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# A Machine-Oriented Logic based on the Resolution Principle.

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... Inference plays an important role in the context of Max-SAT solving (Li et al., 2007;Narodytska & Bacchus, 2014;Abramé & Habet, 2014) and this has led to an increasing interest in studying proof systems for Max-SAT in the literature (Larrosa & Heras, 2005;Bonet, Levy, & Manyà, 2006Larrosa & Rollon, 2020a, 2020bBonet & Levy, 2020;Filmus, Mahajan, Sood, & Vinyals, 2020;Cherif, Habet, & Py, 2022). In particular, Max-SAT resolution (Larrosa & Heras, 2005;Bonet et al., 2006Bonet et al., , 2007 is one of the first known complete systems for Max-SAT and is a natural extension of the resolution rule (Robinson, 1965) used in the context of SAT. Max-SAT resolution proofs are more constrained than their SAT counterparts as the premise clauses are replaced by the conclusions when applying Max-SAT resolution. ...
... On the other hand, to prove that a CNF formula is unsatisfiable, we need to refute the existence of a model. A well-known SAT refutation system is based on an inference rule for SAT called resolution (Robinson, 1965). The resolution rule, defined below, deduces a clause called resolvent which can be added to the formula. ...
... Definition 1 (Resolution (Robinson, 1965)). Given two clauses c 1 = (x ∨ A) and c 2 = (x ∨ B), the resolution rule deduces a third additional clause as follows: ...
Article
Current Max-SAT solvers are able to efficiently compute the optimal value of an input instance but they do not provide any certificate of its validity. In this paper, we present a tool, called MS-Builder, which generates certificates for the Max-SAT problem in the particular form of a sequence of equivalence-preserving transformations. To generate a certificate, MS-Builder iteratively calls a SAT oracle to get a SAT resolution refutation which is handled and adapted into a sound refutation for Max-SAT. In particular, we prove that the size of the computed Max-SAT refutation is linear with respect to the size of the initial refutation if it is semi-read-once, tree-like regular, tree-like or semi-tree-like. Additionally, we propose an extendable tool, called MS-Checker, able to verify the validity of any Max-SAT certificate using Max-SAT inference rules. Both tools are evaluated on the unweighted and weighted benchmark instances of the 2020 Max-SAT Evaluation.
... The laws, the axioms, and, the processes, of such an Algebra will be identical in their whole extent with the laws, the axioms, and, the processes of an Algebra of Logic. [7,[37][38]. ...
... A clausal sentence is either a literal, an atomic sentence or a negation of an atomic sentence, or a disjunction of literals, e.g., p, ¬p. If p and q are logical constants, then the following three are clausal sentences: p, ¬p, ¬p ∨ q (for a complete discussion, see [37]). ...
... First-order logic programming [Apt, 1990, Lloyd, 1994] is a declarative paradigm based on formal logic, that views computing as a procedure to find a proof for a logical theory, i.e: a set of logical sentences. A program encodes this logical theory and the proof is typically produced -if finding one is possible -using the resolution principle [Robinson, 1965], which provides a single rule of deductive inference that is sound and complete for proving statements constructed using the syntax of logic programs. The most CHAPTER 2. BACKGROUND NOTIONS widespread syntax for logic programming is known as Horn logic [Horn, 1951], a Turing complete [Tärnlund, 1977] restricted (in the form of rules it allows) version of first-order predicate logic. ...
... Of course, generating T ∞ P is computationally expensive and not the only method to determine if P |= A. Many strategies and algorithms have been developed in the literature, most notably following the resolution principle [Robinson, 1965]. The most popular variation of the resolution principle is called selective linear definite (SLD) resolution [Kowalski and Kuehner, 1971] and is the one used by Prolog. ...
Thesis
Machine ethics is an uprising sub-field of artificial intelligence fueled by the interest and concerns about the deployment of automated agents in our everyday life. As these agents gain independence from human intervention and make decisions with possible impact on human welfare, real concerns are rising across domains.Due to those reasons, various approaches have been proposed to imbue automated agents with ethical considerations. Several research currents have developed models stemming from psychology and philosophy in an effort to adapt decision-making algorithms to consider ethical values so that the impact of agents on people is bounded and guided by these notions.Most of these approaches consist of either reasoning and applying a set of well-known ethical restrictions, also known as principles (top-down), or inferring them based on carefully crafted datasets through learning algorithms (bottom-up).In this thesis, we look at the problem of implementing these ethical principles in the context of tasks involving sequences of interdependent decisions, i.e: automated planning. We show how certain notions can be modeled using preference-based frameworks, as in top-down approaches, and how these preferences can be inferred from a corpus of data like bottom-up methodologies, to develop a hybrid approach that can be applied to planning problems. An implementation for each facet of our approach is provided in order to test our ideas in practical scenarios.
... The notions of read-once, tree-like and dag-like refutations also apply to CNF formulas [4]. In case of CNF formulas refutations are typically studied in the resolution refutation system [22]. It is important to note that both tree-like and dag-like refutations are complete in that every unsatisfiable CNF formula and linear system has a tree-like refutation and a dag-like refutation. ...
... 19: Φ ← Make-2CNF (U, d). 20: if (Φ is satisfiable) then 21: v ← satisfying assignment to Φ. 22: else Φ has a resolution refutation R. 23: return (The integer refutation corresponding to R (see Sect. 6)) 24: for i = 1 . . . n do 25: if d i ∈ Z then 26: y i ← d i . ...
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This paper is concerned with the design and analysis of a bit-scaling based algorithm for the problem of checking integer feasibility in a system of unit two variable per inequality (UTVPI) constraints (IF). The insights developed during algorithm design result in new techniques for extracting refutations of integer feasibility in such systems. Recall that a UTVPI constraint is a linear constraint of the form: $$a_i\cdot x_i+a_j \cdot x_j \le b_{ij}$$, where $$a_i, a_j \in \{0,1,-1\}$$ and $$b_{ij} \in {\mathbb {Z}}$$. These constraints arise in a number of application domains including but not limited to program verification (array bounds checking and abstract interpretation), operations research (packing and covering), and logic programming. Over the years, several algorithms have been proposed for the IF problem. Most of these algorithms are based on two inference rules, viz. the transitive rule and the tightening rule. None of these algorithms are bit-scaling. In other words, the running times of these algorithms are parameterized only by the number of variables and the number of constraints in the UTVPI constraint system (UCS) and not by the sizes of input constants. We introduce a novel algorithm for the IF problem, which is based on a collection of new insights. These insights are used to design a new bit-scaling algorithm for IF that runs in $$O(\sqrt{n}\cdot m \cdot \log _2 C)$$ time, where n denotes the number of variables, m denotes the number of constraints, and C denotes the absolute value of the most negative constant defining the UCS. An interesting consequence of our research is the development of techniques for extracting refutations of integer infeasibility in UCSs that are linearly feasible. If the UCS is linearly feasible, then our algorithm creates a 2CNF formula. The UCS has an integer refutation (i.e., does not have a lattice point) if and only if the created 2CNF formula has a resolution refutation (i.e., is unsatisfiable).
... Figure 13 presents a unification algorithm U , which takes a constraint and produces a substitution θ and a pattern π . The algorithm can be understood as extending Robinson's unification algorithm (Robinson, 1965) to handle variational types and dynamic types and to support error tolerance. To support error tolerance, the unification not only returns a substitution but also a typing pattern. ...
... Int. Case (e) unifies two static types and is delegated to Robinson's unification algorithm (Robinson, 1965). Case (f) unifies two function types by unifying their respective argument and return types. ...
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Gradual typing allows programs to enjoy the benefits of both static typing and dynamic typing. While it is often desirable to migrate a program from more dynamically typed to more statically typed or vice versa, gradual typing itself does not provide a way to facilitate this migration. This places the burden on programmers who have to manually add or remove type annotations. Besides the general challenge of adding type annotations to dynamically typed code, there are subtle interactions between these annotations in gradually typed code that exacerbate the situation. For example, to migrate a program to be as static as possible, in general, all possible combinations of adding or removing type annotations from parameters must be tried out and compared. In this paper, we address this problem by developing migrational typing , which efficiently types all possible ways of replacing dynamic types with fully static types for a gradually typed program. The typing result supports automatically migrating a program to be as static as possible or introducing the least number of dynamic types necessary to remove a type error. The approach can be extended to support user-defined criteria about which annotations to modify. We have implemented migrational typing and evaluated it on large programs. The results show that migrational typing scales linearly with the size of the program and takes only 2–4 times longer than plain gradual typing.
... The following proposition is known as the resolution technique in the literature, which was first proved in (Robinson 1965), and then used in many SAT algorithms. Definition 1. (Resolution on a variable) Let F be a CNFformula containing a variable x. ...
... We may always use F \x to denote the CNF-formula after resolving a variable x in F . Proposition 1. (Robinson 1965) Let F be a CNF-formula containing a variable x and F \x be the CNF-formula after resolving on variable x. Then F has a satisfying assignment if and only if F \x does. ...
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We show that the CNF satisfiability problem can be solved O^*(1.2226^m) time, where m is the number of clauses in the formula, improving the known upper bounds O^*(1.234^m) given by Yamamoto 15 years ago and O^*(1.239^m) given by Hirsch 22 years ago. By using an amortized technique and careful case analysis, we successfully avoid the bottlenecks in previous algorithms and get the improvement.
... The Davis-Putnam algorithm [Davis et al. (1960)] and DPLL backtracking algorithm [Davis et al. (1962)] were among the earliest effective algorithms for propositional knowledge bases. Thereafter Robinson developed the full resolution rule [Robinson (1965)]. Due to the close relation between propositional inference and the satisfiability problem (SAT), all the algorithms developed for SAT are actually working for propositional inference either [Gu et al. (1997)]. ...
... Yani, tümevarım çıkarım kuralları tümdengelim çıkarım kurallarının tersi olarak elde edilebilir. Bu düşünceyle, (Muggleton ve Buntine, 1988) ters çözümlemeyi, (Robinson, 1965) tarafından bulunan bir tümdengelimli çıkarım kuralı olan çözümlemenin (resolotion) tersi olarak sunmuşlardır. Ters çözümleme, günümüzde aşağıdan yukarıya TMP"de öne çıkan bir genelleştirme operatörüdür. ...
Thesis
Word Sense Disambiguation (WSD) is one of the important processes needed for natural language processing applications and is defined as determining the sense of a multi-sense word in a given context. The aim of this thesis is to report on the performance results achieved by applying some WSD approaches to Turkish texts and present the evaluations made using results. In the study, firstly an unsupervised corpus based WSD application developed with collocation knowledge is presented and then its results are evaluated. Afterwards, an alternative WSD application developed considering the insufficiency of in the performance results achieved in that application is accounted for in detail. This latter application rests on Inductive Logic Programming (ILP), which is a method that circumvents the incapability of traditional machine learning approaches in employing background knowledge and making deductive inferences. To this effect, the topic of ILP is given a detailed explanation and its applicability to WSD is demonstrated with empirical results obtained using Turkish data.
... The origin of unification theory is usually attributed to Julia Robinson [50]. The classical syntactic unification problem is as follows: given two term s, t (built from function symbols and variables), find a unifier for them, that is, a uniform replacement of the variables occurring in s and t by other terms that makes s and t identical. ...
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In this paper, we study projective algebras in varieties of (bounded) commutative integral residuated lattices. We make use of a well-established construction in residuated lattices, the ordinal sum, and the order property of divisibility. Via the connection between projective and splitting algebras, we show that the only finite projective algebra in $$\mathsf {{FL}_{ew}}$$ is the two-element Boolean algebra. Moreover, we show that several interesting varieties have the property that every finitely presented algebra is projective, such as locally finite varieties of hoops. Furthermore, we show characterization results for finite projective Heyting algebras, and finitely generated projective algebras in locally finite varieties of bounded hoops and BL-algebras. Finally, we connect our results with the algebraic theory of unification.
... Resolution, which is a way to tell whether a propositional formula is satisfiable and to prove that a first-order formula is unsatisfiable, is a rule of inference leading to a refutation theorem-proving technique for sentences in CPL and FOL. It can be traced back to (Davis & Putnam, 1960), and then improved in 1965 by John Alan Robinson (Robinson, 1965). ...
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Computation tree logic (CTL) is an essential specification language in the field of formal verification. In systems design and verification, it is often important to update existing knowledge with new attributes and subtract the irrelevant content while preserving the given properties on a known set of atoms. Under the scenario, given a specification, the weakest sufficient condition (WSC) and the strongest necessary condition (SNC) are dual concepts and very informative in formal verification. In this article, we generalize our previous results (i.e., the decomposition, homogeneity properties, and the representation theorem) on forgetting in bounded CTL to the unbounded one. The cost we pay is that, unlike the bounded case, the result of forgetting in CTL may no longer exist. However, SNC and WSC can be obtained by the new forgetting machinery we are presenting. Furthermore, we complement our model-theoretic approach with a resolution-based method to compute forgetting results in CTL. This method is currently the only way to compute forgetting results for CTL and temporal logic. The method always terminates and is sound. That way, we set up the resolution-based approach for computing WSC and SNC in CTL.
... The challenge of converting data into simple and humanly comprehensible logic has been addressed in many areas of previous research. In the early 1960s, work on formal logic led to the inception of logical programming and rule-learning algorithms [1][2][3][4][5][6][7][8]. The latter-including algorithms such as Corels [9], Slipper [10], Skope-Rules [11], RuleFit [12], LRI [13], MLRules [14], and more-often rely on greedy approaches to extract short Boolean expressions from more complex models (such as large decision trees). ...
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Artificial intelligence and machine learning have demonstrated remarkable results in science and applied work. However, present AI models, developed to be run on computers but used in human-driven applications, create a visible disconnect between AI forms of processing and human ways of discovering and using knowledge. In this work, we introduce a new concept of “Human Knowledge Models” (HKMs), designed to reproduce human computational abilities. Departing from a vast body of cognitive research, we formalized the definition of HKMs into a new form of machine learning. Then, by training the models with human processing capabilities, we learned human-like knowledge, that humans can not only understand, but also compute, modify, and apply. We used several datasets from different applied fields to demonstrate the advantages of HKMs, including their high predictive power and resistance to noise and overfitting. Our results proved that HKMs can efficiently mine knowledge directly from the data and can compete with complex AI models in explaining the main data patterns. As a result, our study reveals the great potential of HKMs, particularly in the decision-making applications where “black box” models cannot be accepted. Moreover, this improves our understanding of how well human decision-making, modeled by HKMs, can approach the ideal solutions in real-life problems.
... As a consequence, new complete inference systems for MaxSAT have had to be defined [20]. They are MaxSAT extensions of either the resolution rule [33] or semantic tableaux [11,15,34]. ...
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We define a new MaxSAT tableau calculus based on resolution. Given a multiset of propositional clauses ϕ, we prove that the calculus is sound in the sense that if the minimum number of contradictions derived among the branches of a completed tableau for ϕ is m, then the minimum number of unsatisfied clauses in ϕ is m. We also prove that it is complete in the sense that if the minimum number of unsatisfied clauses in ϕ is m, then the minimum number of contradictions among the branches of any completed tableau for ϕ is m. Moreover, we describe how to extend the proposed calculus to solve Weighted Partial MaxSAT.
... The most widely used propositional proof system is RES (Robinson 1965). A proof in RES is a sequence of clauses C 1 , . . . ...
Article
Modern complete SAT solvers almost uniformly implement variations of the clause learning framework introduced by Grasp and Chaff. The success of these solvers has been theoretically explained by showing that the clause learning framework is an implementation of a proof system which is as poweful as resolution. However, exponential lower bounds are known for resolution, which suggests that significant advances in SAT solving must come from implementations of more powerful proof systems. We present a clause learning SAT solver that uses extended resolution. It is based on a restriction of the application of the extension rule. This solver outperforms existing solvers on application instances from recent SAT competitions as well as on instances that are provably hard for resolution.
... This iteration continues until a clause is generated (typically the empty clause for refutation theorem proving) that signals a proof has been found, the set of clauses is saturated, or a timeout is reached. For more details on saturation (Robinson 1965) and saturation-calculi, we refer the reader to (Bachmair and Ganzinger 1998). ...
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Automated theorem provers have traditionally relied on manually tuned heuristics to guide how they perform proof search. Deep reinforcement learning has been proposed as a way to obviate the need for such heuristics, however, its deployment in automated theorem proving remains a challenge. In this paper we introduce TRAIL, a system that applies deep reinforcement learning to saturation-based theorem proving. TRAIL leverages (a) a novel neural representation of the state of a theorem prover and (b) a novel characterization of the inference selection process in terms of an attention-based action policy. We show through systematic analysis that these mechanisms allow TRAIL to significantly outperform previous reinforcement-learning-based theorem provers on two benchmark datasets for first-order logic automated theorem proving (proving around 15% more theorems).
... Clausal programs are sets of clauses. Robinson (1965) showed that a single rule of inference (the resolution principle) is both sound and refutation complete for clausal logic. However, reasoning about full clausal logic is computationally expensive (Nienhuys-Cheng & Wolf, 1997). ...
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Inductive logic programming (ILP) is a form of machine learning. The goal of ILP is to induce a hypothesis (a set of logical rules) that generalises training examples. As ILP turns 30, we provide a new introduction to the field. We introduce the necessary logical notation and the main learning settings; describe the building blocks of an ILP system; compare several systems on several dimensions; describe four systems (Aleph, TILDE, ASPAL, and Metagol); highlight key application areas; and, finally, summarise current limitations and directions for future research.
... Robinson created the original unification algorithm [12]. The significance of this algorithm was described by [8] with: ...
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The Robinson unification algorithm has exponential worst case behavior. This triggered the development of (semi-)linear versions around 1976 by Martelli and Montanari as well as by Paterson and Wegman (J Comput Syst Sci 16(2):158–167, 1978, https://doi.org/10.1016/0022-0000(78)90043-0). Another version emerged by Baader and Snyder around 2001. While these versions are distinctly faster on larger input pairs, the Robinson version still does better than them on small-sized inputs. This paper describes yet another (semi-)linear version that is faster and challenges also the Robinson version on small-sized inputs. All versions have been implemented and compared against each other on different types and sizes of input pairs.
... The main challenge of using either the chase algorithm [2,67], or resolution [81] or semantic tableau [50] to develop satisfiability checking procedures is to ensure that the procedure terminates [21,35,36,58,59,62,63,70]. In developing resolution-based decision procedure the key issue is to ensure a finitely bounded search space. This can be achieved by defining a finitely bounded clausal class to which any BCQ problem can be mapped to and all inferences produce conclusions that belong to this class. ...
Preprint
Answering Boolean conjunctive query over logical constraints is an essential problem in knowledge representation. Other problems in computer science such as constraint satisfaction and homomorphism problems can also be seen as Boolean conjunctive query answering problems. This paper develops saturation-based Boolean conjunctive query answering and rewriting procedures for the guarded, the loosely guarded and the clique guarded fragments. We improve existing resolution-based decision procedures for the guarded and the loosely guarded fragments, and devise a saturation-based approach deciding Boolean conjunctive query answering problems for the guarded, the loosely guarded and the clique guarded fragments. Based on the saturation-based query answering procedure, we introduce a novel saturation-based query rewriting setting that aims to back-translate the saturated clausal set derived from saturation-based query answering procedures, to a (Skolem-symbol-free) first-order formula, and devise a saturation-based query rewriting procedures for all these guarded fragments. Unlike mainstream query answering and rewriting approaches, our procedures derive a compact saturation that is reusable even if the data changes. This paper lays the theoretical foundations for the first practical Boolean conjunctive query answering and the first saturation-based Boolean conjunctive query rewriting procedures for the guarded, the loosely guarded and the clique guarded fragments.
... Unification is a key mechanism in resolution [41] and paramodulation-based [36] theorem proving. Since Plotkin's work [40] on equational unification, i.e., E-unification modulo an equational theory E, it is widely used for increased effectiveness. ...
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ResearchGate has not been able to resolve any references for this publication.


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CALLING SNPS afther aligmnet with BBMap
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6.6 years ago

Well... My god have to fix a lot things before Im able to call my snps

I got my bam file directly from fastq using BBMap (thanks to genomax)

Of course my bam file is not sorted and no index, so I do...

samtools sort out.bam sorted.bam
samtools index sorted.bam sorted.bai

Then I continue at next step

java -jar /home/cri/Desktop/GATK/GenomeAnalysisTK.jar -T RealignerTargetCreator -R hg19.fa -I sorted.bam  -o sorted.intervals

and I get this error message

Error details: SAM file doesn't have any read groups defined in the header.  The GATK no longer supports SAM files without read groups

and I use this code to fix it

java -jar /home/cri/Desktop/picard1/AddOrReplaceReadGroups.jar I=sorted.bam O=header.bam RGLB=LIBRARY RGPL="Ion Torrent" RGPU=RUN RGSM=SAMPLE RGCN=BCM

Then I try again

java -jar /home/cri/Desktop/GATK/GenomeAnalysisTK.jar -T RealignerTargetCreator -R hg19.fa -I header.bam  -o header.intervals

Then I get another error (i jump to error to error... ^" )

 ERROR MESSAGE: Invalid command line: Cannot process the provided BAM/CRAM file(s) because they were not indexed.  The GATK does offer limited processing of unindexed BAM/CRAMs in --unsafe mode, but this feature is unsupported -- use it at your own risk!

I realize my bai file was called before sorted.bai so I rename to header.bai...didn't work either, then I index again my header.bam

samtools index header.bam header.bai

and Identify target regions for realignment finally works....

why I need to do twice the index?

bbmap SNP • 3.2k views
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GATK is the only program that requires read groups and there is no getting around that requirement. Next time add the readgroups before sorting and indexing.

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Thanks to all for the tips :) (I put in my notes)

My pc works now on step: Local realignment around indels :))))

java -Xmx4g -Djava.io.tmpdir=/tmp \

-jar GenomeAnalysisTK.jar \

-I input.marked.bam \

-R hg19.fa \

-T IndelRealigner \

-targetIntervals input.bam.list \

-o input.marked.realigned.bam

seems is going to take looong time.... my god need lot patient to do this kind of job...or maybe a better pc :P

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Yes it will take a while. A few steps in GATK can be sped up by throwing more cores at it (and of course there is the trick of splitting up the analysis by chromosome after alignment and running the pipeline).

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Thanks Chris

ok..so I can split the jobb, smart...If the power electricity go out at home I will cry, I will see if find the code how to do it!

I have now 14h left! My pc just have 4cores amd, I invested most at ram (32gb) and I wait until they will pay me at work (this is Spain...) so I can look for a better CPU :)

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If they don't pay you the least they can do is give you access to a server/cluster at work? You really should be doing this type of work on a remote machine that is beefy and is on a UPS, if you have unreliable power at home.

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well..Nearly three days took for do realignment around indels...

Now Im stock at this step

1) Count covariates:

java -Xmx4g -jar /home/cri/Desktop/GATK/GenomeAnalysisTK.jar \
-l INFO \
-R hg19.fa \
--DBSNP dbsnp132.txt \
-T CountCovariates \
-cov QualityScoreCovariate \
-cov CycleCovariate \
-cov DinucCovariate \
-recalFile input.recal_data.csv

I get this error

ERROR MESSAGE: Walker CountCovariates is no longer available in the GATK; it has been deprecated since version 2.0 (use BaseRecalibrator instead; see documentation for usage)

So I check for next code:

 java -jar /home/cri/Desktop/GATK/GenomeAnalysisTK.jar \
-T BaseRecalibrator \
-R hg19.fa \
-knownSites latest_dbsnp.vcf \
-o recal_data.table

ERROR MESSAGE: Could not read file /media/cri/CRIS_DATA/ULT/OTRO COPIA/otro/latest_dbsnp.vcf because file 'latest_dbsnp.vcf' does not exist

Im bit losted how to get that vcf file

I go to

Thanks

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Which guide are you following for this exercise? Can you post a link?

Also use ADD REPLY when posting additional information like this. SUBMIT ANSWER should only be used for new answers for the original question.

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I did wrong....sorry

Thats the main pipeline I follow, but at the end many steps has been modified like use BBMap...

Now I down the file from here, dunno if its the right one

ftp://ftp.ncbi.nih.gov/snp/organisms/human_9606/VCF/GATK/

Thanks!

(I fixed one of the error message....I did mistake with copy and paste)

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Entering edit mode
6.6 years ago
Chris Fields ★ 2.2k

I agree with genomax. Also something to consider, many aligners have an option to do this when you initially run the alignment. bwa mem for instance has the -R option:

       -R STR        read group header line such as '@RG\tID:foo\tSM:bar' [null]

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BBMap also has this option, with the "rgid=" flag for setting the readgroup ID, and similar flags for setting other readgroup fields. I don't think readgroups are used by many tools other than GATK, though. Maybe Picard.

Incidentally, BBMap can also generate a shellscript for you to produce a sorted, indexed bam file (assuming samtools is installed)... "bamscript=bamscript.sh" (or bs for short - "bs=bs.sh"). So you can run this command:

bbmap.sh stuff bs=bs.sh; sh bs.sh

...and when it completes there will be a sorted, indexed bam file + bai file. I added that because it's a common task and I find it hard to remember the syntax.

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Thanks Brian

Next time I try with another fastq file I will put that commands on BBMap, for me all of this is new so I learn on a hard way ^^" I appreciate all the help.

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Posts

## Fun Stuff

I was reading Kate Nowak’s blog f(t) yesterday and followed a link to the Phillips Exeter Academy website where they’ve posted a pretty amazing collection of problem sets for all levels of math from Pre-algebra to Calculus (and beyond!).

I’ve only just begun to work on some of these, but I’m having a good time so far.

Can you find a fraction so that the difference between the fraction and its reciprocal is exactly equal to 1?

That is $\frac ab - \frac ba=1$

The problem gave the example that $\frac85 - \frac58 = \frac{39}{40}$

D’oh – off by $\frac{1}{40}$

Then the question asks – Can you find another fraction that gets closer than this?

I approached this from a couple of different ways – first I took the original equation $\frac ab - \frac ba=1$ and created a common denominator to get $\frac{a^2 - b^2}{ab} = 1$ or $a^2 - b^2 = ab$ and $a^2 - ab - b^2 = 0$.

I didn’t pursue this past that point, but did come back to it later.

Next, I broke out the spreadsheet and set it up to take all the numbers from 1-30 and create fractions and their reciprocals from these and subtract them.

Looking at all that, I noticed a few places where the difference between 1 and the $\frac ab - \frac ba$ was smaller than $\frac{1}{40}$.

This happened for $\frac {13}{8} - \frac {8}{13}$ and $\frac {21}{13} - \frac {13}{21}$.  Then it was time for class.

The numbers in the fractions that were getting close to 1 had caught my eye yesterday and this morning when I came in, I sat down with it again and saw that they were all consecutive Fibonacci numbers.  So, I made a new spreadsheet with Fibonacci numbers and the fractions and reciprocals and noticed that the difference $\frac ab - \frac ba$ was approaching 1.

At some point yesterday afternoon I went to Wolfram Alpha and typed in $a^2 - ab - b^2 = 0$ just to see what I would get and it provided a relationship between the two variables that comes from treating one of the variables as a constant so that

$a=\frac b2 (1 \pm \sqrt{5})$

In other words a and b cannot both be integers!  And of course, some of you may have picked up on this sooner than I did – the number that produces the exact value of 1 is phi – The Golden Ratio.

$\phi - \frac {1}{\phi}=1$.