A second— and more central— challenge is improvingsymbolic AI systems’ ability to learn sophisticated patternsand categories from examples represented with relationalknowledge. For example, symbolic AI systems consumerelational knowledge to induce logic programs, performcomparative analysis (e.g., McLure, Friedman, and For-bus, 2015), induce qualitative and quantitative models (e.g.,Friedman, Taylor, and Forbus, 2009), and perform case-based reasoning. These techniques compute mappings, hy-potheses, and abstractions over specific symbols and state-ments within the relational knowledge, but most of these ap-proaches are less effective at automatically learning and ex-pressing categories involving arbitrary many symbols in ag-
gregations, sequences, and cycles. This is a representational(or re-representational) challenge for cognitive systems.
This paper presents path-specific isomorphic abstraction(ψ-abstraction) designed to address the above scalabilityand representational challenges. ψ-abstraction traversesrelational knowledge as a graph and selectively general-izes isomorphic subgraphs using an enhanced SAGE (Se-quential Analogical Generalization of Exemplars) algorithm(McLure, Friedman, and Forbus, 2015). This compactlyrepresents arbitrarily-long relational paths and cycles (e.g.,of temporal, causal, flow, or spatial relations) within a ma-trix of isomorphic entities and values. ψ-abstraction thenaugments the initial relational knowledge to explicitly de-scribe this matrix as an ordered sequence of entities, andit describes bounded qualitative relationships over general-ized qualitative or numerical values. Finally, ψ-abstractionoptionally removes the generalized graph structure, leavingthe generalized, compressed sequence description and qual-itative descriptions.
Rerepresenting the graph with ψ-abstraction jointly im-proves similarity-based reasoning and compacts the repre-sentation. The according claims of this paper are as follows:
1. ψ-abstraction reduces the relational knowledge requiredto express graphs with repetitive or sequential structure,with potentially lossless compression.
2. ψ-abstraction increases the structural similarity of cate-gory exemplars that involve sequences and cycles.
We support these claims with experimental results of ψ-abstraction on fourteen examples spanning diverse domains:2D spatial representations; 3D spatial representations; tem-poral calendar concepts; abstract feedback systems; narra-tives; human anatomy; and cell signalling. We measureinformation theoretic compression, sequence identification,and the increased structural similarity of within-category ex-amples (e.g., two stacks of blocks, with different height) andacross-category examples (e.g., a stack and a ladder) afterprocessing them with ψ-abstraction. Since ψ-abstraction in-creases within-category and across-category similarity, it isa promising knowledge representation strategy for learning-by-generalization as well as learning with near-misses.
We begin by describing background and related work andthen we describe the ψ-abstraction approach and our exper-iments to support the above claims. We close with a discus-
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(d) (e)
Figure 1: The input, intermediate products, and final product of ψ-abstraction: (a) relational knowledge describing a six-rungladder displayed as a graph; (b) binary projection of the relational knowledge; (c) binary projection of just the above relation;(d) generalization along the above binary projection; and (e) ψ-abstracted graph with generalized (green) structure, sequenceannotation (yellow), and qualitative descriptions of generalized quantity behaviors (orange).
sion of the implications and future work, including develop-ing additional strategies and constraints for ψ-abstraction.
Background & Related WorkGraph rewriting and graph mining have been research top-ics in computer science for decades, and these techniqueshave been used to optimize cybersecurity, compilers, andprovenance-tracking.
In the graph compression research, previous work focuseson contracting graphs by pairwise chunking (e.g., Matsudaet al., 2000), which iteratively compresses the graph at sin-gle points. By comparison, ψ-abstraction iteratively gener-alizes subgraphs of larger relational graphs along a singlepath, leaving the vertices behind in case they comprise thesubgraph of the next iteration. It later summarizes the gen-eralized subgraphs into a sequential structure.
In graph-mining research, machine learning techniquesuse kernel methods (e.g., Horvath, Gartner, and Wrobel,2004) to allow classifiers to select the most discriminatingstructures in graph-mining and machine learning settings.Kernel methods and ψ-abstraction both perform subgraphdiscovery, but ψ-abstraction exploits paths of common sub-graphs to directly rewrite (and optionally, prune) the graphalong paths.ψ-abstraction builds on previous research on compu-
tational models of structure-mapping that have producedtechniques for describing and quantifying isomorphic (i.e.,similarly-structured) portions of relational knowledge across
cases (Falkenhainer, Forbus, and Gentner, 1989) and withincases (Ferguson, 1994). These techniques are based onthe psychological Structure-Mapping Theory of analogy andsimilarity (Gentner, 1983). Structure-Mapping Theory holdsthat people match entities and relationships between rela-tional representations (a base and a target representation)when they reason by analogy, with three constraints:
1. One-to-one: An entity or relation in the base can map to atmost one in the target, and the reverse for target-to-base.
2. Parallel connectivity: If a base relation maps to a targetrelation, so must each of their arguments, in order.
3. Tiered identicality: A relation can only map to anotherrelation with the same predicate.
In graph theory terms, the first (one-to-one) constraint isshared with definition of an isomorphism. The second (par-allel connectivity) constraint states that if we treat a rela-tional graph as a directed graph where relations are nodesthat have index-labeled edges to their arguments, then if tworelation nodes are isomorphic, their child nodes must alsobe isomorphic. Applied recursively, the first two constraintsstate that isomorphic (i.e., mapped) nodes must have isomor-phic reachable subgraphs.
Computational structure-mapping techniques supportanalogical generalization (Friedman, Taylor, and Forbus,2009, Kuehne et al., 2000) and inter-category comparativeanalysis (McLure, Friedman, and Forbus, 2015) with en-couraging results. However, the constraints of structure-
mapping, which make it tractable and practical, also presentchallenges in certain cases. For example, consider a spatialdescription of a ladder with six rungs and another of a ladderwith twelve rungs. Structure-mapping alone may easily mapthe side-rails of each ladder together, but due to the one-to-one constraint, at most half the rungs on the 12-rung ladderwill have corresponding rungs. Intuitively, people probablymentally rerepresent the knowledge to account for repetitionprior to performing similarity-based reasoning.
Rerepresentation techniques allow us to preservestructure-mapping constraints with more practical results.Previous work in computational rerepresentation has (1)changed the granularity/connectivity of knowledge repre-sentation while maintaining semantics (Yan, Forbus, andGentner, 2003) or (2) relaxed semantics while maintaininggranularity/connectivity (Falkenhainer, 1988). However, theimplementations of these techniques do not directly solvethe ladder example above.
The MAGI algorithm, also based on structure-mapping,detects self-similarity in relational representations by com-puting and then decomposing global analogical mappingsover the entire case/graph (Ferguson, 1994). MAGI may in-deed identify the rungs as similar components, but it doesnot perform the generalization, compression, and qualitativesummaries of ψ-abstraction. Further, ψ-abstraction does notneed to construct a global mapping, and its time complexityis decoupled from the size of the entire relational descrip-tion, as we describe below.
ApproachHere we describe the algorithms and representations of ψ-abstraction. While ψ-abstraction is not a cognitive modelper se, it builds on the same key principle of previ-ous structure-mapping research: reasoning is guided bythe [graph-based] structure of relational knowledge. ψ-abstraction employs this principle at a different granularity,by traversing the knowledge graph itself, along paths andcycles, to perform similarity-based generalization and graphsummarization.ψ-abstraction takes structured relational knowledge as in-
put. Consider the predicate calculus statement:(valueOf (YPosFn rung-2a) 1).This statement includes a relation valueOf associating
two arguments: a functional term (YPosFn rung-2a)referring to the y-position of the entity symbol rung-2a;and the number 1. This and other statements can be unam-biguously represented in a directed graph. Figure 1(a) illus-trates such a graph, describing a six-rung ladder, that showsthis relation (at the top of the graph) and others. We refer tothis as a relational graph.
We draw relational graphs with squares denoting atomic(i.e., symbol or numerical) nodes, ovals denoting non-atomic (i.e., relational or functional) nodes, and top-levelassertions marked with * (e.g., the valueOf formula isasserted at the top-level but the YPosFn formula is not).Edges are labeled by argument index, starting with zero. Therelational graph is acyclic, since no formula can contain it-self as an argument.
From any given entity, e.g., rung-2a, we can identify allstatements mentioning it by regressing (i.e., traversing back-ward) along edges to identify all top-level assertions. Fromany top-level assertion, we can identify all atomic and non-atomic formulas it includes by traversing forward to com-pute the reachable subgraph. When we perform these opera-tions in succession, i.e., finding all statements mentioning anentity and then finding all formulas in those statements, wecall it the relevant subgraph of an entity.1 Importantly, com-puting the relevant subgraph of an entity does not involveiterating over all statements, so the time complexity is basedsolely on the number and structure of statements containingthe entity.
Identify the abstraction path. ψ-abstraction first identi-fies productive paths or cycles to traverse for abstractionpurposes. It computes the binary projection of the rela-tional graph, as shown in Figure 1(b). This graph is com-prised of all entities or functional formulas that are associ-ated with other entities or functional formulas (not includ-ing numerical terms) in the relational graph. Edges be-tween the entities are predicate-argument combinations link-ing the entities. For example, rung-2a is linked withrung-1a with above[0,1] since (above rung-2arung-1a) appears in the relational graph. Similarly,rung-2a is linked with rung-4a with between[1,2]since (between rung-3a rung-2a rung-4a) ap-pears in the relational graph.ψ-abstraction then computes the longest traversal or sim-
ple cycle in the graph using any single edge label. In thiscase, it is the above[0,1] label. Figure 1(c) shows theabove[0,1] subgraph of Figure 1(b). This will be thepath ψ-abstraction traverses to absract the relational graph.This is a domain-general heuristic for identifying the ab-straction path; however, domain-specific or culture-specificstrategies (e.g., following causal links, following tempo-ral relations, or following specific spatial relations) may beused as well, obviating this path-selection step. Note thatthis is the only time ψ-abstraction accesses or traverses theentire graph; it never accesses the entire graph to performstructure-mapping or subsequent analysis.
Generalize the abstraction path. After ψ-abstraction se-lects the abstraction path, it iterates over each entry (i.e., ver-tex) in the path (which may be a cycle). For each entry in thepath, ψ-abstraction generalizes the path with the followingoperations:
1. Identify the next formula f .2. Compute the relevant subgraph Gf for f .3. If no generalization exists yet, set the generalization toGf ; otherwise:
(a) Compare Gf with the existing generalization viastructure-mapping to compute the best mapping m,where f is constrained to map to the previous formula.
(b) Merge Gf into the generalization along m with theSAGE merge operation (McLure, Friedman, and For-bus, 2015).
1The result is equivalent to the CaseFn of the entity in dy-namic case construction (Mostek, Forbus, and Meverden, 2000).
The result of generalizing the relational graph in Fig-ure 1(a) along the path shown in Figure 1(c) is the gen-eralization shown in Figure 1(d). ψ-abstraction general-izations include the generalized relations (shown in green),where brightness indicates frequency. ψ-abstraction gen-eralizations also retain the sequence of values of eachgeneralized entity. For instance, note that the formerly-numerical valueOf arguments are generalized entity (i.e.,GE-<nbr>) nodes. Other generalized entities, e.g.,currterm-path-<nbr>, have been renamed by ψ-abstraction, which we describe later. Beneath generalizedentities are the columns of numerical or symbolic values thatassumed that node-space in the graph.
Each row across columns describes a single iteration inψ-abstraction’s generalization, e.g., each rung in the se-quence: rung-5a has ZPosFn 6 and is above rung-4a;rung-4a has ZPosFn 4 and is above rung-3a; and soon. We call this an isomorphic matrix, or more succinctly,an isomatrix. Each column in the isomatrix is an atomicisomorphism, and each row contains each record/iterationacross atomic isomorphisms. This preserves numerical andsymbolic data without duplicating the relational data, how-ever it is lossy with respect to retaining which relations areincomplete over which iterations.2
Note that some entities, including the siderails sr-a andsl-a were not generalized against other entities, since theywere mapped against themselves for the entire sequence ofrungs.
Summarize the abstraction path. ψ-abstraction summa-rizes the graph based on the generalization and associatedisomatrix. It first compares other columns in the isomatrixagainst the abstraction path. If those sequences can be pro-duced with a single left- or right-shift (for acyclic paths)or a single left- or right-rotation (for cyclic paths), thenψ-abstraction will assert that they are the previous or nextterm in the sequence. ψ-abstraction renames generalized en-tity corresponding to the abstraction path currterm it re-names the generalized entities corresponding to the previousterm prevterm and next term nextterm accordingly, asshown in Figure 1(d).
If there are other columns of generalized entities that are(1) not quantity values and (2) not next or previous terms,ψ-abstraction returns to its generalization stage and appendsthis path to its abstraction path, so that each generalizationiteration computes the relevant subgraph of the next itemin every path sequence. It iterates until no more paths ofentities can be generalized.ψ-abstraction then removes completely-generalized enti-
ties, such that no more statements describe the entity in therelational graph that are not included in the generalization,to produce the ψ-abstracted relational graph in Figure 1(e).The ψ-abstracted graph contains generalized (green) struc-ture as well as unabstracted (in white) structure from theoriginal graph. It describes the generalized path(s) with theyellow, dashed summary knowledge in Figure 1(e). This
2For instance, if leftOf was not asserted for some rung withthe right siderail sr-a, the isomatrix would not retain which rungthis was.
includes annotating which entity is the current (and option-ally, next and previous) term, whether the path is cyclic oracyclic, and listing the entities in the sequence.ψ-abstraction performs ordered qualitative analysis of the
various quantity behaviors over the isomatrix, and describesthis with the orange, dashed summary knowledge in Fig-ure 1(e). This asserts that for the (sub)sequence (spanrung-5a rung-0a), the current term’s y-position and x-position are constant, but the z-position is strictly decreasingover the sequence.
This is a more satisfying description of a ladder forexemplar-based learning and reasoning, and ψ-abstractionhas constructed it in a domain-general manner, by travers-ing and generalizing a path in the original relational graph.The isomatrix contains information for reasoning about eachrung in isolation, and for partial reconstruction of the origi-nal representation if necessary.
EvaluationWe next describe two ψ-abstraction experiments, spanningfourteen examples over multiple domains. We relate eachexperiment to the overall claims of this paper, including ψ-abstraction’s ability to abstract relational knowledge and itseffect on structural similarity.
Demonstrating ψ-abstraction across domains. In ourfirst experiment, we ψ-abstract relational graphs in multi-ple domains to demonstrate ψ-abstraction’s ability to sum-marize different graph structures and multi-sequence expan-sions. This includes (1) a spatial description of a square spi-ral, (2) a description of the human circulatory system withqualitative O2 concentrations from (Friedman and Forbus,2011), (3) activation of RAF, MEK, and MAPK for cell sig-naling, and (4) a summarized plot of the children’s book“The Spiffiest Giant in Town.”
For each input graph, we run ψ-abstraction as described inthe previous section. For the circulatory system example andthe children’s book, ψ-abstraction repeats its generalizationstage at least three times, having found more abstractablesequences. Results are shown in Figure 2: Figure 2(top)illustrates the spiral, its relational graph, its generalizationand isomatrix, and the resulting ψ-abstracted graph. Fig-ure 2(bottom) illustrates the resulting ψ-abstracted graphsof the remaining three examples. Of the remaining three,only the circulatory system example (Figure 2e) has quan-tity data, so ψ-abstraction asserts thatO2 decreases from theleft atrium (lt-a2) to the right ventricle (rt-v2), and in-creases in the complement of the cycle. This is due to O2
consumption in the body and O2 infusion in the lungs, re-spectively.
Three of the ψ-abstracted graphs Figure 2(e-g) containmore than one sequence. This is because ψ-abstractionidentified multiple distinct sequences in the isomatrix:in Figure 2(e), ψ-abstraction identified distinct sequencesof FluidFlow instances, ContainedFluid instances,path instances, and container instances; in Figure 2(f), ψ-abstraction identified sequences of activating entities, ac-tivated entities, Activation instances, and base com-pounds; and in Figure 2(g), ψ-abstraction identifies se-
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(a) (b) (c) (d)
Figure 2: At top, results of abstracting a spatial description of a square spiral: (a) sketch; (b) initial relational graph; (c) gen-eralization matrix; and (d) ψ-abstracted graph. At bottom, three ψ-abstracted graphs: (e) double-loop model of the circulatorysystem with oxygen concentration; (f) activation of RAF, MEK, and MAPK; and (g) the plot of the children’s book “TheSpiffiest Giant in Town.”
quences of clothing (given by the giant), animals (recipientsof the clothing), missing entities (mitigated by the clothing),and the associated Encounter events. All of these se-quences are joined by generalized structure describing howthe different sequences combine semantically.
Theψ-abstracted graphs yielded compression ratios of 5.1for the spiral, 1.24 for the circulatory system, 1.21 for thecell signaling, and 1.18 for the narrative, including qualita-tive descriptions and sequence descriptions. Intuitively, thecompression rate increases with the length of the sequence,all else being equal.
Evaluating ψ-abstraction’s effect on similarity. In thesecond experiment, we compare pairs of relational graphswith structure-mapping to compute an original mapping,and then we compare the respective ψ-abstracted graphswith structure-mapping to compute a abstracted mapping.We compare the original and abstracted mappings subjec-tively for practicality and objectively using normalized sim-ilarity scores.
The pairs of examples used for this experiment include:(1) expert vs. novice circulatory system descriptions (with-out O2); (2) ladders of different sizes; (3) seasons vs.months of the year with their avg. Minneapolis tempera-tures; (4) stacks of blocks of different sizes and dimensions;and (5) a four-node feedback cycle of viral videos vs. athree-node feedback cycle of mosquito bites.
Figure 3 illustrates the results in the above order, includ-
ing the original graphs with an original mapping (at left),and the resulting ψ-abstracted graphs with an abstractedmapping (at right). Many of the original mappings aretoo large to read (due to having more entities and not asmuch overlap), but they are included for coarse size com-parison. The white nodes of the mapping indicate overlap-ping/isomorphic structure, and the colored nodes indicatestructure from a single (base or target) graph. Boxed col-ored entities or values indicate that the two entities or valuescorrespond in the mapping at a given GE-<nbr> spot.
ψ-abstraction’s effect on similarity is most easily no-ticed in the feedback cycle comparison (Figure 3, bot-tom): in the original mapping, (LikesFn video) andits corresponding qualitative proportionality qprop rela-tions map to nothing, due to structure-mapping’s one-to-one constraint. In the abstracted mapping, everything corre-sponds, as a cycle of qualitative proportionality. Similarly, inthe seasons-to-months comparison, the abstracted mappingmaps the two cyclic sequences, their qualitative temperaturechanges, and all other nodes except CalendarSeasonand CalendarMonth assertions.
ψ-abstraction increased the similarity of the circulatorymapping by 29.9% (final similarity 0.87), the ladders map-ping by 37% (final similarity 1.0), the seasons-to-monthsmapping by 67% (final similarity 0.98), the stacks mappingby 42.9% (final similarity 1.0), and the feedbacks mappingby 23.4% (final similarity 1.0). We also compared an orig-
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6-Rung Ladder 12-Rung Ladder 6-Rung Ladder
Mpls Seasons & Temp Mpls Months & Temp Mpls Seasons & Temp Mpls Months & Temp
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12-Rung Ladder
Figure 3: Table of ψ-abstraction results, displaying (at left) the two input representations and their resulting analogical mapping,and (at right) the same inputs after ψ-abstraction, and their resulting analogical mapping. Smaller mappings with more (white)overlapping structure and corresponding (boxed) entities indicate higher similarity.
inal stack and ladder with a ψ-abstracted stack and ladder(i.e., an inter-category comparison), for a similarity increaseof 140% (final ψ-abstracted similarity 0.75). This similarityincrease is due to structure-mapping utilizing the sequenceobjects, abstracted category structure, and qualitative de-scriptions in its mapping rather than putting specific Blockand Rung instances in correspondence.
Conclusion & Future WorkThis paper presented the ψ-abstraction approach to general-izing, compressing, and summarizing relational graphs. Wehave supported our claim that ψ-abstraction compresses se-quences of isomorphic subgraphs into summary descriptionson multiple examples across domains. We also supportedour claim that ψ-abstraction increases the structural simi-larity of same-category exemplars when the category per-mits arbitrary repetition. We reported that ψ-abstraction in-creases inter-category similarity, e.g., of stacks and ladders.One objection is that ψ-abstraction dilutes the practicalityof similarity with its summarization, but recent work in con-cept learning with near misses (e.g., McLure, Friedman, andForbus, 2015) suggests otherwise, since increasing inter-category similarity makes subtle-but-important differencesapparent during comparative analysis. Indeed, ψ-abstractionmakes these subtle differences apparent: the siderails of theladder are unmapped, the fact that the blocks touch (and therungs don’t) is explicit, and the categories (i.e., Rung andBlock) are unmapped. These are logical criteria for differ-entiating these categories.
In addition to aiding machine learning techniques, ψ-abstraction extracts and describes actionable sequences forgoal-directed reasoning. For example, ψ-abstraction de-scribes explicit causal chains for reasoning about interven-tion, explicit temporal chains for planning and scheduling,and sequential structural relationships, e.g., for determininghow to appropriately add more blocks to a stack or continuethe pattern of a spiral.
We differentiated ψ-abstraction from other graph rewrit-ing/mining in our discussion of related work, but these mightbe integrated into ψ-abstraction for breadth and efficiency.For instance, kernel methods that utilize natural graph pat-terns (e.g., Horvath, Gartner, and Wrobel, 2004) may effi-ciently identify or prioritize opportunities for ψ-abstraction.
We demonstrated ψ-abstraction as a proof-of-concept onexamples without noisy numerical data or distracting entitiesand relations, when data in the real world is not so charita-ble. ψ-abstraction’s qualitative summarization is sensitive tosome noise by asserting monotonic quantity changes in ad-dition to strict increasing/decreasing changes, but other ap-proaches for computing qualitative behaviors from numer-ical data (e.g., Zabkar et al., 2013) might be more robust.Aside from the numerical data, distracting entities, e.g., abox the ladder is resting upon, would— at present— inter-fere with ψ-abstraction’s generalization, e.g., by includingthe box in its path-based compression of the rungs. We be-lieve we could trivially constrain ψ-abstraction’s path traver-sal to only include nodes (i.e., entities) of identical cate-gories. This means ψ-abstraction would traverse each en-
tity along the above relation path that shares identical isarelations (e.g., Rung instances) and stop when it reaches anon-category member. Constraining ψ-abstraction and iden-tifying novel traversal strategies is near-term future work.
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